## ‘Jargon’

“Generally, the likelihood function is the joint density (or probability function for discrete variables), de fined as a function of the unknown parameters…” (math stuff omitted)

[…]

“This estimator is rarely available, since the second derivatives of the log-likelihood function are often complicated nonlinear functions of the data whose exact expected values will be unknown. The estimator is

positive de
finite provided θ(0) is identi
ed, and therefore usually positive de
finite in fi
nite samples.”

I just got thinking. This makes perfect sense, I understand what’s being said. But how many assumptions about prior knowledge does it take to get a likelihood of 0,5 of someone understanding all of this? Regular reader ‘Plamus’ knows this stuff, I’m sure about that, but is he even in the majority or not of my readers? How many such sentences would I meet if I started reading a book on histology? (I know the answer to that and my respect for doctors went up after that.)

It’s not the marginal piece of information that’s a problem. It’s all the stuff you need to know and remember in order to understand the marginal piece of information.

In your opinion: What math-courses at which level would make understanding the sentence easy?

Comment by Stefan | September 14, 2010 |

For many people, there’s no way that sentence would ever get easy. ‘IQ-and-stuff-like-that’ is a big issue here.

Given that’s not an issue… You need some linear algebra (how vectors and matrixes work and how to work with them) and calculus (preferably multivariable calculus), at the very least enough to know what a Hessian is and how to calculate it. You need to know some statistics, know how distributions work, how to set up a likelihood function and work with it, arguably some stuff about asymptotics. University level maths and statistics, nothing less than that, and not just 1 or 2 courses. Where I study, people who’d encounter a sentence like that one would probably have something like at minimum ~6-7-8 courses in university level statistics and mathematics behind them (plus A-level HS maths), plus some courses making use of that stuff in praxis. How much it’d take to have something like that be easy for you would greatly depend on your endowments and your starting point. I don’t know your background or the motivation behind the question, that would also be helpful.

Razib Khan conducted a reader survey in February, where one of the questions was: “Highest Level of Math Completed”.

Link here: http://www.gnxp.com/blog/2009/02/gnxp-survey-results.php

The options were: Pre-Algebra, Algebra, Geometry, Algebra II, Pre-Calculus, Calculus, Differential Equations, Linear Algebra, Multivariable Calculus, Higher than Multivariable Calculus and Have Math Degree. We have multivariable calculus before linear algebra, and to use numerical methods to solve non-linear differential equations is also much more difficult than most of the multivariable calculus stuff you’re likely to encounter in such a course – so the ordering is not all that strict.. To understand the statements above, you’re at least at linear algebra plus calculus, preferably multivarible calculus. Plus some statistics on top of the maths.

As an aside, in general I’d also say that even if you understand the stuff, statistics is rarely easy, just like logic isn’t easy. That’s part of why it takes so much time to learn the stuff, because it doesn’t come natural to most of us – our brains didn’t evolve to (consciously) do that kind of stuff. It’s like solving differential equations: Almost everybody can catch a ball if you thrown it to them, but ask them to solve the math stuff they do with their eyes and hands in real time afterwards with a pen and paper, and more than 9 out of 10 will have no clue what to do. Those who can do it with pen and paper could have caught another 50 balls in the time that it takes them to write the stuff down. They could have caught (/tens of?/hundreds of?) thousands of balls in the time it took them to learn how to actually do the calculation on paper. Oh yeah – of course it’s also worth remembering that cats and insects can actually solve those or some similar differential equations too, and quite a bit faster than we can.

Comment by US | September 14, 2010 |

I’ve just begun learning the linear algebra part (chem engineering), so the question was posed to gauge the journey ahead. Thanks for the reply!

How much do you estimate knowing math at that level contributes to your overall or everyday “mental acuity”?

Comment by Stefan | September 14, 2010 |

Chem engineering is quite a different education, so I’d be cautious about ‘gauging the journey ahead’ too much by using econ math and statistics; it’s likely quite different stuff your education will emphasize, and there’s no reason why a chemical engineer should know what the above sentences mean, just as there’s no reason why I should know what, say, an α-helix is or what doctors use haematoxylin for (I don’t know anything about chem engineering, so I can’t even use an example from your own field…).

I have no idea how to answer your last question. The best answer I can come up with right now is probably this: I think knowing math at that level makes a lot of people more stupid, because it makes them think that they’re much smarter than they really are.

Comment by US | September 14, 2010 |

I heartily agree with US’s points. As usual, I’d like to add a few random observations, and expound on a few of the points.

Yes, the excerpt makes sense to me, I have studied all of this back in the day. Still, if I had to work the details of the argument, I would probably have to hit a reference book of the web for a refresher on a some of the finer points of statistical parameterization. US overestimates me somewhat – and maybe not me per se, but the limitations of human memory. You may have done dozens of practice problems on, say, random variable transformation, but if you do not use it often enough and/or are not in the process of actively studying it, you lose the skill. To me, the more important thing is to keep the skill at a level where you can regain it with minimal review. I was doing just this today at work – working out how many validation cycles I had to put a certain model through; normally this is not an issue, since computing power is cheap, but in this case it may be the difference between a quad-core computer crunching regressions for a day, or for a couple of weeks.

Which brings us to a second issue. There are different levels of competence. Take, for example, the transportation problem. In the case of a 3×3 matrix, it’s fairly easy to do, 4×4 is a pain in the derriere, I have done 5×5 manually, because I had a sadistic professor in quantitative methods back in the day, who did not trust computers, because “they round after a certain decimal position”. Right now, I could not do the problem manually without some review; I know, however, enough about the setup to be able to plug it into a statistical package, which will give me the results in no time flat, and to interpret the results. Am I competent in it or not? These days, the purely analytical solutions to problems are often of interest to theoreticians, and software solves the problems for you (quant methods, brute force, heuristics, etc.), if you know enough to tell it what to do.

Also, I believe that economics in particular, has become way too dependent on mathematical methods. Those give you elegant analytical solutions to models, get you published, enhance your career – and yet are mostly totally impractical because of the simplifying assumptions you must make in order to be able to set up a model that is amenable at all to such solutions. This is not true of physics, chemistry, engineering, even CS and biology – these guys need exact solutions, or things melt, blow up, collapse, poison people, or hang up.

As to Stefan’s last question, I’ll second US’s reply. At that level, the benefit to “mental acuity” is likely small. There is, however, a level somewhat below this one that benefits you immensely. It’s also a level that the vast majority (I’ll guestimate 95%+) of people not only do not reach, but are not aware exists. Delineating this valuable “package” is hard, but I would put there the fundamentals of logic and linear algebra, calculus up to differentiation (with some multivariate elements), probability, and a smorgasbord of statistical techniques/concepts – z, t, F, and chi-square distributions with related hypothesis testing (both parametric and non-parametric), regressions, principal component analysis, time series, autocorrelation… I would NOT put there, for example, integration and differential equations, advanced linear algebra, and a whole host of statistical methods that tell you basically how to solve particular problems, but add little to conceptual knowledge. I am fully aware that, being a number-cruncher by trade, I have “déformation professionnelle”, and probably overvalue some of these concepts, just as military and law-enforcement people cannot believe how we can live unaware of the dangers they see, doctors cringe at the harm we cause ourselves, etc. Every expert tends to see his or her area as existential meta-knowledge – the stuff you need to know/do in order to survive and be able to make use of others’ not-so-critical knowledge. It’s human nature – so fraught with biases.

Good luck with linear algebra, Stefan.

Comment by Plamus | September 15, 2010 |

Thanks for the comment Plamus.

“Also, I believe that economics in particular, has become way too dependent on mathematical methods. Those give you elegant analytical solutions to models, get you published, enhance your career – and yet are mostly totally impractical because of the simplifying assumptions you must make in order to be able to set up a model that is amenable at all to such solutions.”

I’ve remarked upon that problem myself from time to time. It’s a far bigger problem in macro than in the rest of the field, at least it looks that way to me. The macro models are in general so complex that most people don’t get how they work, yet most of them actually rely on assumptions so simplistic (/and stupid) that if people actually could understand how the models worked in detail, they’d lose most, if not all, their respect for that part of the field. As it is now, all they see is a lot of fancy math.

Also a ‘good luck’ to Stefan from me.

Comment by US | September 15, 2010 |

Thanks both of you.

This is gold, Plamus:

“Delineating this valuable “package” is hard, but I would put there the fundamentals of logic and linear algebra, calculus up to differentiation (with some multivariate elements), probability, and a smorgasbord of statistical techniques/concepts – z, t, F, and chi-square distributions with related hypothesis testing (both parametric and non-parametric), regressions, principal component analysis, time series, autocorrelation.”

Comment by Stefan | September 15, 2010 |