## The Mathematical Challenge of Large Networks

This is another one of the aforementioned lectures I watched a while ago, but had never got around to blogging:

…

If I had to watch this one again, I’d probably skip most of the second half; it contains highly technical coverage of topics in graph theory, and it was very difficult for me to follow (but I did watch it to the end, just out of curiosity).

The lecturer has put up a ~500 page publication on these and related topics, which is available here, so if you want to know more that’s an obvious place to go have a look. A few other relevant links to stuff mentioned/covered in the lecture:

Szemerédi regularity lemma.

Graphon.

Turán’s theorem.

Quantum graph.

## Quantifying tradeoffs between fairness and accuracy in online learning

…

From a brief skim of this paper, which is coauthored by the guy giving this lecture, it looked to me like it covers many of the topics discussed in the lecture. So if you’re unsure as to whether or not to watch the lecture (…or if you want to know more about this stuff after you’ve watched the lecture) you might want to have a look at that paper. Although the video is long for a single lecture I would note that the lecture itself lasts only approximately one hour; the last 10 minutes are devoted to Q&A.

## Biodemography of aging (IV)

My working assumption as I was reading part two of the book was that I would not be covering that part of the book in much detail here because it would simply be too much work to make such posts legible to the readership of this blog. However I then later, while writing this post, had the thought that given that almost nobody reads along here anyway (I’m not complaining, mind you – this is how I like it these days), the main beneficiary of my blog posts will always be myself, which lead to the related observation/notion that I should not be limiting my coverage of interesting stuff here simply because some hypothetical and probably nonexistent readership out there might not be able to follow the coverage. So when I started out writing this post I was working under the assumption that it would be my last post about the book, but I now feel sure that if I find the time I’ll add at least one more post about the book’s statistics coverage. On a related note I am explicitly making the observation here that this post was written for *my* benefit, not yours. You can read it if you like, or not, but it was not really written for you.

I have added bold a few places to emphasize key concepts and observations from the quoted paragraphs and in order to make the post easier for me to navigate later (all the italics below are on the other hand those of the authors of the book).

…

“** Biodemography** is a multidisciplinary branch of science that unites under its umbrella various analytic approaches aimed at integrating biological knowledge and methods and traditional demographic analyses to shed more light on variability in mortality and health across populations and between individuals.

**is a special subfield of biodemography that focuses on understanding the impact of processes related to aging on health and longevity.”**

*Biodemography of aging*“Mortality rates as a function of age are a cornerstone of many demographic analyses. The longitudinal **age** **trajectories of biomarkers** add a new dimension to the traditional demographic analyses: the mortality rate becomes a function of not only age but also of these biomarkers (with additional dependence on a set of sociodemographic variables). Such analyses should incorporate dynamic characteristics of trajectories of biomarkers to evaluate their impact on mortality or other outcomes of interest. Traditional analyses using baseline values of biomarkers (e.g., Cox proportional hazards or logistic regression models) do not take into account these dynamics. One approach to the evaluation of the impact of biomarkers on mortality rates is to use the Cox proportional hazards model with time-dependent covariates; this approach is used extensively in various applications and is available in all popular statistical packages. In such a model, the biomarker is considered a time-dependent covariate of the hazard rate and the corresponding regression parameter is estimated along with standard errors to make statistical inference on the direction and the significance of the effect of the biomarker on the outcome of interest (e.g., mortality). However, **the choice of the analytic approach should not be governed exclusively by its simplicity or convenience of application. It is essential to consider whether the method gives meaningful and interpretable results relevant to the research agenda. In the particular case of biodemographic analyses, the Cox proportional hazards model with time-dependent covariates is not the best choice.**”

“Longitudinal studies of aging present special methodological challenges due to inherent characteristics of the data that need to be addressed in order to avoid biased inference. The challenges are related to the fact that the populations under study (aging individuals) experience substantial **dropout rates** related to death or poor health and often have co-morbid conditions related to the disease of interest. The standard assumption made in longitudinal analyses (although usually not explicitly mentioned in publications) is that dropout (e.g., death) is not associated with the outcome of interest. While this can be safely assumed in many general longitudinal studies (where, e.g., the main causes of dropout might be the administrative end of the study or moving out of the study area, which are presumably not related to the studied outcomes), the very nature of the longitudinal outcomes (e.g., measurements of some physiological biomarkers) analyzed in a longitudinal study of aging assumes that they are (at least hypothetically) related to the process of aging. Because the process of aging leads to the development of diseases and, eventually, death, in longitudinal studies of aging an assumption of non-association of the reason for dropout and the outcome of interest is, at best, risky, and usually is wrong. As an illustration, we found that the average trajectories of different physiological indices of individuals dying at earlier ages markedly deviate from those of long-lived individuals, both in the entire Framingham original cohort […] and also among carriers of specific alleles […] In such a situation, **panel compositional changes due to attrition** affect the averaging procedure and modify the averages in the total sample. Furthermore, biomarkers are subject to **measurement error** and random biological variability. They are usually collected intermittently at examination times which may be sparse and typically biomarkers are not observed at event times. It is well known in the statistical literature that ignoring measurement errors and biological variation in such variables and using their observed “raw” values as time-dependent covariates in a Cox regression model may lead to biased estimates and incorrect inferences […] **Standard methods of survival analysis such as the Cox proportional hazards model** (Cox 1972) **with time-dependent covariates should be avoided in analyses of biomarkers measured with errors** because they can lead to biased estimates.”

“Statistical methods aimed at analyses of time-to-event data jointly with longitudinal measurements have become known in the mainstream biostatistical literature as “**joint models for longitudinal and time-to-event data**” (“survival” or “failure time” are often used interchangeably with “time-to-event”) or simply “**joint models**.” This is an active and fruitful area of biostatistics with an explosive growth in recent years. […] The standard joint model consists of two parts, the first representing the dynamics of longitudinal data (which is referred to as the “longitudinal sub-model”) and the second one modeling survival or, generally, time-to-event data (which is referred to as the “survival sub-model”). […] Numerous extensions of this basic model have appeared in the joint modeling literature in recent decades, providing great flexibility in applications to a wide range of practical problems. […] The standard parameterization of the joint model (11.2) assumes that the risk of the event at age t depends on the current “true” value of the longitudinal biomarker at this age. While this is a reasonable assumption in general, it may be argued that additional dynamic characteristics of the longitudinal trajectory can also play a role in the risk of death or onset of a disease. For example, if two individuals at the same age have exactly the same level of some biomarker at this age, but the trajectory for the first individual increases faster with age than that of the second one, then the first individual can have worse survival chances for subsequent years. […] Therefore, extensions of the basic parameterization of joint models allowing for dependence of the risk of an event on such dynamic characteristics of the longitudinal trajectory can provide additional opportunities for comprehensive analyses of relationships between the risks and longitudinal trajectories. Several authors have considered such extended models. […] **joint models are computationally intensive** and are sometimes prone to convergence problems [however such] models provide more efficient estimates of the effect of a covariate […] on the time-to-event outcome in the case in which there is […] an effect of the covariate on the longitudinal trajectory of a biomarker. This means that** analyses of longitudinal and time-to-event data in joint models may require smaller sample sizes to achieve comparable statistical power** **with analyses based on time-to-event data alone** (Chen et al. 2011).”

“To be useful as a tool for biodemographers and gerontologists who seek biological explanations for observed processes, models of longitudinal data should be based on realistic assumptions and reflect relevant knowledge accumulated in the field. An example is the shape of the risk functions. Epidemiological studies show that **the conditional hazards of health and survival events considered as functions of risk factors often have U- or J-shapes** […], so a model of aging-related changes should incorporate this information. In addition, risk variables, and, what is very important, their effects on the risks of corresponding health and survival events, experience aging-related changes and these can differ among individuals. […] An important class of models for joint analyses of longitudinal and time-to-event data incorporating a stochastic process for description of longitudinal measurements uses an epidemiologically-justified assumption of a quadratic hazard (i.e., U-shaped in general and J-shaped for variables that can take values only on one side of the U-curve) considered as a function of physiological variables. **Quadratic hazard models** have been developed and intensively applied in studies of human longitudinal data”.

“Various approaches to statistical model building and data analysis that incorporate unobserved heterogeneity are ubiquitous in different scientific disciplines. **Unobserved heterogeneity** in models of health and survival outcomes can arise because there may be relevant risk factors affecting an outcome of interest that are either unknown or not measured in the data. Frailty models introduce the concept of unobserved heterogeneity in survival analysis for time-to-event data. […] Individual age trajectories of biomarkers can differ due to various observed as well as unobserved (and unknown) factors and such individual differences propagate to differences in risks of related time-to-event outcomes such as the onset of a disease or death. […] The joint analysis of longitudinal and time-to-event data is the realm of a special area of biostatistics named “joint models for longitudinal and time-to-event data” or simply “joint models” […] Approaches that incorporate heterogeneity in populations through random variables with continuous distributions (as in the standard joint models and their extensions […]) assume that the risks of events and longitudinal trajectories follow similar patterns for all individuals in a population (e.g., that biomarkers change linearly with age for all individuals). Although such homogeneity in patterns can be justifiable for some applications, generally this is a rather strict assumption […] A population under study may consist of subpopulations with distinct patterns of longitudinal trajectories of biomarkers that can also have different effects on the time-to-event outcome in each subpopulation. When such subpopulations can be defined on the base of observed covariate(s), one can perform stratified analyses applying different models for each subpopulation. However, observed covariates may not capture the entire heterogeneity in the population in which case it may be useful to conceive of the population as consisting of *latent* subpopulations defined by unobserved characteristics. Special methodological approaches are necessary to accommodate such hidden heterogeneity. Within the joint modeling framework, a special class of models, **joint latent class models**, was developed to account for such heterogeneity […] The joint latent class model has three components. First, it is assumed that a population consists of a fixed number of (latent) subpopulations. The latent class indicator represents the latent class membership and the probability of belonging to the latent class is specified by a multinomial logistic regression function of observed covariates. It is assumed that individuals from different latent classes have different patterns of longitudinal trajectories of biomarkers and different risks of event. The key assumption of the model is conditional independence of the biomarker and the time-to-events given the latent classes. Then the class-specific models for the longitudinal and time-to-event outcomes constitute the second and third component of the model thus completing its specification. […] **the latent class stochastic process model** […] provides a useful tool for dealing with unobserved heterogeneity in joint analyses of longitudinal and time-to-event outcomes and taking into account hidden components of aging in their joint influence on health and longevity. This approach is also helpful for sensitivity analyses in applications of the original stochastic process model. We recommend starting the analyses with the original stochastic process model and estimating the model ignoring possible hidden heterogeneity in the population. Then the latent class stochastic process model can be applied to test hypotheses about the presence of hidden heterogeneity in the data in order to appropriately adjust the conclusions if a latent structure is revealed.”

“**The longitudinal genetic-demographic model** (or the genetic-demographic model for longitudinal data) […] combines three sources of information in the likelihood function: (1) follow-up data on survival (or, generally, on some time-to-event) for genotyped individuals; (2) (cross-sectional) information on ages at biospecimen collection for genotyped individuals; and (3) follow-up data on survival for non-genotyped individuals. […] Such joint analyses of genotyped and non-genotyped individuals can result in substantial improvements in statistical power and accuracy of estimates compared to analyses of the genotyped subsample alone if the proportion of non-genotyped participants is large. Situations in which genetic information cannot be collected for all participants of longitudinal studies are not uncommon. They can arise for several reasons: (1) the longitudinal study may have started some time before genotyping was added to the study design so that some initially participating individuals dropped out of the study (i.e., died or were lost to follow-up) by the time of genetic data collection; (2) budget constraints prohibit obtaining genetic information for the entire sample; (3) some participants refuse to provide samples for genetic analyses. Nevertheless, even when genotyped individuals constitute a majority of the sample or the entire sample, application of such an approach is still beneficial […] **The genetic stochastic process model** […] adds a new dimension to genetic biodemographic analyses, combining information on longitudinal measurements of biomarkers available for participants of a longitudinal study with follow-up data and genetic information. Such **joint analyses of different sources of information** collected in both genotyped and non-genotyped individuals allow for more efficient use of the research potential of longitudinal data which otherwise remains underused when only genotyped individuals or only subsets of available information (e.g., only follow-up data on genotyped individuals) are involved in analyses. Similar to the longitudinal genetic-demographic model […], **the benefits of combining data** on genotyped and non-genotyped individuals in the genetic SPM come from the presence of common parameters describing characteristics of the model for genotyped and non-genotyped subsamples of the data. This takes into account the knowledge that the non-genotyped subsample is a mixture of carriers and non-carriers of the same alleles or genotypes represented in the genotyped subsample and applies the ideas of heterogeneity analyses […] When the non-genotyped subsample is substantially larger than the genotyped subsample, these joint analyses can lead to a noticeable increase in the power of statistical estimates of genetic parameters compared to estimates based only on information from the genotyped subsample. **This approach is applicable not only to genetic data but to any discrete time-independent variable that is observed only for a subsample of individuals in a longitudinal study.**”

“Despite an existing tradition of interpreting differences in the shapes or parameters of the mortality rates (survival functions) resulting from the effects of exposure to different conditions or other interventions in terms of characteristics of individual aging, this practice has to be used with care. This is because such characteristics are difficult to interpret in terms of properties of external and internal processes affecting the chances of death. An important question then is: What kind of mortality model has to be developed to obtain parameters that are biologically interpretable? The purpose of this chapter is to describe an approach to mortality modeling that represents mortality rates in terms of parameters of physiological changes and declining health status accompanying the process of aging in humans. […] **A traditional (demographic) description of changes in individual health/survival status is performed using a continuous-time random Markov process** with a finite number of states, and age-dependent transition intensity functions (transitions rates). Transitions to the absorbing state are associated with death, and the corresponding transition intensity is a mortality rate. Although such a description characterizes connections between health and mortality, it does not allow for studying factors and mechanisms involved in the aging-related health decline. Numerous epidemiological studies provide compelling evidence that health transition rates are influenced by a number of factors. Some of them are fixed at the time of birth […]. Others experience stochastic changes over the life course […] **The presence of** such **randomly changing influential factors violates the Markov assumption, and makes the description of aging-related changes in health status more complicated.** […] The age dynamics of influential factors (e.g., physiological variables) in connection with mortality risks has been described using a stochastic process model of human mortality and aging […]. Recent extensions of this model have been used in analyses of longitudinal data on aging, health, and longevity, collected in the Framingham Heart Study […] This model and its extensions are described in terms of **a Markov stochastic process satisfying a diffusion-type stochastic differential equation.** The stochastic process is stopped at random times associated with individuals’ deaths. […] When an individual’s health status is taken into account, the coefficients of the stochastic differential equations become dependent on values of the **jumping process.** This dependence violates the Markov assumption and renders the conditional Gaussian property invalid. So the description of this (continuously changing) component of aging-related changes in the body also becomes more complicated. Since studying age trajectories of physiological states in connection with changes in health status and mortality would provide more realistic scenarios for analyses of available longitudinal data, it would be a good idea to find an appropriate mathematical description of the joint evolution of these interdependent processes in aging organisms. For this purpose,** we propose a comprehensive model of human aging, health, and mortality in which the Markov assumption is fulfilled by a two-component stochastic process consisting of jumping and continuously changing processes. The jumping component is used to describe relatively fast changes in health status occurring at random times, and the continuous component describes relatively slow stochastic age-related changes of individual physiological states. **[…] The use of stochastic differential equations for random continuously changing covariates has been studied intensively in the analysis of longitudinal data […] Such a description is convenient since it captures the feedback mechanism typical of biological systems reflecting regular aging-related changes and takes into account the presence of random noise affecting individual trajectories. It also captures the dynamic connections between aging-related changes in health and physiological states, which are important in many applications.”

## Random stuff

It’s been a long time since I last posted one of these posts, so a great number of links of interest has accumulated in my bookmarks. I intended to include a large number of these in this post and this of course means that I surely won’t cover each specific link included in this post in anywhere near the amount of detail it deserves, but that can’t be helped.

…

i. Autism Spectrum Disorder Grown Up: A Chart Review of Adult Functioning.

“For those diagnosed with ASD in childhood, most will become adults with a significant degree of disability […] Seltzer et al […] concluded that, despite considerable heterogeneity in social outcomes, “few adults with autism live independently, marry, go to college, work in competitive jobs or develop a large network of friends”. However, the trend within individuals is for some functional improvement over time, as well as a decrease in autistic symptoms […]. Some authors suggest that a sub-group of 15–30% of adults with autism will show more positive outcomes […]. Howlin et al. (2004), and Cederlund et al. (2008) assigned global ratings of social functioning based on achieving independence, friendships/a steady relationship, and education and/or a job. These two papers described respectively 22% and 27% of groups of higher functioning (IQ above 70) ASD adults as attaining “Very Good” or “Good” outcomes.”

“[W]e evaluated the adult outcomes for 45 individuals diagnosed with ASD prior to age 18, and compared this with the functioning of 35 patients whose ASD was identified after 18 years. Concurrent mental illnesses were noted for both groups. […] Comparison of adult outcome within the group of subjects diagnosed with ASD prior to 18 years of age showed significantly poorer functioning for those with co-morbid Intellectual Disability, *except in the domain of establishing intimate relationships* [my emphasis. To make this point completely clear, one way to look at these results is that apparently in the domain of partner-search autistics diagnosed during childhood are doing so badly in general that being intellectually disabled on top of being autistic is apparently conferring no additional disadvantage]. Even in the normal IQ group, the mean total score, i.e. the sum of the 5 domains, was relatively low at 12.1 out of a possible 25. […] Those diagnosed as adults had achieved significantly more in the domains of education and independence […] Some authors have described a subgroup of 15–27% of adult ASD patients who attained more positive outcomes […]. Defining an arbitrary adaptive score of 20/25 as “Good” for our normal IQ patients, 8 of thirty four (25%) of those diagnosed as adults achieved this level. Only 5 of the thirty three (15%) diagnosed in childhood made the cutoff. (The cut off was consistent with a well, but not superlatively, functioning member of society […]). None of the Intellectually Disabled ASD subjects scored above 10. […] All three groups had a high rate of co-morbid psychiatric illnesses. Depression was particularly frequent in those diagnosed as adults, consistent with other reports […]. Anxiety disorders were also prevalent in the higher functioning participants, 25–27%. […] Most of the higher functioning ASD individuals, whether diagnosed before or after 18 years of age, were functioning well below the potential implied by their normal range intellect.”

Related papers: Social Outcomes in Mid- to Later Adulthood Among Individuals Diagnosed With Autism and Average Nonverbal IQ as Children, Adults With Autism Spectrum Disorders.

…

ii. Premature mortality in autism spectrum disorder. This is a Swedish matched case cohort study. Some observations from the paper:

“The aim of the current study was to analyse all-cause and cause-specific mortality in ASD using nationwide Swedish population-based registers. A further aim was to address the role of intellectual disability and gender as possible moderators of mortality and causes of death in ASD. […] Odds ratios (ORs) were calculated for a population-based cohort of ASD probands (*n* = 27 122, diagnosed between 1987 and 2009) compared with gender-, age- and county of residence-matched controls (*n* = 2 672 185). […] During the observed period, 24 358 (0.91%) individuals in the general population died, whereas the corresponding figure for individuals with ASD was 706 (2.60%; OR = 2.56; 95% CI 2.38–2.76). Cause-specific analyses showed elevated mortality in ASD for almost all analysed diagnostic categories. Mortality and patterns for cause-specific mortality were partly moderated by gender and general intellectual ability. […] Premature mortality was markedly increased in ASD owing to a multitude of medical conditions. […] Mortality was significantly elevated in both genders relative to the general population (males: OR = 2.87; females OR = 2.24)”.

“Individuals in the control group died at a mean age of 70.20 years (s.d. = 24.16, median = 80), whereas the corresponding figure for the entire ASD group was 53.87 years (s.d. = 24.78, median = 55), for low-functioning ASD 39.50 years (s.d. = 21.55, median = 40) and high-functioning ASD 58.39 years (s.d. = 24.01, median = 63) respectively. […] Significantly elevated mortality was noted among individuals with ASD in all analysed categories of specific causes of death except for infections […] ORs were highest in cases of mortality because of diseases of the nervous system (OR = 7.49) and because of suicide (OR = 7.55), in comparison with matched general population controls.”

…

iii. Adhesive capsulitis of shoulder. This one is related to a health scare I had a few months ago. A few quotes:

“**Adhesive capsulitis** (also known as **frozen shoulder**) is a painful and disabling disorder of unclear cause in which the shoulder capsule, the connective tissue surrounding the glenohumeral joint of the shoulder, becomes inflamed and stiff, greatly restricting motion and causing chronic pain. Pain is usually constant, worse at night, and with cold weather. Certain movements or bumps can provoke episodes of tremendous pain and cramping. […] People who suffer from adhesive capsulitis usually experience severe pain and sleep deprivation for prolonged periods due to pain that gets worse when lying still and restricted movement/positions. The condition can lead to depression, problems in the neck and back, and severe weight loss due to long-term lack of deep sleep. People who suffer from adhesive capsulitis may have extreme difficulty concentrating, working, or performing daily life activities for extended periods of time.”

Some other related links below:

The prevalence of a diabetic condition and adhesive capsulitis of the shoulder.

“Adhesive capsulitis is characterized by a progressive and painful loss of shoulder motion of unknown etiology. Previous studies have found the prevalence of adhesive capsulitis to be slightly greater than 2% in the general population. However, the relationship between adhesive capsulitis and diabetes mellitus (DM) is well documented, with the incidence of adhesive capsulitis being two to four times higher in diabetics than in the general population. It affects about 20% of people with diabetes and has been described as the most disabling of the common musculoskeletal manifestations of diabetes.”

Adhesive Capsulitis (review article).

“Patients with type I diabetes have a 40% chance of developing a frozen shoulder in their lifetimes […] Dominant arm involvement has been shown to have a good prognosis; associated intrinsic pathology or insulin-dependent diabetes of more than 10 years are poor prognostic indicators.^{15} Three stages of adhesive capsulitis have been described, with each phase lasting for about 6 months. The first stage is the freezing stage in which there is an insidious onset of pain. At the end of this period, shoulder ROM [range of motion] becomes limited. The second stage is the frozen stage, in which there might be a reduction in pain; however, there is still restricted ROM. The third stage is the thawing stage, in which ROM improves, but can take between 12 and 42 months to do so. Most patients regain a full ROM; however, 10% to 15% of patients suffer from continued pain and limited ROM.”

Musculoskeletal Complications in Type 1 Diabetes.

“The development of periarticular thickening of skin on the hands and limited joint mobility (cheiroarthropathy) is associated with diabetes and can lead to significant disability. The objective of this study was to describe the prevalence of cheiroarthropathy in the well-characterized Diabetes Control and Complications Trial/Epidemiology of Diabetes Interventions and Complications (DCCT/EDIC) cohort and examine associated risk factors […] This cross-sectional analysis was performed in 1,217 participants (95% of the active cohort) in EDIC years 18/19 after an average of 24 years of follow-up. Cheiroarthropathy — defined as the presence of any one of the following: adhesive capsulitis, carpal tunnel syndrome, flexor tenosynovitis, Dupuytren’s contracture, or a positive prayer sign [related link] — was assessed using a targeted medical history and standardized physical examination. […] Cheiroarthropathy was present in 66% of subjects […] Cheiroarthropathy is common in people with type 1 diabetes of long duration (∼30 years) and is related to longer duration and higher levels of glycemia. Clinicians should include cheiroarthropathy in their routine history and physical examination of patients with type 1 diabetes because it causes clinically significant functional disability.”

Musculoskeletal disorders in diabetes mellitus: an update.

“Diabetes mellitus (DM) is associated with several musculoskeletal disorders. […] The exact pathophysiology of most of these musculoskeletal disorders remains obscure. Connective tissue disorders, neuropathy, vasculopathy or combinations of these problems, may underlie the increased incidence of musculoskeletal disorders in DM. The development of musculoskeletal disorders is dependent on age and on the duration of DM; however, it has been difficult to show a direct correlation with the metabolic control of DM.”

Rheumatic Manifestations of Diabetes Mellitus.

Prevalence of symptoms and signs of shoulder problems in people with diabetes mellitus.

Musculoskeletal Disorders of the Hand and Shoulder in Patients with Diabetes.

“In addition to micro- and macroangiopathic complications, diabetes mellitus is also associated with several musculoskeletal disorders of the hand and shoulder that can be debilitating (1,2). Limited joint mobility, also termed diabetic hand syndrome or cheiropathy (3), is characterized by skin thickening over the dorsum of the hands and restricted mobility of multiple joints. While this syndrome is painless and usually not disabling (2,4), other musculoskeletal problems occur with increased frequency in diabetic patients, including Dupuytren’s disease [“Dupuytren’s disease […] may be observed in up to 42% of adults with diabetes mellitus, typically in patients with long-standing T1D” – link], carpal tunnel syndrome [“The prevalence of [carpal tunnel syndrome, CTS] in patients with diabetes has been estimated at 11–30 % […], and is dependent on the duration of diabetes. […] Type I DM patients have a high prevalence of CTS with increasing duration of disease, up to 85 % after 54 years of DM” – link], palmar flexor tenosynovitis or trigger finger [“The incidence of trigger finger [/stenosing tenosynovitis] is 7–20 % of patients with diabetes comparing to only about 1–2 % in nondiabetic patients” – link], and adhesive capsulitis of the shoulder (5–10). The association of adhesive capsulitis with pain, swelling, dystrophic skin, and vasomotor instability of the hand constitutes the “shoulder-hand syndrome,” a rare but potentially disabling manifestation of diabetes (1,2).”

“The prevalence of musculoskeletal disorders was greater in diabetic patients than in control patients (36% vs. 9%, P < 0.01). Adhesive capsulitis was present in 12% of the diabetic patients and none of the control patients (P < 0.01), Dupuytren’s disease in 16% of diabetic and 3% of control patients (P < 0.01), and flexor tenosynovitis in 12% of diabetic and 2% of control patients (P < 0.04), while carpal tunnel syndrome occurred in 12% of diabetic patients and 8% of control patients (P = 0.29). Musculoskeletal disorders were more common in patients with type 1 diabetes than in those with type 2 diabetes […]. Forty-three patients [out of 100] with type 1 diabetes had either hand or shoulder disorders (37 with hand disorders, 6 with adhesive capsulitis of the shoulder, and 10 with both syndromes), compared with 28 patients [again out of 100] with type 2 diabetes (24 with hand disorders, 4 with adhesive capsulitis of the shoulder, and 3 with both syndromes, P = 0.03).”

Association of Diabetes Mellitus With the Risk of Developing Adhesive Capsulitis of the Shoulder: A Longitudinal Population-Based Followup Study.

“A total of 78,827 subjects with at least 2 ambulatory care visits with a principal diagnosis of DM in 2001 were recruited for the DM group. The non-DM group comprised 236,481 age- and sex-matched randomly sampled subjects without DM. […] During a 3-year followup period, 946 subjects (1.20%) in the DM group and 2,254 subjects (0.95%) in the non-DM group developed ACS. The crude HR of developing ACS for the DM group compared to the non-DM group was 1.333 […] the association between DM and ACS may be explained at least in part by a DM-related chronic inflammatory process with increased growth factor expression, which in turn leads to joint synovitis and subsequent capsular fibrosis.”

It is important to note when interpreting the results of the above paper that these results are based on Taiwanese population-level data, and type 1 diabetes – which is obviously the high-risk diabetes subgroup in this particular context – is rare in East Asian populations (as observed in Sperling *et al.*, “A child in Helsinki, Finland is almost 400 times more likely to develop diabetes than a child in Sichuan, China”. Taiwanese incidence of type 1 DM in children is estimated at ~5 in 100.000).

…

iv. Parents who let diabetic son starve to death found guilty of first-degree murder. It’s been a while since I last saw one of these ‘boost-your-faith-in-humanity’-cases, but they in my impression do pop up every now and then. I should probably keep at hand one of these articles in case my parents ever express worry to me that they weren’t good parents; they could have done *a lot* worse…

…

v. Freedom of medicine. One quote from the conclusion of Cochran’s post:

“[I]t is surely possible to materially improve the efficacy of drug development, of medical research as a whole. We’re doing better than we did 500 years ago – although probably worse than we did 50 years ago. But I would approach it by learning as much as possible about medical history, demographics, epidemiology, evolutionary medicine, theory of senescence, genetics, etc. Read Koch, not Hayek. There is no royal road to medical progress.”

I agree, and I was considering including some related comments and observations about health economics in this post – however I ultimately decided against doing that in part because the post was growing unwieldy; I might include those observations in another post later on. Here’s another somewhat older Westhunt post I at some point decided to bookmark – I in particular like the following neat quote from the comments, which expresses a view I have of course expressed myself in the past here on this blog:

“When you think about it, falsehoods, stupid crap, make the best group identifiers, because anyone might agree with you when you’re obviously right. Signing up to clear nonsense is a better test of group loyalty. A true friend is with you when you’re wrong. Ideally, not just wrong, but barking mad, rolling around in your own vomit wrong.”

…

vi. Economic Costs of Diabetes in the U.S. in 2012.

“Approximately 59% of all health care expenditures attributed to diabetes are for health resources used by the population aged 65 years and older, much of which is borne by the Medicare program […]. The population 45–64 years of age incurs 33% of diabetes-attributed costs, with the remaining 8% incurred by the population under 45 years of age. The annual attributed health care cost per person with diabetes […] increases with age, primarily as a result of increased use of hospital inpatient and nursing facility resources, physician office visits, and prescription medications. Dividing the total attributed health care expenditures by the number of people with diabetes, we estimate the average annual excess expenditures for the population aged under 45 years, 45–64 years, and 65 years and above, respectively, at $4,394, $5,611, and $11,825.”

“Our logistic regression analysis with NHIS data suggests that diabetes is associated with a 2.4 percentage point increase in the likelihood of leaving the workforce for disability. This equates to approximately 541,000 working-age adults leaving the workforce prematurely and 130 million lost workdays in 2012. For the population that leaves the workforce early because of diabetes-associated disability, we estimate that their average daily earnings would have been $166 per person (with the amount varying by demographic). Presenteeism accounted for 30% of the indirect cost of diabetes. The estimate of a 6.6% annual decline in productivity attributed to diabetes (in excess of the estimated decline in the absence of diabetes) equates to 113 million lost workdays per year.”

…

…

viii. Effect of longer term modest salt reduction on blood pressure: Cochrane systematic review and meta-analysis of randomised trials. Did I blog this paper at some point in the past? I could not find any coverage of it on the blog when I searched for it so I decided to include it here, even if I have a nagging suspicion I may have talked about these findings before. What did they find? The short version is this:

“A modest reduction in salt intake for four or more weeks causes significant and, from a population viewpoint, important falls in blood pressure in both hypertensive and normotensive individuals, irrespective of sex and ethnic group. Salt reduction is associated with a small physiological increase in plasma renin activity, aldosterone, and noradrenaline and no significant change in lipid concentrations. These results support a reduction in population salt intake, which will lower population blood pressure and thereby reduce cardiovascular disease.”

…

ix. Some wikipedia links:

Heroic Age of Antarctic Exploration (featured).

Kuiper belt (featured).

Treason (one quote worth including here: “Currently, the consensus among major Islamic schools is that apostasy (leaving Islam) is considered treason and that the penalty is death; this is supported not in the Quran but in the Hadith.^{[42]}^{[43]}^{[44]}^{[45]}^{[46]}^{[47]}“).

File:World map of countries by number of cigarettes smoked per adult per year.

Savant syndrome (“It is estimated that 10% of those with autism have some form of savant abilities”). A small sidenote of interest to Danish readers: The Danish Broadcasting Corporation recently featured a series about autistics with ‘special abilities’ – the show was called ‘The hidden talents’ (De skjulte talenter), and after multiple people had nagged me to watch it I ended up deciding to do so. Most of the people in that show presumably had some degree of ‘savantism’ combined with autism at the milder end of the spectrum, i.e. Asperger’s. I was somewhat conflicted about what to think about the show and did consider blogging it in detail (in Danish?), but I decided against it. However I do want to add here to Danish readers reading along who’ve seen the show that they would do well to repeatedly keep in mind that a) the great majority of autistics do not have abilities like these, b) many autistics with abilities like these presumably do quite poorly, and c) that many autistics have even greater social impairments than do people like e.g. (the very likeable, I have to add…) Louise Wille from the show).

Black Death (“Over 60% of Norway’s population died in 1348–1350”).

Renault FT (“among the most revolutionary and influential tank designs in history”).

Weierstrass function (“an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere”).

Void coefficient. (“a number that can be used to estimate how much the reactivity of a nuclear reactor changes as voids (typically steam bubbles) form in the reactor moderator or coolant. […] Reactivity is directly related to the tendency of the reactor core to change power level: if reactivity is positive, the core power tends to increase; if it is negative, the core power tends to decrease; if it is zero, the core power tends to remain stable. […] A positive void coefficient means that the reactivity increases as the void content inside the reactor increases due to increased boiling or loss of coolant; for example, if the coolant acts as a neutron absorber. If the void coefficient is large enough and control systems do not respond quickly enough, this can form a positive feedback loop which can quickly boil all the coolant in the reactor. This happened in the RBMK reactor that was destroyed in the Chernobyl disaster.”).

Gregor MacGregor (featured) (“a Scottish soldier, adventurer, and confidence trickster […] MacGregor’s Poyais scheme has been called one of the most brazen confidence tricks in history.”).

## Random Stuff

i. On the youtube channel of the Institute for Advanced Studies there has been a lot of activity over the last week or two (far more than 100 new lectures have been uploaded, and it seems new uploads are still being added at this point), and I’ve been watching a few of the recently uploaded astrophysics lectures. They’re quite technical, but you can watch them and follow enough of the content to have an enjoyable time despite not understanding everything:

This is a good lecture, very interesting. One major point made early on: “the take-away message is that the most common planet in the galaxy, at least at shorter periods, are planets for which there is no analogue in the solar system. The most common kind of planet in the galaxy is a planet with a radius of two Earth radii.” Another big take-away message is that small planets seem to be quite common (as noted in the conclusions, “16% of Sun-like stars have an Earth-sized planet”).

…

Of the lectures included in this post this was the one I liked the least; there are too many (‘obstructive’) questions/interactions between lecturer and attendants along the way, and the interactions/questions are difficult to hear/understand. If you consider watching both this lecture and the lecture below, I would say that it would probably be wise to watch the lecture below this one before you watch this one; I concluded that in retrospect some of the observations made early on in the lecture below would have been useful to know about before watching this lecture. (The first half of the lecture below was incidentally to me somewhat easier to follow than was the second half, but especially the first half hour of it is really quite good, despite the bad start (which one can always blame on Microsoft…)).

…

…

ii. Words I’ve encountered recently (…or ‘recently’ – it’s been a while since I last posted one of these lists): Divagations, periphrasis, reedy, architrave, sett, pedipalp, tout, togs, edentulous, moue, tatty, tearaway, prorogue, piscine, fillip, sop, panniers, auxology, roister, prepossessing, cantle, catamite, couth, ordure, biddy, recrudescence, parvenu, scupper, husting, hackle, expatiate, affray, tatterdemalion, eructation, coppice, dekko, scull, fulmination, pollarding, grotty, secateurs, bumf (I must admit that I like this word – it seems fitting, somehow, to use that word for this concept…), durophagy, randy, (brief note to self: Advise people having children who ask me about suggestions for how to name them against using this name (or variants such as Randi), it does not seem like a great idea), effete, apricity, sororal, bint, coition, abaft, eaves, gadabout, lugubriously, retroussé, landlubber, deliquescence, antimacassar, inanition.

…

iii. “The point of rigour is *not* to destroy all intuition; instead, it should be used to destroy *bad* intuition while clarifying and elevating *good* intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.” (Terry Tao, There’s more to mathematics than rigour and proofs)

…

iv. A century of trends in adult human height. A figure from the paper (Figure 3 – Change in adult height between the 1896 and 1996 birth cohorts):

(Click to view full size. WordPress seems to have changed the way you add images to a blog post – if this one is even so annoyingly large, I apologize, I have tried to minimize it while still retaining detail, but the original file is huge). An observation from the paper:

“Men were taller than women in every country, on average by ~11 cm in the 1896 birth cohort and ~12 cm in the 1996 birth cohort […]. In the 1896 birth cohort, the male-female height gap in countries where average height was low was slightly larger than in taller nations. In other words, at the turn of the 20th century, men seem to have had a relative advantage over women in undernourished compared to better-nourished populations.”

…

v. I found this paper, on Exercise and Glucose Metabolism in Persons with Diabetes Mellitus, interesting in part because I’ve been very surprised a few times by offhand online statements made by diabetic athletes, who had observed that their blood glucose really didn’t drop all that fast during exercise. Rapid and annoyingly large drops in blood glucose during exercise have been a really consistent feature of my own life with diabetes during adulthood. It seems that there may be big inter-individual differences in terms of the effects of exercise on glucose in diabetics. From the paper:

“Typically, prolonged moderate-intensity aerobic exercise (i.e., 30–70% of one’s VO_{2}max) causes a reduction in glucose concentrations because of a failure in circulating insulin levels to decrease at the onset of exercise.^{12} During this type of physical activity, glucose utilization may be as high as 1.5 g/min in adolescents with type 1 diabetes^{13} and exceed 2.0 g/min in adults with type 1 diabetes,^{14} an amount that quickly lowers circulating glucose levels. Persons with type 1 diabetes have large interindividual differences in blood glucose responses to exercise, although some intraindividual reproducibility exists.^{15} The wide ranging glycemic responses among individuals appears to be related to differences in pre-exercise blood glucose concentrations, the level of circulating counterregulatory hormones and the type/duration of the activity.^{2}“

## Random stuff

I find it difficult to find the motivation to finish the half-finished drafts I have lying around, so this will have to do. Some random stuff below.

i.

(15.000 views… In some sense that seems really ‘unfair’ to me, but on the other hand I doubt neither Beethoven nor Gilels care; they’re both long dead, after all…)

…

ii. New/newish words I’ve encountered in books, on vocabulary.com or elsewhere:

Agley, peripeteia, dissever, halidom, replevin, socage, organdie, pouffe, dyarchy, tauricide, temerarious, acharnement, cadger, gravamen, aspersion, marronage, adumbrate, succotash, deuteragonist, declivity, marquetry, machicolation, recusal.

…

iii. A lecture:

It’s been a long time since I watched it so I don’t have anything intelligent to say about it now, but I figured it might be of interest to one or two of the people who still subscribe to the blog despite the infrequent updates.

…

iv. A few wikipedia articles (I won’t comment much on the contents or quote extensively from the articles the way I’ve done in previous wikipedia posts – the links shall have to suffice for now):

Russian political jokes. Some of those made me laugh (e.g. this one: “A judge walks out of his chambers laughing his head off. A colleague approaches him and asks why he is laughing. “I just heard the funniest joke in the world!” “Well, go ahead, tell me!” says the other judge. “I can’t – I just gave someone ten years for it!”).

Political mutilation in Byzantine culture.

…

v. World War 2, if you think of it as a movie, has a highly unrealistic and implausible plot, according to this amusing post by Scott Alexander. Having recently read a rather long book about these topics, one aspect I’d have added had I written the piece myself would be that an additional factor making the setting seem even more implausible is how so many presumably quite smart people were so – what at least in retrospect seems – *unbelievably stupid* when it came to Hitler’s ideas and intentions before the war. Going back to Churchill’s own life I’d also add that if you were to make a movie about Churchill’s life during the war, which you could probably relatively easily do if you were to just base it upon his own copious and widely shared notes, then it could probably be made into a quite decent movie. His own comments, remarks, and observations certainly made for a great book.

## A few lectures

Below are three new lectures from the Institute of Advanced Study. As far as I’ve gathered they’re all from an IAS symposium called ‘Lens of Computation on the Sciences’ – all three lecturers are computer scientists, but you don’t have to be a computer scientist to watch these lectures.

Should computer scientists and economists band together more and try to use the insights from one field to help solve problems in the other field? Roughgarden thinks so, and provides examples of how this might be done/has been done. Applications discussed in the lecture include traffic management and auction design. I’m not sure how much of this lecture is easy to follow for people who don’t know anything about either topic (i.e., computer science and economics), but I found it not too difficult to follow – it probably helped that I’ve actually done work on a few of the things he touches upon in the lecture, such as basic auction theory, the fixed point theorems and related proofs, basic queueing theory and basic discrete maths/graph theory. Either way there are certainly much more technical lectures than this one available at the IAS channel.

…

I don’t have Facebook and I’m not planning on ever getting a FB account, so I’m not really sure I care about the things this guy is trying to do, but the lecturer does touch upon some interesting topics in network theory. Not a great lecture in my opinion and occasionally I think the lecturer ‘drifts’ a bit, talking without saying very much, but it’s also not a terrible lecture. A few times I was really annoyed that you can’t see where he’s pointing that damn laser pointer, but this issue should not stop you from watching the video, especially not if you have an interest in analytical aspects of how to approach and make sense of ‘Big Data’.

…

I’ve noticed that Scott Alexander has said some nice things about Scott Aaronson a few times, but until now I’ve never actually read any of the latter guy’s stuff or watched any lectures by him. I agree with Scott (Alexander) that Scott (Aaronson) is definitely a smart guy. This is an interesting lecture; I won’t pretend I understood all of it, but it has some thought-provoking ideas and important points in the context of quantum computing and it’s actually a quite entertaining lecture; I was close to laughing a couple of times.

## Quotes

i. “By all means think yourself big but don’t think everyone else small” (‘Notes on Flyleaf of Fresh ms. Book’, *Scott’s Last Expedition*. See also this).

ii. “The man who knows everyone’s job isn’t much good at his own.” (-ll-)

iii. “It is amazing what little harm doctors do when one considers all the opportunities they have” (Mark Twain, as quoted in the Oxford Handbook of Clinical Medicine, p.595).

iv. “A** **first-rate theory predicts; a second-rate theory forbids and a third-rate theory explains after the event.” (Aleksander Isaakovich Kitaigorodski)

v. “[S]ome of the most terrible things in the world are done by people who think, genuinely think, that they’re doing it for the best” (Terry Pratchett, Snuff).

vi. “That was excellently observ’d, say I, when I read a Passage in an Author, where his Opinion agrees with mine. When we differ, there I pronounce him to be mistaken.” (Jonathan Swift)

vii. “Death is nature’s master stroke, albeit a cruel one, because it allows genotypes space to try on new phenotypes.” (Quote from the Oxford Handbook of Clinical Medicine, p.6)

*
*viii. “The purpose of models is not to fit the data but to sharpen the questions.” (Samuel Karlin)

ix. “We may […] view set theory, and mathematics generally, in much the way in which we view theoretical portions of the natural sciences themselves; as comprising truths or hypotheses which are to be vindicated less by the pure light of reason than by the indirect systematic contribution which they make to the organizing of empirical data in the natural sciences.” (Quine)

x. “At root what is needed for scientific inquiry is just receptivity to data, skill in reasoning, and yearning for truth. Admittedly, ingenuity can help too.” (-ll-)

xi. “A statistician carefully assembles facts and figures for others who carefully misinterpret them.” (Quote from *Mathematically Speaking – A Dictionary of Quotations*, p.329. Only source given in the book is: “Quoted in Evan Esar, *20,000 Quips and Quotes*“)

xii. “A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances.” (Quote from *Mathematically Speaking – A Dictionary of Quotations*, p. 328. The source provided is: “Elements of Statistics, Part I, Chapter I (p.4)”).

xiii. “We own to small faults to persuade others that we have not great ones.” (Rochefoucauld)

xiv. “There is more self-love than love in jealousy.” (-ll-)

xv. “We should not judge of a man’s merit by his great abilities, but by the use he makes of them.” (-ll-)

xvi. “We should gain more by letting the world see what we are than by trying to seem what we are not.” (-ll-)

xvii. “Put succinctly, a prospective study looks for the effects of causes whereas a retrospective study examines the causes of effects.” (Quote from p.49 of *Principles of Applied Statistics*, by Cox & Donnelly)

xviii. “… he who seeks for methods without having a definite problem in mind seeks for the most part in vain.” (David Hilbert)

xix. “Give every man thy ear, but few thy voice” (Shakespeare).

xx. “Often the fear of one evil leads us into a worse.” (Nicolas Boileau-Despréaux)

## The Nature of Statistical Evidence

Here’s my goodreads review of the book.

As I’ve observed many times before, a wordpress blog like mine is not a particularly nice place to cover mathematical topics involving equations and lots of Greek letters, so the coverage below will be more or less purely conceptual; don’t take this to mean that the book doesn’t contain formulas. Some parts of the book look like this:

That of course makes the book hard to blog, also for other reasons than just the fact that it’s typographically hard to deal with the equations. In general it’s hard to talk about the content of a book like this one without going into *a lot* of details outlining how you get from A to B to C – usually you’re only really interested in C, but you need A and B to make sense of C. At this point I’ve sort of concluded that when covering books like this one I’ll only cover some of the main themes which are easy to discuss in a blog post, and I’ve concluded that I should skip coverage of (potentially important) points which might also be of interest if they’re difficult to discuss in a small amount of space, which is unfortunately often the case. I should perhaps observe that although I noted in my goodreads review that in a way there was a bit too much philosophy and a bit too little statistics in the coverage for my taste, you should definitely not take that objection to mean that this book is full of fluff; a lot of that philosophical stuff is ‘formal logic’ type stuff and related comments, and the book in general is quite dense. As I also noted in the goodreads review I didn’t read this book as carefully as I might have done – for example I skipped a couple of the technical proofs because they didn’t seem to be worth the effort – and I’d probably need to read it again to fully understand some of the minor points made throughout the more technical parts of the coverage; so that’s of course a related reason why I don’t cover the book in a great amount of detail here – it’s hard work just to read the damn thing, to talk about the technical stuff in detail here as well would definitely be overkill even if it would surely make me understand the material better.

I have added some observations from the coverage below. I’ve tried to clarify beforehand which question/topic the quote in question deals with, to ease reading/understanding of the topics covered.

…

On how statistical methods are related to experimental science:

“statistical methods have aims similar to the process of experimental science. But statistics is not itself an experimental science, it consists of models of how to do experimental science. Statistical theory is a logical — mostly mathematical — discipline; its findings are not subject to experimental test. […] The primary sense in which statistical theory is a science is that it guides and explains statistical methods. A sharpened statement of the purpose of this book is to provide explanations of the senses in which some statistical methods provide scientific evidence.”

On mathematics and axiomatic systems (the book goes into much more detail than this):

“It is not sufficiently appreciated that a link is needed between mathematics and methods. Mathematics is not about the world until it is interpreted and then it is only about models of the world […]. No contradiction is introduced by either interpreting the same theory in different ways or by modeling the same concept by different theories. […] In general, a primitive undefined term is said to be **interpreted** when a meaning is assigned to it and when all such terms are interpreted we have an **interpretation** of the axiomatic system. It makes no sense to ask which is the correct interpretation of an axiom system. This is a primary strength of the axiomatic method; we can use it to organize and structure our thoughts and knowledge by simultaneously and economically treating all interpretations of an axiom system. It is also a weakness in that failure to define or interpret terms leads to much confusion about the implications of theory for application.”

It’s all about models:

“The scientific method of theory checking is to compare predictions deduced from a theoretical model with observations on nature. Thus science must predict what happens in nature but it need not explain why. […] whether experiment is consistent with theory is relative to accuracy and purpose. All theories are simplifications of reality and hence no theory will be expected to be a perfect predictor. Theories of statistical inference become relevant to scientific process at precisely this point. […] Scientific method is a practice developed to deal with experiments on **nature. **Probability theory is a deductive study of the properties of **models **of such experiments. All of the theorems of probability are results about models of experiments.”

But given a frequentist interpretation you can test your statistical theories with the real world, right? Right? Well…

“How might we check the long run stability of relative frequency? If we are to compare mathematical theory with experiment then only finite sequences can be observed. But for the Bernoulli case, the event that frequency approaches probability is stochastically independent of any sequence of finite length. […] Long-run stability of relative frequency cannot be checked experimentally. There are neither theoretical nor empirical guarantees that, a priori, one can recognize experiments performed under uniform conditions and that under these circumstances one *will* obtain stable frequencies.” [related link]

What should we expect to get out of mathematical and statistical theories of inference?

“What can we expect of a theory of statistical inference? We can expect an internally consistent explanation of why certain conclusions follow from certain data. The theory will not be about inductive rationality but about a *model *of inductive rationality. Statisticians are used to thinking that they apply their logic to models of the physical world; less common is the realization that their logic itself is only a model. Explanation will be in terms of introduced concepts which do not exist in nature. Properties of the concepts will be derived from assumptions which merely seem reasonable. This is the only sense in which the axioms of any mathematical theory are true […] We can expect these concepts, assumptions, and properties to be intuitive but, unlike natural science, they cannot be checked by experiment. Different people have different ideas about what “seems reasonable,” so we can expect different explanations and different properties. We should not be surprised if the theorems of two different theories of statistical evidence differ. If two models had no different properties then they would be different versions of the same model […] We should not expect to achieve, by mathematics alone, a single coherent theory of inference, for mathematical truth is conditional and the assumptions are not “self-evident.” Faith in a set of assumptions would be needed to achieve a single coherent theory.”

On disagreements about the nature of statistical evidence:

“The context of this section is that there is disagreement among experts about the nature of statistical evidence and consequently much use of one formulation to criticize another. Neyman (1950) maintains that, from his behavioral hypothesis testing point of view, Fisherian significance tests do not express evidence. Royall (1997) employs the “law” of likelihood to criticize hypothesis as well as significance testing. Pratt (1965), Berger and Selke (1987), Berger and Berry (1988), and Casella and Berger (1987) employ Bayesian theory to criticize sampling theory. […] Critics assume that their findings are about evidence, but they are at most about models of evidence. Many theoretical statistical criticisms, when stated in terms of evidence, have the following outline: According to model A, evidence satisfies proposition P. But according to model B, which is correct since it is derived from “self-evident truths,” P is not true. Now evidence can’t be two different ways so, since B is right, A must be wrong. Note that the argument is symmetric: since A appears “self-evident” (to adherents of A) B must be wrong. But both conclusions are invalid since evidence can be modeled in different ways, perhaps useful in different contexts and for different purposes. From the observation that P is a theorem of A but not of B, all we can properly conclude is that A and B are different models of evidence. […] The common practice of using one theory of inference to critique another is a misleading activity.”

Is mathematics a science?

“Is mathematics a science? It is certainly systematized knowledge much concerned with structure, but then so is history. Does it employ the scientific method? Well, partly; hypothesis and deduction are the essence of mathematics and the search for counter examples is a mathematical counterpart of experimentation; but the question is not put to nature. Is mathematics about nature? In part. The hypotheses of most mathematics are suggested by some natural primitive concept, for it is difficult to think of interesting hypotheses concerning nonsense syllables and to check their consistency. However, it often happens that as a mathematical subject matures it tends to evolve away from the original concept which motivated it. Mathematics in its purest form is probably not natural science since it lacks the experimental aspect. Art is sometimes defined to be creative work displaying form, beauty and unusual perception. By this definition pure mathematics is clearly an art. On the other hand, applied mathematics, taking its hypotheses from real world concepts, is an attempt to describe nature. Applied mathematics, without regard to experimental verification, is in fact largely the “conditional truth” portion of science. If a body of applied mathematics has survived experimental test to become trustworthy belief then it is the essence of natural science.”

Then what about statistics – is statistics a science?

“Statisticians can and do make contributions to subject matter fields such as physics, and demography but statistical theory and methods proper, distinguished from their findings, are not like physics in that they are not about nature. […] Applied statistics is natural science but the findings are about the subject matter field not statistical theory or method. […] Statistical theory helps with how to do natural science but it is not itself a natural science.”

…

I should note that I am, and have for a long time been, in broad agreement with the author’s remarks on the nature of science and mathematics above. Popper, among many others, discussed this topic a long time ago e.g. in The Logic of Scientific Discovery and I’ve basically been of the opinion that (‘pure’) mathematics is not science (‘but rather ‘something else’ … and that doesn’t mean it’s not useful’) for probably a decade. I’ve had a harder time coming to terms with how precisely to deal with statistics in terms of these things, and in that context the book has been conceptually helpful.

Below I’ve added a few links to other stuff also covered in the book:

Propositional calculus.

Kolmogorov’s axioms.

Neyman-Pearson lemma.

Radon-Nikodyn theorem. (not covered in the book, but the necessity of using ‘a Radon-Nikodyn derivative’ to obtain an answer to a question being asked was remarked upon at one point, and I had no clue what he was talking about – it seems that the stuff in the link was what he was talking about).

A very specific and relevant link: Berger and Wolpert (1984). The stuff about Birnbaum’s argument covered from p.24 (p.40) and forward is covered in some detail in the book. The author is critical of the model and explains in the book in some detail why that is. See also: *On the foundations of statistical inference* (Birnbaum, 1962).

## A few lectures

This one was mostly review for me, but there was also some new stuff and it was a ‘sort of okay’ lecture even if I was highly skeptical about a few points covered. I was debating whether to even post the lecture on account of those points of contention, but I figured that by adding a few remarks below I could justify doing it. So below a few skeptical comments relating to content covered in the lecture:

a) 28-29 minutes in he mentions that the cutoff for hypertension in diabetics is a systolic pressure above 130. Here opinions definitely differ, and opinions about treatment cutoffs differ; in the annual report from the Danish Diabetes Database they follow up on whether hospitals and other medical decision-making units are following guidelines (I’ve talked about the data on the blog, e.g. here), and the BP goal of involved decision-making units evaluated is currently whether diabetics with systolic BP above 140 receive antihypertensive treatment. This recent Cochrane review concluded that: “At the present time, evidence from randomized trials does not support blood pressure targets lower than the standard targets in people with elevated blood pressure and diabetes” and noted that: “The effect of SBP targets on mortality was compatible with both a reduction and increase in risk […] Trying to achieve the ‘lower’ SBP target was associated with a significant increase in the number of other serious adverse events”.

b) Whether retinopathy screenings should be conducted yearly or biennially is also contested, and opinions differ – this is not mentioned in the lecture, but I sort of figure maybe it should have been. There’s some evidence that annual screening is better (see e.g. this recent review), but the evidence base is not great and clinical outcomes do not seem to differ much in general; as noted in the review, “Observational and economic modelling studies in low-risk patients show little difference in clinical outcomes between screening intervals of 1 year or 2 years”. To stratify based on risk seems desirable from a cost-effectiveness standpoint, but how to stratify optimally seems to not be completely clear at the present point in time.

c) The Somogyi phenomenon is highly contested, and I was very surprised about his coverage of this topic – ‘he’s a doctor lecturing on this topic, he should know better’. As the wiki notes: “Although this theory is well known among clinicians and individuals with diabetes, there is little scientific evidence to support it.” I’m highly skeptical, and I seriously question the advice of lowering insulin in the context of morning hyperglycemia. As observed in *Cryer’s* text: “there is now considerable evidence against the Somogyi hypothesis (Guillod et al. 2007); morning hyperglycemia is the result of insulin lack, not post-hypoglycemic insulin resistance (Havlin and Cryer 1987; Tordjman et al. 1987; Hirsch et al. 1990). There is a dawn phenomenon—a growth hormone–mediated increase in the nighttime to morning plasma glucose concentration (Campbell et al. 1985)—but its magnitude is small (Periello et al. 1991).”

…

I decided not to embed this lecture in the post mainly because the resolution is unsatisfactorily low so that a substantial proportion of the visual content is frankly unintelligible; I figured this would bother others more than it did me and that a semi-satisfactory compromise solution in terms of coverage would be to link to the lecture, but not embed it here. You can hear what the lecturer is saying, which was enough for me, but you can’t make out stuff like effect differences, p-values, or many of the details in the graphic illustrations included. Despite the title of the lecture on youtube, the lecture actually mainly consists of a brief overview of pharmacological treatment options for diabetes.

…

…

If you want to skip the introduction, the first talk/lecture starts around 5 minutes and 30 seconds into the video. Note that despite the long running time of this video the lectures themselves only take about 50 minutes in total; the rest of it is post-lecture Q&A and discussion.

## Mathematically Speaking

This is a book full of quotes on the topic of mathematics. As is always the case for books full of quotations, most of the quotes in this book aren’t very good, but occasionally you come across a quote or two that enable you to justify reading on. I’ll likely include some of the good/interesting quotes in the book in future ‘quotes’ posts. Below I’ve added some sample quotes from the book. I’ve read roughly three-fifths of the book so far and I’m currently hovering around a two-star rating on goodreads.

…

“Since authors seldom, if ever, say what they mean, the following glossary is offered to neophytes in mathematical research to help them understand the language that surrounds the formulas …

ANALOGUE. This is an a. of: I have to have *some* excuse for publishing it.

APPLICATIONS. This is of interest in a.: I have to have *some* excuse for publishing it.

COMPLETE. The proof is now c.: I can’t finish it. […]

DIFFICULT. This problem is d.: I don’t know the answer. (*Cf.* Trivial)

GENERALITY. Without loss of g.: I have done an easy special case. […]

INTERESTING. X’s paper is I.: I don’t understand it.

KNOWN. This is a k. result but I reproduce the proof for convenience of the reader: My paper isn’t long enough. […]

NEW. This was proved by X but the following n. proof may present points of interest: I can’t understand X.

NOTATION. To simplify the n.: It is too much trouble to change now.

OBSERVED. It will be o. that: I hope you have not noticed that.

OBVIOUS. It is o.: I can’t prove it.

READER. The details may be left to the r.: I can’t do it. […]

STRAIGHTFORWARD. By a s. computation: I lost my notes.

TRIVIAL. This problem is t.: I know the answer (*Cf.* Difficult).

WELL-KNOWN. The result is w.: I can’t find the reference.” (Pétard, H. [Pondiczery, E.S.]).

Here are a few quotes similar to the ones above, provided by a different, unknown source:

“BRIEFLY: I’m running out of time, so I’ll just write and talk faster. […]

HE’S ONE OF THE GREAT LIVING MATHEMATICIANS: He’s written 5 papers and I’ve read 2 of them. […]

I’VE HEARD SO MUCH ABOUT YOU: Stalling a minute may give me time to recall who you are. […]

QUANTIFY: I can’t find anything wrong with your proof except that it won’t work if x is a moon of Jupiter (popular in applied math courses). […]

SKETCH OF A PROOF: I couldn’t verify all the details, so I’ll break it down into the parts I couldn’t prove.

YOUR TALK WAS VERY INTERESTING: I can’t think of anything to say about your talk.” (‘Unknown’)

“Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines which conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view.” (Carl Boyer)

“It’s the nature of mathematics to pose more problems than it can solve.” (Ivars Peterson)

“the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed […] The chief merit of mathematization is that it compels us to become conscious of what we are assuming.” (Bertrand de Jouvenel)

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” (Albert Einstein)

“[Mathematics] includes much that will neither hurt one who does not know it nor help one who does.” (J. B. Mencke)

“Pure mathematics consists entirely of asseverations to the extent that, if such and such a proposition is true of *anything*, then such and such another proposition is true of *anything*. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … *If* our hypothesis is about *anything*, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” (Bertrand Russell)

“Mathematical rigor is like clothing; in its style it ought to suit the occasion, and it diminishes comfort and restricts freedom of movement if it is either too loose or too tight.” (G. F. Simmons).

“at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up … In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.” (John von Neumann)

## Random stuff/Open Thread

i. A lecture on mathematical proofs:

…

ii. “In the fall of 1944, only seven percent of all bombs dropped by the Eighth Air Force hit within 1,000 feet of their aim point.”

From wikipedia’s article on Strategic bombing during WW2. The article has a lot of stuff. The ‘RAF estimates of destruction of “built up areas” of major German cities’ numbers in the article made my head spin – they didn’t bomb the Germans back to the stone age, but they sure tried. Here’s another observation from the article:

“After the war, the U.S. Strategic Bombing Survey reviewed the available casualty records in Germany, and concluded that official German statistics of casualties from air attack had been too low. The survey estimated that at a minimum 305,000 were killed in German cities due to bombing and estimated a minimum of 780,000 wounded. Roughly 7,500,000 German civilians were also rendered homeless.” (The German population at the time was roughly 70 million).

…

iii. Also war-related: Eddie Slovik:

“**Edward Donald “Eddie” Slovik** (February 18, 1920 – January 31, 1945) was a United States Army soldier during World War II and the only American soldier to be court-martialled and executed for desertion since the American Civil War.^{[1]}^{[2]}

Although over 21,000 American soldiers were given varying sentences for desertion during World War II, including 49 death sentences, Slovik’s was the only death sentence that was actually carried out.^{[1]}^{[3]}^{[4]}

During World War II, 1.7 million courts-martial were held, representing one third of all criminal cases tried in the United States during the same period. Most of the cases were minor, as were the sentences.^{[2]} Nevertheless, a clemency board, appointed by the Secretary of War in the summer of 1945, reviewed all general courts-martial where the accused was still in confinement.^{[2]}^{[5]} That Board remitted or reduced the sentence in 85 percent of the 27,000 serious cases reviewed.^{[2]} The death penalty was rarely imposed, and those cases typically were for rapes or murders. […] In France during World War I from 1917 to 1918, the United States Army executed 35 of its own soldiers, but all were convicted of rape and/or unprovoked murder of civilians and not for military offenses.^{[13]} During World War II in all theaters of the war, the United States military executed 102 of its own soldiers for rape and/or unprovoked murder of civilians, but only Slovik was executed for the military offense of desertion.^{[2]}^{[14]} […] of the 2,864 army personnel tried for desertion for the period January 1942 through June 1948, 49 were convicted and sentenced to death, and 48 of those sentences were voided by higher authority.”

What motivated me to read the article was mostly curiosity about how many people were actually executed for deserting during the war, a question I’d never encountered any answers to previously. The US number turned out to be, well, let’s just say it’s lower than I’d expected it would be. American soldiers who chose to desert during the war seem to have had much, much better chances of surviving the war than had soldiers who did not. Slovik was not a lucky man. On a related note, given numbers like these I’m really surprised desertion rates were not much higher than they were; presumably community norms (”desertion = disgrace’, which would probably rub off on other family members…’) played a key role here.

…

iv. Chess and infinity. I haven’t posted this link before even though the thread is a few months old, and I figured that given that I just had a conversation on related matters in the comment section of SCC (here’s a link) I might as well repost some of this stuff here. Some key points from the thread (I had to make slight formatting changes to the quotes because wordpress had trouble displaying some of the numbers, but the content is unchanged):

u/TheBB:

“Shannon has estimated the number of possible legal *positions* to be about 10^{43}. The number of legal *games* is quite a bit higher, estimated by Littlewood and Hardy to be around 10^{10^5} (commonly cited as 10^{10^50} perhaps due to a misprint). This number is so large that it can’t really be compared with anything that is not combinatorial in nature. It is far larger than the number of subatomic particles in the observable universe, let alone stars in the Milky Way galaxy.

As for your bonus question, a typical chess game today lasts about 40 to 60 moves (let’s say 50). Let us say that there are 4 reasonable candidate moves in any given position. I suspect this is probably an underestimate if anything, but let’s roll with it. That gives us about 4^{2×50} ≈ 10^{60} games that might reasonably be played by good human players. If there are 6 candidate moves, we get around 10^{77}, which is in the neighbourhood of the number of particles in the observable universe.”

u/Wondersnite:

“To put 10^{10^5} into perspective:

There are 10^{80} protons in the Universe. Now imagine inside each proton, we had a whole entire Universe. Now imagine again that inside each proton inside each Universe inside each proton, you had another Universe. If you count up all the protons, you get (10^{80} )^{3} = 10^{240,} which is nowhere near the number we’re looking for.

You have to have Universes inside protons all the way down to **1250 steps** to get the number of legal chess games that are estimated to exist. […]

Imagine that every single subatomic particle in the entire observable universe was a supercomputer that analysed a possible game in a single Planck unit of time (10^{-43} seconds, the time it takes light in a vacuum to travel 10^{-20} times the width of a proton), and that every single subatomic particle computer was running from the beginning of time up until the heat death of the Universe, 10^{1000} years ≈ 10^{11} × 10^{1000} seconds from now.

Even in these ridiculously favorable conditions, we’d only be able to calculate

10^{80} × 10^{43} × 10^{11} × 10^{1000} = 10^{1134}

possible games. Again, this doesn’t even come close to 10^{10^5} = 10^{100000} .

Basically, if we ever solve the game of chess, it definitely won’t be through brute force.”

…

v. An interesting resource which a friend of mine recently shared with me and which I thought I should share here as well: Nature Reviews – Disease Primers.

…

vi. Here are some words I’ve recently encountered on vocabulary.com: augury, spangle, imprimatur, apperception, contrition, ensconce, impuissance, acquisitive, emendation, tintinnabulation, abalone, dissemble, pellucid, traduce, objurgation, lummox, exegesis, probity, recondite, impugn, viscid, truculence, appurtenance, declivity, adumbrate, euphony, educe, titivate, cerulean, ardour, vulpine.

## Wikipedia articles of interest

i. Lock (water transport). *Zumerchik and Danver’s book* covered this kind of stuff as well, sort of, and I figured that since I’m not going to blog the book – for reasons provided in my goodreads review here – I might as well add a link or two here instead. The words ‘sort of’ above are in my opinion justified because the book coverage is so horrid you’d never even know what a lock is used for from reading that book; you’d need to look that up elsewhere.

On a related note there’s a lot of stuff in that book about the history of water transport etc. which you probably won’t get from these articles, but having a look here will give you some idea about which sort of topics many of the chapters of the book are dealing with. Also, stuff like this and this. The book coverage of the latter topic is incidentally *much*, *much* more detailed than is that wiki article, and the article – as well as many other articles about related topics (economic history, etc.) on the wiki, to the extent that they even exist – could clearly be improved greatly by adding content from books like this one. However I’m not going to be the guy doing that.

…

ii. Congruence (geometry).

…

iii. Geography and ecology of the Everglades (featured).

I’d note that this is a topic which seems to be reasonably well covered on wikipedia; there’s for example also a ‘good article’ on the Everglades and a featured article about the Everglades National Park. A few quotes and observations from the article:

“The **geography and ecology of the Everglades** involve the complex elements affecting the natural environment throughout the southern region of the U.S. state of Florida. Before drainage, the Everglades were an interwoven mesh of marshes and prairies covering 4,000 square miles (10,000 km^{2}). […] Although sawgrass and sloughs are the enduring geographical icons of the Everglades, other ecosystems are just as vital, and the borders marking them are subtle or nonexistent. Pinelands and tropical hardwood hammocks are located throughout the sloughs; the trees, rooted in soil inches above the peat, marl, or water, support a variety of wildlife. The oldest and tallest trees are cypresses, whose roots are specially adapted to grow underwater for months at a time.”

“A vast marshland could only have been formed due to the underlying rock formations in southern Florida.^{[15]} The floor of the Everglades formed between 25 million and 2 million years ago when the Florida peninsula was a shallow sea floor. The peninsula has been covered by sea water at least seven times since the earliest bedrock formation. […] At only 5,000 years of age, the Everglades is a young region in geological terms. Its ecosystems are in constant flux as a result of the interplay of three factors: the type and amount of water present, the geology of the region, and the frequency and severity of fires. […] Water is the dominant element in the Everglades, and it shapes the land, vegetation, and animal life of South Florida. The South Florida climate was once arid and semi-arid, interspersed with wet periods. Between 10,000 and 20,000 years ago, sea levels rose, submerging portions of the Florida peninsula and causing the water table to rise. Fresh water saturated the limestone, eroding some of it and creating springs and sinkholes. The abundance of fresh water allowed new vegetation to take root, and through evaporation formed thunderstorms. Limestone was dissolved by the slightly acidic rainwater. The limestone wore away, and groundwater came into contact with the surface, creating a massive wetland ecosystem. […] Only two seasons exist in the Everglades: wet (May to November) and dry (December to April). […] The Everglades are unique; no other wetland system in the world is nourished primarily from the atmosphere. […] Average annual rainfall in the Everglades is approximately 62 inches (160 cm), though fluctuations of precipitation are normal.”

“Between 1871 and 2003, 40 tropical cyclones struck the Everglades, usually every one to three years.”

“Islands of trees featuring dense temperate or tropical trees are called tropical hardwood hammocks.^{[38]} They may rise between 1 and 3 feet (0.30 and 0.91 m) above water level in freshwater sloughs, sawgrass prairies, or pineland. These islands illustrate the difficulty of characterizing the climate of the Everglades as tropical or subtropical. Hammocks in the northern portion of the Everglades consist of more temperate plant species, but closer to Florida Bay the trees are tropical and smaller shrubs are more prevalent. […] Islands vary in size, but most range between 1 and 10 acres (0.40 and 4.05 ha); the water slowly flowing around them limits their size and gives them a teardrop appearance from above.^{[42]} The height of the trees is limited by factors such as frost, lightning, and wind: the majority of trees in hammocks grow no higher than 55 feet (17 m). […] There are more than 50 varieties of tree snails in the Everglades; the color patterns and designs unique to single islands may be a result of the isolation of certain hammocks.^{[44]} […] An estimated 11,000 species of seed-bearing plants and 400 species of land or water vertebrates live in the Everglades, but slight variations in water levels affect many organisms and reshape land formations.”

“Because much of the coast and inner estuaries are built by mangroves—and there is no border between the coastal marshes and the bay—the ecosystems in Florida Bay are considered part of the Everglades. […] Sea grasses stabilize sea beds and protect shorelines from erosion by absorbing energy from waves. […] Sea floor patterns of Florida Bay are formed by currents and winds. However, since 1932, sea levels have been rising at a rate of 1 foot (0.30 m) per 100 years.^{[81]} Though mangroves serve to build and stabilize the coastline, seas may be rising more rapidly than the trees are able to build.^{[82]}”

…

iv. Chang and Eng Bunker. Not a long article, but interesting:

“**Chang** (Chinese: 昌; pinyin: *Chāng*; Thai: จัน, *Jan*, rtgs: Chan) and **Eng** (Chinese: 恩; pinyin: *Ēn*; Thai: อิน *In*) **Bunker** (May 11, 1811 – January 17, 1874) were Thai-American conjoined twin brothers whose condition and birthplace became the basis for the term “Siamese twins”.^{[1]}^{[2]}^{[3]}”

I loved some of the implicit assumptions in this article: “Determined to live as normal a life they could, Chang and Eng settled on their small plantation and bought slaves to do the work they could not do themselves. […] Chang and Adelaide [his wife] would become the parents of eleven children. Eng and Sarah [‘the other wife’] had ten.”

A ‘normal life’ indeed… The women the twins married were incidentally sisters who ended up disliking each other (I can’t imagine why…).

…

v. Genie (feral child). This is a very long article, and you should be warned that many parts of it may not be pleasant to read. From the article:

“**Genie** (born 1957) is the pseudonym of a feral child who was the victim of extraordinarily severe abuse, neglect and social isolation. Her circumstances are prominently recorded in the annals of abnormal child psychology.^{[1]}^{[2]} When Genie was a baby her father decided that she was severely mentally retarded, causing him to dislike her and withhold as much care and attention as possible. Around the time she reached the age of 20 months Genie’s father decided to keep her as socially isolated as possible, so from that point until she reached 13 years, 7 months, he kept her locked alone in a room. During this time he almost always strapped her to a child’s toilet or bound her in a crib with her arms and legs completely immobilized, forbade anyone from interacting with her, and left her severely malnourished.^{[3]}^{[4]}^{[5]} The extent of Genie’s isolation prevented her from being exposed to any significant amount of speech, and as a result she did not acquire language during childhood. Her abuse came to the attention of Los Angeles child welfare authorities on November 4, 1970.^{[1]}^{[3]}^{[4]}

In the first several years after Genie’s early life and circumstances came to light, psychologists, linguists and other scientists focused a great deal of attention on Genie’s case, seeing in her near-total isolation an opportunity to study many aspects of human development. […] In early January 1978 Genie’s mother suddenly decided to forbid all of the scientists except for one from having any contact with Genie, and all testing and scientific observations of her immediately ceased. Most of the scientists who studied and worked with Genie have not seen her since this time. The only post-1977 updates on Genie and her whereabouts are personal observations or secondary accounts of them, and all are spaced several years apart. […]

Genie’s father had an extremely low tolerance for noise, to the point of refusing to have a working television or radio in the house. Due to this, the only sounds Genie ever heard from her parents or brother on a regular basis were noises when they used the bathroom.^{[8]}^{[43]} Although Genie’s mother claimed that Genie had been able to hear other people talking in the house, her father almost never allowed his wife or son to speak and viciously beat them if he heard them talking without permission. They were particularly forbidden to speak to or around Genie, so what conversations they had were therefore always very quiet and out of Genie’s earshot, preventing her from being exposed to any meaningful language besides her father’s occasional swearing.^{[3]}^{[13]}^{[43]} […] Genie’s father fed Genie as little as possible and refused to give her solid food […]

In late October 1970, Genie’s mother and father had a violent argument in which she threatened to leave if she could not call her parents. He eventually relented, and later that day Genie’s mother was able to get herself and Genie away from her husband while he was out of the house […] She and Genie went to live with her parents in Monterey Park.^{[13]}^{[20]}^{[56]} Around three weeks later, on November 4, after being told to seek disability benefits for the blind, Genie’s mother decided to do so in nearby Temple City, California and brought Genie along with her.^{[3]}^{[56]}

On account of her near-blindness, instead of the disabilities benefits office Genie’s mother accidentally entered the general social services office next door.^{[3]}^{[56]} The social worker who greeted them instantly sensed something was not right when she first saw Genie and was shocked to learn Genie’s true age was 13, having estimated from her appearance and demeanor that she was around 6 or 7 and possibly autistic. She notified her supervisor, and after questioning Genie’s mother and confirming Genie’s age they immediately contacted the police. […]

Upon admission to Children’s Hospital, Genie was extremely pale and grossly malnourished. She was severely undersized and underweight for her age, standing 4 ft 6 in (1.37 m) and weighing only 59 pounds (27 kg) […] Genie’s gross motor skills were extremely weak; she could not stand up straight nor fully straighten any of her limbs.^{[83]}^{[84]} Her movements were very hesitant and unsteady, and her characteristic “bunny walk”, in which she held her hands in front of her like claws, suggested extreme difficulty with sensory processing and an inability to integrate visual and tactile information.^{[62]} She had very little endurance, only able to engage in any physical activity for brief periods of time.^{[85]} […]

Despite tests conducted shortly after her admission which determined Genie had normal vision in both eyes she could not focus them on anything more than 10 feet (3 m) away, which corresponded to the dimensions of the room she was kept in.^{[86]} She was also completely incontinent, and gave no response whatsoever to extreme temperatures.^{[48]}^{[87]} As Genie never ate solid food as a child she was completely unable to chew and had very severe dysphagia, completely unable to swallow any solid or even soft food and barely able to swallow liquids.^{[80]}^{[88]} Because of this she would hold anything which she could not swallow in her mouth until her saliva broke it down, and if this took too long she would spit it out and mash it with her fingers.^{[50]} She constantly salivated and spat, and continually sniffed and blew her nose on anything that happened to be nearby.^{[83]}^{[84]}

Genie’s behavior was typically highly anti-social, and proved extremely difficult for others to control. She had no sense of personal property, frequently pointing to or simply taking something she wanted from someone else, and did not have any situational awareness whatsoever, acting on any of her impulses regardless of the setting. […] Doctors found it extremely difficult to test Genie’s mental age, but on two attempts they found Genie scored at the level of a 13-month-old. […] When upset Genie would wildly spit, blow her nose into her clothing, rub mucus all over her body, frequently urinate, and scratch and strike herself.^{[102]}^{[103]} These tantrums were usually the only times Genie was at all demonstrative in her behavior. […] Genie clearly distinguished speaking from other environmental sounds, but she remained almost completely silent and was almost entirely unresponsive to speech. When she did vocalize, it was always extremely soft and devoid of tone. Hospital staff initially thought that the responsiveness she did show to them meant she understood what they were saying, but later determined that she was instead responding to nonverbal signals that accompanied their speaking. […] Linguists later determined that in January 1971, two months after her admission, Genie only showed understanding of a few names and about 15–20 words. Upon hearing any of these, she invariably responded to them as if they had been spoken in isolation. Hospital staff concluded that her active vocabulary at that time consisted of just two short phrases, “stop it” and “no more”.^{[27]}^{[88]}^{[99]} Beyond negative commands, and possibly intonation indicating a question, she showed no understanding of any grammar whatsoever. […] Genie had a great deal of difficulty learning to count in sequential order. During Genie’s stay with the Riglers, the scientists spent a great deal of time attempting to teach her to count. She did not start to do so at all until late 1972, and when she did her efforts were extremely deliberate and laborious. By 1975 she could only count up to 7, which even then remained very difficult for her.”

“From January 1978 until 1993, Genie moved through a series of at least four additional foster homes and institutions. In some of these locations she was further physically abused and harassed to extreme degrees, and her development continued to regress. […] Genie is a ward of the state of California, and is living in an undisclosed location in the Los Angeles area.^{[3]}^{[20]} In May 2008, ABC News reported that someone who spoke under condition of anonymity had hired a private investigator who located Genie in 2000. She was reportedly living a relatively simple lifestyle in a small private facility for mentally underdeveloped adults, and appeared to be happy. Although she only spoke a few words, she could still communicate fairly well in sign language.^{[3]}“

## A few lectures

(This was a review lecture for me as I read a textbook on these topics a few months back going into quite a lot more detail – the post I link to has some relevant links if you’re curious to explore this topic further).

…

A few relevant links: Group (featured), symmetry group, Cayley table, Abelian group, Symmetry groups of Platonic solids, dual polyhedron, Lagrange’s theorem (group theory), Fermat’s little theorem. I think he was perhaps trying to cover a little bit too much ground in too little time by bringing up the RSA algorithm towards the end, but I’m sort of surprised how many people disliked the video; I don’t think it’s that bad.

…

The beginning of the lecture has a lot of remarks about Fourier‘s life which are in some sense not ‘directly related’ to the mathematics, and so if this is what you’re most interested in knowing more about you can probably skip the first 11 minutes or so of the lecture without missing out on much. The lecture is very non-technical compared to coverage like this, this, and this (…or this).

…

I think one thing worth mentioning here is that the lecturer is the author of a rather amazing book on the topic he talks about in the lecture.

## Wikipedia articles of interest

i. Invasion of Poland. I recently realized I had no idea e.g. how long it took for the Germans and Soviets to defeat Poland during WW2 (the answer is 1 month and five days). The Germans attacked more than two weeks before the Soviets did. The article has lots of links, like most articles about such topics on wikipedia. Incidentally the question of why France and Britain applied a double standard and only declared war on Germany, and not the Soviet Union, is discussed in much detail in the links provided by u/OldWorldGlory here.

…

ii. Huaynaputina. From the article:

“A few days before the eruption, someone reported booming noise from the volcano and fog-like gas being emitted from its crater. The locals scrambled to appease the volcano, preparing girls, pets, and flowers for sacrifice.”

This makes sense – what else would one do in a situation like that? Finding a few virgins, dogs and flowers seems like the sensible approach – yes, you have to love humans and how they always react in sensible ways to such crises.

I’m not really sure the rest of the article is really all that interesting, but I found the above sentence both amusing and depressing enough to link to it here.

…

iii. Albert Pierrepoint. This guy killed hundreds of people.

On the other hand people were fine with it – it was his job. Well, sort of, this is actually slightly complicated. (“Pierrepoint was often dubbed the Official Executioner, despite there being no such job or title”).

Anyway this article is clearly the story of a guy who achieved his childhood dream – though unlike other children, he did not dream of becoming a fireman or a pilot, but rather of becoming the Official Executioner of the country. I’m currently thinking of using Pierrepoint as the main character in the motivational story I plan to tell my nephew when he’s a bit older.

…

iv. Second Crusade (featured). Considering how many different ‘states’ and ‘kingdoms’ were involved, a surprisingly small amount of people were actually fighting; the article notes that “[t]here were perhaps 50,000 troops in total” on the Christian side when the attack on Damascus was initiated. It wasn’t enough, as the outcome of the crusade was a decisive Muslim victory in the ‘Holy Land’ (Middle East).

…

v. 0.999… (featured). This thing is equal to one, but it can sometimes be really hard to get even very smart people to accept this fact. Lots of details and some proofs presented in the article.

…

vi. Shapley–Folkman lemma (‘good article’ – but also a somewhat technical article).

…

vii. Multituberculata. This article is not that special, but I add it here also because I think it ought to be and I’m actually sort of angry that it’s not; sometimes the coverage provided on wikipedia simply strikes me as grossly unfair, even if this is perhaps a slightly odd way to think about stuff. As pointed out in the article (Agustí points this out in his book as well), “The multituberculates existed for about 120 million years, and are often considered the most successful, diversified, and long-lasting mammals in natural history.” Yet notice how much (/little) coverage the article provides. Now compare the article with this article, or this.

## Introduction to Systems Analysis: Mathematically Modeling Natural Systems (I)

“This book was originally developed alongside the lecture Systems Analysis at the Swiss Federal Institute of Technology (ETH) Zürich, on the basis of lecture notes developed over 12 years. The lecture, together with others on analysis, differential equations and linear algebra, belongs to the basic mathematical knowledge imparted on students of environmental sciences and other related areas at ETH Zürich. […] The book aims to be more than a mathematical treatise on the analysis and modeling of natural systems, yet a certain set of basic mathematical skills are still necessary. We will use linear differential equations, vector and matrix calculus, linear algebra, and even take a glimpse at nonlinear and partial differential equations. Most of the mathematical methods used are covered in the appendices. Their treatment there is brief however, and without proofs. Therefore it will not replace a good mathematics textbook for someone who has not encountered this level of math before. […] The book is firmly rooted in the algebraic formulation of mathematical models, their analytical solution, or — if solutions are too complex or do not exist — in a thorough discussion of the anticipated model properties.”

…

I finished the book yesterday – here’s my goodreads review (note that the first link in this post was not to the goodreads profile of the book for the reason that goodreads has listed the book under the wrong title). I’ve never read a book about ‘systems analysis’ before, but as I also mention in the goodreads review it turned out that much of this stuff was stuff I’d seen before. There are 8 chapters in the book. Chapter one is a brief introductory chapter, the second chapter contains a short overview of mathematical models (static models, dynamic models, discrete and continuous time models, stochastic models…), the third chapter is a brief chapter about static models (the rest of the book is about dynamic models, but they want you to at least know the difference), the fourth chapter deals with linear (differential equation) models with one variable, chapter 5 extends the analysis to linear models with several variables, chapter 6 is about non-linear models (covers e.g. the Lotka-Volterra model (of course) and the Holling-Tanner model (both were covered in Ecological Dynamics, in much more detail)), chapter 7 deals briefly with time-discrete models and how they are different from continuous-time models (I liked Gurney and Nisbet’s coverage of this stuff a lot better, as that book had a lot more details about these things) and chapter 8 concludes with models including both a time- and a space-dimension, which leads to coverage of concepts such as mixing and transformation, advection, diffusion and exchange in a model context.

How to derive solutions to various types of differential equations, how to calculate eigenvalues and what these tell you about the model dynamics (and how to deal with them when they’re imaginary), phase diagrams/phase planes and topographical maps of system dynamics, fixed points/steady states and their properties, what’s an attractor?, what’s hysteresis and in which model contexts might this phenomenon be present?, the difference between homogeneous and non-homogeneous differential equations and between first order- and higher-order differential equations, which role do the initial conditions play in various contexts?, etc. – it’s this kind of book. Applications included in the book are varied; some of the examples are (as already mentioned) derived from the field of ecology/mathematical biology (there are also e.g. models of phosphate distribution/dynamics in lakes and models of fish population dynamics), others are from chemistry (e.g. models dealing with gas exchange – Fick’s laws of diffusion are e.g. covered in the book, and they also talk about e.g. Henry’s law), physics (e.g. the harmonic oscillator, the Lorenz model) – there are even a few examples from economics (e.g. dealing with interest rates). As they put it in the introduction, “Although most of the examples used here are drawn from the environmental sciences, this book is not an introduction to the theory of aquatic or terrestrial environmental systems. Rather, a key goal of the book is to demonstrate the virtually limitless practical potential of the methods presented.” I’m not sure if they succeeded, but it’s certainly clear from the coverage that you can use the tools they cover in a lot of different contexts.

I’m not quite sure how much mathematics you’ll need to know in order to read and understand this book on your own. In the coverage they seem to me to assume some familiarity with linear algebra, multi-variable calculus, complex analysis (/related trigonometry) (perhaps also basic combinatorics – for example factorials are included without comments about how they work). You should probably take the authors at their words when they say above that the book “will not replace a good mathematics textbook for someone who has not encountered this level of math before”. A related observation is also that regardless of whether you’ve seen this sort of stuff before or not, this is probably not the sort of book you’ll be able to read in a day or two.

I think I’ll try to cover the book in more detail (with much more specific coverage of some main points) tomorrow.

## Some links (Open Thread?)

It’s been quite a while since the last time I posted a ‘here’s some interesting stuff I’ve found online’-post, so I’ll do that now even though I actually don’t spend much time randomly looking around for interesting stuff online these days. I added some wikipedia links I’d saved for a ‘wikipedia articles of interest’-post because it usually takes quite a bit of time to write a standard wikipedia post (as it takes time to figure out what to include and what not to include in the coverage) and I figured that if I didn’t add those links here I’d never get around to blogging them.

i. Battle of Dyrrhachium. Found via this link, which has a lot of stuff.

ii. An AMA by someone who claims to have succeeded in faking his own death.

iii. I found this article about the so-called “Einstellung” effect in chess interesting. I’m however not sure how important this stuff really is. I don’t think it’s sub-optimal for a player to spend a significant amount of time in positions like the ones they analyzed on ideas that don’t work, because usually you’ll only have to spot one idea that does to win the game. It’s obvious that one can argue people spend ‘too much’ time looking for a winning combination in positions where *by design* no winning combinations exist, but the fact of the matter is that in positions where ‘familiar patterns’ pop up winning resources often *do* exist, and you don’t win games by overlooking those or by failing to spend time looking for them; occasional suboptimal moves in some contexts may be a reasonable price to pay for increasing your likelihood of finding/playing the best/winning moves when those do exist. Here’s a slightly related link dealing with the question of the potential number of games/moves in chess. Here’s a good wiki article about pawn structures, and here’s one about swindles in chess. I incidentally very recently became a member of the ICC, and I’m frankly impressed with the player pool – which is huge and includes some really strong players (players like Morozevich and Tomashevsky seem to play there regularly). Since I started out on the site I’ve already beaten 3 IMs in bullet and lost a game against Islandic GM Henrik Danielsen. The IMs I’ve beaten were far from the strongest players in the player pool, but in my experience you don’t get to play titled players nearly as often as that on other sites if you’re at my level.

iv. A picture of the Andromeda galaxy. A *really big* picture. Related link here.

v. You may already have seen this one, but in case you have not: A Philosopher Walks Into A Coffee Shop. More than one of these made me laugh out loud. If you like the post you should take a look at the comments as well, there are some brilliant ones there as well.

vi. Amdahl’s law.

vii. Eigendecomposition of a matrix. On a related note I’m currently reading Imboden and Pfenninger’s Introduction to Systems Analysis (which goodreads for some reason has listed under a wrong title, as the goodreads book title is really the subtitle of the book), and today I had a look at the wiki article on Jacobian matrices and determinants for that reason (the book is about as technical as you’d expect from a book with a title like that).

viii. If you’ve been wondering how I’ve found the quotes I’ve posted here on this blog (I’ve posted roughly 150 posts with quotes so far), links like these are very useful.

## Recountings: Conversations with MIT Mathematicians

This post will be brief but I thought that since it’s been a while since I last posted anything and since I just finished reading this book, I wanted to add a few remarks about it here while it was still ‘fresh in my mind’. I’m gradually coming to the conclusion that if I’m to blog all the books I’m reading in the amount of detail I’d ideally like to, I’ll have to read a lot less. This option does not appeal to me; I’d rather provide limited coverage of a book I’ve actually read than not read a book in order to provide more extensive coverage of another book.

Anyway, the book is a rather nice collection of interviews with mathematicians from MIT’s ‘early days’ (in some sense at least – MIT is a rather old institution, but at least some of the people interviewed in this book came along during the days before MIT was what it is today), who talk about the history of the mathematics department of MIT, and other stuff – the people interviewed include an Abel Prize winner and a few people who’ve been members of the Institute for Advanced Study, a former MacArthur Fellow, as well as a guy who used to be on the selection committee for the MacArthur Foundation. All of them are *really, really smart*, and some of them have lived quite interesting lives. To the extent that these guys aren’t impressive enough on their own, some of them also knew some people most non-mathematicians have probably heard about – this book includes contributions from people who were friends of people like John Nash, Grothendieck, Shannon, Minsky, and Chomsky, and they are people who’ve met and talked to people like John von Neumann, Oppenheimer, Weyl, Heisenberg, and Albert Einstein. They talk a little bit about their work and the history of the mathematics department, but they also talk about other stuff as well; there are various amusing anecdotes along the way (for example one interviewee tells the story about the time he lectured in a gorilla suit at MIT), there are stories about the private parties and social lives of the MIT staff during the fifties (and later), we get some personal stories about mathematicians who fled Europe when the Nazis started to cause trouble, and there are stories about student protests in the late sixties and how they were dealt with – the books spans widely. There was some repetition across the interviews (various people answering similar questions in similar ways), and there was more talk about ‘administrative matters’ than I’d have liked – probably a natural consequence of the fact that a few of them (3? *At least* three of the contributors..) were former department heads – which is part of why I didn’t give it five stars, but it’s really a quite nice book. I may or may not blog it later in more detail.

## Statistical Models for Proportions and Probabilities

“Most elementary statistics books discuss inference for proportions and probabilities, and the primary readership for this monograph is the student of statistics, either at an advanced undergraduate or graduate level. As some of the recommended so-called ‘‘large-sample’’ rules in textbooks have been found to be inappropriate, this monograph endeavors to provide more up-to-date information on these topics. I have also included a number of related topics not generally found in textbooks. The emphasis is on model building and the estimation of parameters from the models.

It is assumed that the reader has a background in statistical theory and inference and is familiar with standard univariate and multivariate distributions, including conditional distributions.”

…

The above quote is from the the book‘s preface. The book is highly technical – here’s a screencap of a page roughly in the middle:

I think the above picture provides some background as to why I do not think it’s a good idea to provide detailed coverage of the book here. Not all pages are that bad, but this *is* a book on mathematical statistics. The technical nature of the book made it difficult for me to know how to rate it – I like to ask myself when reading books like this one if I would be able to spot an error in the coverage. In some contexts here I clearly would not be able to do that (given the time I was willing to spend on the book), and when that’s the case I always feel hesitant about rating(/’judging’) books of this nature. I should note that there are pretty much no spelling/formatting errors, and the language is easy to understand (‘if you know enough about statistics…’). I did have one major problem with part of the coverage towards the end of the book, but it didn’t much alter my general impression of the book. The problem was that the author seems to apply (/recommend?) a hypothesis-testing framework for model selection, a practice which although widely used is frankly considered bad statistics by Burnham and Anderson in their book on model selection. In the relevant section of the book Seber discusses an approach to modelling which starts out with a ‘full model’ including both primary effects and various (potentially multi-level) interaction terms (he deals specifically with data derived from multiple (independent?) multinomial distributions, but where the data comes from is not really important here), and then he proceeds to use hypothesis tests of whether interaction terms are zero to determine whether or not interactions should be included in the model or not. For people who don’t know, this model selection method is both very commonly used and a very wrong way to do things; using hypothesis testing as a model selection mechanism is a methodologically invalid approach to model selection, something Burnham and Anderson talks a lot about in their book. I assume I’ll be covering Burnham and Anderson’s book in more detail later on here on the blog, so for now I’ll just make this key point here and then return to that stuff later – if you did not understand the comments above you shouldn’t worry too much about it, I’ll go into much more detail when talking about that stuff later. This problem was the only real problem I had with Seber’s book.

Although I’ll not talk a lot about what the book was about (not only because it might be hard for some readers to follow, I should point out, but also because detailed coverage would take a lot more time than I’d be willing to spend on this stuff), I decided to add a few links to relevant stuff he talks about in the book. Quite a few pages in the book are spent on talking about the properties of various distributions, how to estimate key parameters of interest, and how to construct confidence intervals to be used for hypothesis testing in those specific contexts.

Some of the links below deal with stuff covered in the book, a few others however just deal with stuff I had to look up in order to understand what was going on in the coverage:

Inverse sampling.

Binomial distribution.

Hypergeometric distribution.

Multinomial distribution.

Binomial proportion confidence interval. (Coverage of the Wilson score interval, Jeffreys interval, and the Clopper-Pearson interval included in the book).

Fisher’s exact test.

Marginal distribution.

Fischer information.

Moment-generating function.

Factorial moment-generating function.

Delta method.

Multidimensional central limit theorem (the book applies this, but doesn’t really talk about it).

Matrix function.

McNemar’s test.

## Open Thread

It’s been a long time since I had one of these. Questions? Comments? Random observations?

I hate posting posts devoid of content, so here’s some random stuff:

i.

If you think the stuff above is all fun and games I should note that the topic of chiralty, which is one of the things talked about in the lecture above, was actually covered in some detail in Gale’s book, which hardly is a book which spends a great deal of time talking about esoteric mathematical concepts. On a related note, the main reason why I have not blogged that book is incidentally that I lost all notes and highlights I’d made in the first 200 pages of the book when my computer broke down, and I just can’t face reading that book again simply in order to blog it. It’s a good book, with interesting stuff, and I may decide to blog it later, but I don’t feel like doing it at the moment; without highlights and notes it’s a real pain to blog a book, and right now it’s just not worth it to reread the book. Rereading books can be fun – I’ve incidentally been rereading *Darwin* lately and I may decide to blog this book soon; I imagine I might also choose to reread some of Asimov’s books before long – but it’s not much fun if you’re finding yourself having to do it simply because the computer deleted your work.

…

ii. Beyond Power Calculations: Assessing Type S (Sign) and Type M (Magnitude) Errors.

Here’s the abstract:

“Statistical power analysis provides the conventional approach to assess error rates when designing a research study. However, power analysis is flawed in that a narrow emphasis on statistical significance is placed as the primary focus of study design. In noisy, small-sample settings, statistically significant results can often be misleading. To help researchers address this problem in the context of their own studies, we recommend design calculations in which (a) the probability of an estimate being in the wrong direction (*Type S [sign] error*) and (b) the factor by which the magnitude of an effect might be overestimated (*Type M [magnitude] error or exaggeration ratio*) are estimated. We illustrate with examples from recent published research and discuss the largest challenge in a design calculation: coming up with reasonable estimates of plausible effect sizes based on external information.”

If a study has low power, you can get into a lot of trouble. Some problems are well known, others probably aren’t. A bit more from the paper:

“design calculations can reveal three problems:

1. Most obvious, a study with low power is unlikely to “succeed” in the sense of yielding a statistically significant result.

2. It is quite possible for a result to be significant at the 5% level — with a 95% confidence interval that entirely excludes zero — and for there to be a high chance, sometimes 40% or more, that this interval is on the wrong side of zero. Even sophisticated users of statistics can be unaware of this point — that the probability of a Type S error is not the same as the p value or significance level.[3]

3. Using statistical significance as a screener can lead researchers to drastically overestimate the magnitude of an effect (Button et al., 2013).

Design analysis can provide a clue about the importance of these problems in any particular case.”

“Statistics textbooks commonly give the advice that statistical significance is not the same as practical significance, often with examples in which an effect is clearly demonstrated but is very small […]. In many studies in psychology and medicine, however, the problem is the opposite: an estimate that is statistically significant but with such a large uncertainty that it provides essentially no information about the phenomenon of interest. […] There is a range of evidence to demonstrate that it remains the case that too many small studies are done and preferentially published when “significant.” We suggest that one reason for the continuing lack of real movement on this problem is the historic focus on power as a lever for ensuring statistical significance, with inadequate attention being paid to the difficulties of interpreting statistical significance in underpowered studies. Because insufficient attention has been paid to these issues, we believe that too many small studies are done and preferentially published when “significant.” There is a common misconception that if you happen to obtain statistical significance with low power, then you have achieved a particularly impressive feat, obtaining scientific success under difficult conditions.

However, that is incorrect if the goal is scientific understanding rather than (say) publication in a top journal. In fact, statistically significant results in a noisy setting are highly likely to be in the wrong direction and invariably overestimate the absolute values of any actual effect sizes, often by a substantial factor.”

…

iii. I’m sure most people who might be interested in following the match are already well aware that Anand and Carlsen are currently competing for the world chess championship, and I’m not going to talk about that match here. However I do want to mention to people interested in improving their chess that I recently came across this site, and that I quite like it. It only deals with endgames, but endgames are really important. If you don’t know much about endgames you may find the videos available here, here and here to be helpful.

…

iv. A link: Crosss Validated: “Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.”

A friend recently told me about this resource. I knew about the existence of StackExchange, but I haven’t really spent much time there. These days I mostly stick to books and a few sites I already know about; I rarely look for new interesting stuff online. This also means you should not automatically assume I surely already know about X when you’re considering whether to tell me about X in an Open Thread.