Cramer's Rule

Here's a significantly more useful result from Linear Algebra:

Let A be a non-singular n x n matrix and b an n-dimensional real vector.

Let A_i be A with the ith column replaced with b.

If x is a unique solution to Ax = b, then x_i = |A_i| / |A|

Well, useful in the Q-question world where n < 4. After that, taking the determinants gets to be an awful lot of work and you're back to busting out a numerical package.

Wronskian

Following yesterday's intention to catalog named results, let's start with a weird one.

If we have n functions that are n-1 times differentiable on an interval [a,b], the Wronskian is defined as:



So, basically, you create a matrix where each column vector is the successive derivatives of f_i and you take the determinant.

The significance is the following theorem:

If there exists x in [a,b] such that W[f₁, ..., f_n](x) ≠ 0, then f₁, ..., f_n are linearly independent.

Why does that matter? Well, it means they form a basis for the subspace they span, which is sometimes a useful thing to know. (At least, it's useful if you're staring at a Q question that asks you to find the basis of a subspace. Maybe not that useful any other time.)

Apparently, it's also occasionally useful for solving higher order systems of differential equations, though the methods I dug up on line didn't seem a whole lot easier than just solving the equations the normal way. At any rate, not too many people do that anymore. Numerical packages are so powerful now, it's just not worth the effort except in trivial cases. Wronski developed this idea in 1812, a decade before Babbage secured funding for building the world's first computer.

So, probably won't be using this result very much unless it comes up on the Q, which it might.

I should probably also remember that the converse is not necessarily true. You can have independent functions with a Wronskian that is zero everywhere.

Foundation

I've recently come to what should have been a pretty obvious realization. I guess I've known it for some time, but prepping for the Q has brought it more into focus.

Important results get named. Not necessarily after the person who derived or invented them, but named just the same. Minor results just get listed in texts as Theorem 4.3.2 or something like that. Major results may still be listed that way, but then there will be a parenthetic attribution like (Bayes' Rule). I hadn't given much thought to this until I started looking things up on the web. Since a web page discussing some technicality can't waste space deriving all the necessary supporting results, they have to be able to say "By Bayes' Rule, it follows that ..." and so on. This, of course, is not new to the web, academic papers have been doing the same thing for centuries. I just haven't read too many of those in the past 25 years.

Anyway, my point is that it's a great way to quickly go through a text and find all the important results. So, that's what I'm going to do. Over the next few weeks, I'm going to catalog every named result in all the texts I'm studying from. By completely committing those to memory, I'll not only have the foundational results I need for proving results on the Q, I'll also be in a much better spot for discussing thesis points with other faculty.

And, by the way, Bayes' Rule: P(X|Y) = P(Y|X)P(X)/P(Y). I knew that one already.

Reset

Well, I threw in the towel. Work isn't letting up and the Q is only 9 days away. I deferred it to next semester. While I was at it, I also dropped the Cloud Computing course since if I don't carve out some serious study time, I'm going to be right back in this same bind four months from now.

So, this is a pretty big reset. It probably means I won't finish in three years. I always thought that was ambitious, but it seemed doable up until now. I still wouldn't say it's out of the question, but certainly a lot less likely.

That may not be an entirely bad thing. A year ago, I just wanted to get my "union card" and move on. I figured that anybody hiring a new prof in his mid-50's wasn't going to care about the dissertation; they wanted the 30 years of work experience. The PhD was just one of those things you have to do to be in the club.

I still think that's true, but now I want the dissertation to matter. I've enjoyed digging into a topic. I want to do something of substance. That's not going to happen if my primary goal is to get through as quickly as possible. My remaining class is a directed study with my adviser. I'm hoping we really hone in on a productive line of research this fall. I'd like to come out of UMSL with 3-4 publications along with the thesis. From what I can tell, success in academics isn't that much different from success in anything else. Sure, talent counts, but what really matters is that you're willing to do the work. I am.

Some things are going great

If a little pessimism has crept into my posts over the last few weeks, it's because I really did want to put the Q behind me. Actually, that's an understatement. I wanted to crush it and use that result to get the best faculty members on my committee en route to a dissertation of some consequence.

Well, that may still happen, but only if I wait until February to take it. Work has simply been overwhelming and, even if I am able to pass the exam in ten days, it would be "as one escaping through the flames" as the apostle Paul so aptly put it.

However, that work thing has not been entirely one-sided. Yes, it's pretty much destroyed my life in the short term, but that's pretty much the nature of IT work. The only way to distinguish yourself in this field is by being willing to drop everything when the situation calls for it. And, the situation has called for it. As a result, I've scored some pretty big wins at the office:

• Our annual summer release (always our most stressful) went without incident.
• A month after go-live, we've had no significant production issues. I did have to work two weekends (the final nails in the Q prep coffin) to work through some data quality issues, but those were problems at the source, not with our stuff.
• We closed our second successful iteration (of two) on the project to rehost all our reporting to a scalable platform (Hadoop/Impala/AtScale).
• Primarily as a result of the last item, the members of my team are begging to stay rather than be transferred off. We normally rotate assignments so people don't get burned out, but the developers are really eager to move onto the new technology stack.

So, even if it does knock me down a rung at school, the last couple months have certainly improved my standing at work. I was doing pretty well on both fronts coming into the summer. It will be easier to repair damage on the school side.

Less than hoped for

Today, I tried a dry run of the Q. I don't have an actual practice test, but I do have a list of questions that are supposed to be representative.

It didn't go well.

It's not that I didn't know what was being asked or what approach to take. It's just that all the little details that one needs to fill in the steps come so slowly (and sometimes, they escape me altogether). It's the difference between being familiar with the subjects and really knowing them. The only way I know to bridge that gap is lots of drill. With only ten days to go, that's not really an option.

Sufficiently believable

Hey, here's something we haven't seen much of lately: a post about math. Yes, that is still my major and I haven't dropped out. That said, I'm pursuing a Philosopher's Doctorate, thus, rather than actually do any math, I'm just going to talk about it.

I was reviewing data reduction today and was struck by the tortured definition of a sufficient statistic: If T(X) is a sufficient statistic for θ, then any inference about theta should depend on the sample X only through the value T(X). That is, if x and y are two sample points such that T(x) = T(y), then the inference about θ should be the same whether X = x or X = y is observed.

Ok, the authors do give a formal definition that is more concise, but it still reflects this mentality that the world is this objective rules-based machine and we simply need to discover those rules. I don't know any theoretical physicists who believe that (and, yes, I do know several theoretical physicists; they aren't as rare as you might think). But the stats community, which serves primarily the empirical sciences (I'll be generous and include the social sciences in that group) where everything is debatable, still clings to this notion of absolute truth.

Of course, there are plenty of stats folks who don't. They're called Bayesians. Rather than get wrapped up around what θ is (or if it even exists), a Bayesian is only interested in what we believe about θ. A statistic is sufficient for θ if it fully informs our belief. That is, if our posterior belief about θ is the same whether we're given the statistic or the entire sample, then the statistic is sufficient.

The formal definitions drive this home.

Frequentist: A statistic T(X) is a sufficient statistic for θ if P(X|T(X)) does not depend on θ.

Baked into that is the notion of absolute truth. Otherwise, conditioning X on a function of itself makes no sense at all.

Bayesian: A statistic T(X) is a sufficient statistic for θ if P(θ|T(X)) = P(θ|X).

It's a fair debate, but it seems to me that if θ is the thing you care about but don't know, talking about sufficiency in terms of how much more you know after observing the statistic makes a lot more sense.

Just to be clear, Bayesians and frequentists don't actually disagree on what constitutes a sufficient statistic. The two definitions are equivalent. The disagreement is over what a sufficient (or any other) statistic actually tells you about the world.