From: Dan Fabulich (dfabulich@warpmail.net)
Date: Mon Jul 07 2003 - 19:41:07 MDT
Mitchell Porter wrote:
> But what I still don't get is what's wrong with 'orthodox statistics'. I
> obtained my p1/(p1+p2) through frequentist thinking, which is supposed
> to be anathema to Bayesians. On the SL4 list, Ben Goertzel implies that
> "non-Bayesian probabilists" are restricted in their choice of priors:
> http://sl4.org/archive/0305/6820.html
>
> ... but even if what he says is true, I don't see it reflecting on the
> subjectivist-vs-frequentist debate. Both sides need priors to get
> anywhere, and there's nothing in frequentism to *compel* the use of
> Gaussian priors. So, I still don't get the philosophical lesson.
"Bayesian v. frequentist" philosophical arguments are plagued with
vagueness; one feels the vagueness more acutely than in many other
philosophical arguments b/c they both seem to brush up right next to
what's analytically verifiable, and because the math can be so niggling
and error-prone.
(I note, for example, that you happened to misread the problem Eliezer
posed; you solved a perfectly fine problem instead, but not exactly the
problem as posed. That problem has some handy features that which allow
you to simplify and solve the problem just a bit faster. [Though I sure
as hell wouldn't notice them unless I was working on paper.])
[There's something else driving this "argument" as well, IMO. There's a
deep psychological problem with the study of statistics: it's so easy to
make trivial errors with huge ramifications that, once you get kind of
good at it, you realize just how absurdly often people make radical
mistakes in their statistical intuitions; people who do this a lot usually
leave the path of interpretative charity, assuming that their opponents
are probably just making a mistake like everyone else, not that they are
well-informed thinking people with reasonable and possibly serious
objections.]
But, let's get back to your question. There is no hard-and-fast
definition of "frequentist" in this debate, unfortunately. I wish I could
say that Goertzel was "wrong" in his characterization of frequentists, but
I'm afraid I just can't guarantee that. One thing I can say, though, is
that he almost certainly wasn't really talking about the definition of
frequentist/orthodox that Eliezer was using in his bonus question 3:
> 3. Using orthodox methods and an expensive statistics program,
> calculate the probability that, if the hypothesis "I am looking at the
> red barrel is correct", you will see 7 or more red tokens in ten
> samples. Explain why this number is actively harmful to genuine
> statistical understanding in twenty-five words or less.
IMO, the relevant "frequentist" position at which Eliezer was poking fun
here was the position that there are some common-sense events that have no
probability whatsoever; these "singular" events don't have a probability
of 0 or 1, they just don't have "probabilities" at all... according to
this type of frequentist, the whole notion is simply not applicable
sometimes.
A good example might be to ask what the odds might be that a certain
couple will get married. In this case, the frequentist feels obligated to
respond that "probability" is a misnomer here: probability only applies
when there's a large set of things that can be literally counted. If
there were a million pairs of nearly identical couples, THEN the
frequentist would let it count as a "probability", but not otherwise.
Hence, some kind of frequentist might say of Eliezer's question 1, as
posed:
> 1. After repeatedly revising your probability to take each of these
> observations into account, what is your estimated chance that the barrel
> is the one containing mostly blue tokens?
that the question, as stated, doesn't make any sense: there is no such
thing as an estimated probability or prior chance in this case. Hence,
nothing to calculate, and nowhere to begin. (It's hard to say what
exactly a "real" self-described frequentist might say in response to
Eliezer's example, because it's easy to see how the experiment is, at
least, repeatable in principle, as opposed to, say, the married couple
example, which really just can't be repeated with any kind of scientific
rigor as far as we know: lovers attitudes towards each other are
irretrievably dependent on history, so we wouldn't call any repeated
attempts at getting them hitched "repetitions" of the "same" kind of
event.)
But that frequentist WOULD be comfortable answering question 3.
Furthermore, if this frequentist were really hard-core, she might reject
Bayes' Theorem altogether and try to find the frequency by actually
repeating the experiment a bunch of times. That, the frequentist might
insist, would be the only way to guarantee that a meaningful result was
being calculated.
So a frequentist like the kind I'M describing, as OPPOSED to what you did,
would literally write a program (or use an expensive pre-existing one,
depending on how much money was on hand) and do the experiment over and
over again, until an answer to question 3 was found to within some
experimental precision. (Note that talking about "precision" here tends
to require some background assumptions like those Goertzel mentioned in
his answer to your question on SL4. What could that "precision" mean
unless you assumed something Gaussian were going on in the background?)
Looking back at the problem, I have no idea which barrel is red, (since
neither barrel was painted,) but, on the presumption that the barrel with
mostly blue tokens was painted blue, and the other barrel (which you
failed to notice had an EQUAL number of blue and red tokens) must have
been painted red. Neither do I find myself able to say much about how the
frequentist, even my hard-core frequentist, has actively harmed her
statistical understanding... even frequentists can plug numbers into
Bayes as well as anybody else, but sometimes they choose not to on
principle.
But if I had to give an answer, it would be this: "orthodox statisticians
cannot explain WHY the given answer is correct; they couldn't have
predicted the answer, and couldn't therefore have been "surprised" by the
results." [Hence, they have left the path of science.]
I appear to have narrowly met the word limit. Well, hopefully my wordy
answer has served you better than the terse but certainly charming
response Elizer himself probably would have given.
-Dan
-unless you love someone-
-nothing else makes any sense-
e.e. cummings
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