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toto comments on Case study: abuse of frequentist statistics - Less Wrong
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This is going to sound silly, but...could someone explain frequentist statistics to me?
Here's my current understanding of how it works:
We've got some hypothesis H, whose truth or falsity we'd like to determine. So we go out and gather some evidence E. But now, instead of trying to quantify our degree of belief in H (given E) as a conditional probability estimate using Bayes' Theorem (which would require us to know P(H), P(E|H), and P(E|~H)), what we do is simply calculate P(E|~H) (techniques for doing this being of course the principal concern of statistics texts), and then place H into one of two bins depending on whether P(E|~H) is below some threshold number ("p-value") that somebody decided was "low": if P(E|~H) is below that number, we put H into the "accepted" bin (or, as they say, we reject the null hypothesis ~H); otherwise, we put H into the "not accepted" bin (that is, we fail to reject ~H).
Now, if that is a fair summary, then this big controversy between frequentists and Bayesians must mean that there is a sizable collection of people who think that the above procedure is a better way of obtaining knowledge than performing Bayesian updates. But for the life of me, I can't see how anyone could possibly think that. I mean, not only is the "p-value" threshold arbitrary, not only are we depriving ourselves of valuable information by "accepting" or "not accepting" a hypothesis rather than quantifying our certainty level, but...what about P(E|H)?? (Not to mention P(H).) To me, it seems blatantly obvious that an epistemology (and that's what it is) like the above is a recipe for disaster -- specifically in the form of accumulated errors over time.
I know that statisticians are intelligent people, so this has to be a strawman or something. Or at least, there must be some decent-sounding arguments that I haven't heard -- and surely there are some frequentist contrarians reading this who know what those arguments are. So, in the spirit of Alicorn's "Deontology for Cosequentialists" or ciphergoth's survey of the anti-cryonics position, I'd like to suggest a "Frequentism for Bayesians" post -- or perhaps just a "Frequentism for Dummies", if that's what I'm being here.
Is that not precisely the problem? Often, the H you are interested in is so vague ("there is some kind of effect in a certain direction") that it is very difficult to estimate P(E / H) - or even to define it.
OTOH, P(E / ~H) is often very easy to compute from first principles, or to obtain through experiments (since conditions where "the effect" is not present are usually the most common).
Example: I have a coin. I want to know if it is "true" or "biased". I flip it 100 times, and get 78 tails.Now how do I estimate the probability of obtaining this many tails, knowing that the coin is "biased"? How do I even express that analytically? By contrast, it is very easy to compute the probability of this sequence (or any other) with a "non-biased" coin.
So there you have it. The whole concept of "null hypotheses" is not a logical axiom, it simply derives from real-world observation: in the real world, for most of the H we are interested in, estimating P(E / ~H) is easy, and estimating P(E / H) is either hard or impossible.
P(H) is silently set to .5. If you know P(E / ~H), this makes P(E / H) unnecessary to compute the real quantity of interest, P(H / E) / P(~H / E). I think.
There needs to be a post specifically devoted to arguments of the form "It's okay to do things wrong, because doing them right would be hard". I've seen this so many times, in so many places, in so many subjects, that I have to conclude that people just don't see what is wrong with it.
(No, I'm not talking about making simplifying assumptions or idealizations in models. More like presenting a collection of sometimes-useful ad-hoc tricks as a competing theory, which is then argued for as a theory against its competitors on the basis of its being "easier to apply".)
Bayes' Theorem says that P(H|E) = P(H)P(E|H)/P(E). That's, like, the law. You don't get to take P(E|H) out of the equation, or pretend it isn't there, just because it's difficult to estimate. As I've said elsewhere, if you have a belief, then you've done a Bayesian update -- which means you have some assumption about each of those quantities appearing in the formula, whether you choose to confront these assumptions or not.
As a matter of fact, if you find P(E|H) overly difficult to estimate, that means your H isn't paying its rent.