Bill Taylor says...
>> Bayesian would say it makes sense to talk about the "probability" that
>> there is life on Mars, but the frequentist would see that as nonsense.
>
>This is an excellent example.
It's a good example of why frequentism is useless in the real world.
Suppose you are thinking of launching a mission to Mars to bring back
soil samples for the purpose of looking for evidence of life. Now,
people don't just launch billion-dollar missions to check out random
facts. Maybe there is an alien restaurant hidden on the dark side of
the moon. Maybe one of the undiscovered moons of Jupiter is a perfect
cube. Maybe Elvis is alive and living in a secret underwater
headquarters off the cost of Venezuela.
We have to make some kind of decision about what to investigate
based on the cost of the investigation, the likelihood of finding
something interesting, and the value of that thing if we do find
it. But what is the notion of "likelihood" that we use? I claim
it is Bayesian, not frequentist. It's a combination of subjective
notions of likelihood, continually adjusted by ac***ulated evidence.
>Someone noted that this question could be embedded into a wider
>question like "how many Mars-like planets have life on them?"
>But this is very much NOT the same question;
>when we ask the Mars question, we are very definitely thinking
>of "our" Mars, not any other, and ultimately want a reply
>concerning that. Objective statistics has little to say here.
And for similar reasons, objective statistics have little
to say about *anything* in the real world. The statistics
by themselves don't tell us what's a sensible thing to do,
they don't tell us what is likely to happen. The statistics
are only useful if augmented with some guesses or assumptions
that go beyond what the statistics say.
>OC Bayesians admit this, and defend themselves with the comment that
>there is *always* going to be subjectivity in statistics (or anything
>else), and that at least they put it all "up front", in the subjective
>prior, where everyone can see and criticise it. (Though what such,
>criticisms could possibly carry in terms of intellectual weight
>they never specify). IMHO this justification is nonsense, like
>a child painter saying "look at the lovely mess I've made, it's ART".
The frequentist is making the same sorts of messes. The mess
is not *caused* by Bayesianism, it is inherent in trying to
apply mathematics to the real world.
The epistemological situation we are in in the real world is
that not only do we do not know *what* the probabilities are,
we don't even know *whether* things are probabilistic (in the
frequentist sense) at all. It's possible that things are completely
deterministic, and the appearance of randomness is due to complexity,
lack of knowledge of the forces at work, and lack of knowledge of
initial conditions. (Of course, quantum mechanics makes determinism
very implausible). So we don't know enough to know whether
frequentism has *any* application in the real world.
The frequentist arguments against Bayesianism are actually
arguments against using probability theory, period.
>> What paradigm is more popular among economists and other applied
>> scientists?
>
>Orthodox; though this is largely because Bayesians have been very slow
>in developing stats packages that *automatically* require the
>subjective input of the users. There have been some efforts
>in this direction; but it also means users are eventually going
>to wonder "WHY am I being asked all these silly things?"
>It may turn out counter-productive, in the very long run!
The reason they are silly is because in many situations,
the prior probabilities have negligible impact on the posterior
probabilities.
>BTW - I *do* think that Bayesian ideas have a good place!
>Namely, in subjective cir***stances.
I want to know when frequentist ideas ever have a good place.
It seems to me that it is only for simplicity; the frequentist
keeps track of fewer numbers. So computations using frequentism
might be simpler. And, as you say, the frequentist doesn't need
to fill out annoying questionnaires. So from my point of view,
frequentism seems like an ad hoc methodology that attempts to
get good results with less hassle and computation. It's a
heuristic, rather than anything principled. Which is fine,
as long as you realize that it's a heuristic.
>But if there is to be any suggestion of public concerns,
>especially e.g. of medical/pharmaceutical concerns, and other
>dangerous contexts, then a subjective approach is totally
>irresponsible and hateful.
I don't see how frequentism is any less dangerous or irresponsible
in such cases. Yes, if you give somebody a powerful drug, the
drug itself might kill the patient. If you *don't* give the drug,
then the disease might kill him. Whether you are a frequentist
or a Bayesian does not change this dilemma. How is frequentism
a less dangerous approach?
Here's an example
Suppose that there are two competing theories about
some medical condition, and these theories give
different conclusions about what the proper treatment
should be. How do you decide which treatment to give?
The frequentist approach doesn't have an answer. It
can only give odds if we *know* the probabilities,
and only know the probabilities if we know which
theory is correct. You can decide to wait until
one theory or the other is proven correct beyond
reasonable doubt (which I claim is a subjective
criterion, by the way), but your patient may be
dead by then. Alternatively, you can just pick
whichever theory has the most impressive sup****ting
evidence, and go with that one. But what if there
is only a slight advantage for one side over another?
Let's add some more details: Suppose that theory A
implies that the patient's odds will be made slightly
worse by doing procedure P. Suppose that theory B
implies that the patient will almost surely recover if you
do procedure P (and will almost surely die if you
don't). Suppose that theory A has slightly better evidence
in its favor than theory B. Which theory do you go with?
Do you perform the procedure, or not?
The frequentist has no answer in this case. He can just
say: If we do procedure P, you'll either live or die, and
we don't know which.
The Bayesian has an answer in this case: If there is only
slightly better reasons for preferring A over B, then do
the procedure. He computes a certain posterior
probability that A is true, and a certain posterior
probability that B is true, and then uses these probabilities
to compute the probability that the patient will recover
given procedure P, and the probability that he will recover
if he isn't given procedure P. The odds are better with
procedure P.
Frequentism cannot do this sort of weighing of probabilities,
because "theories being true" is *not* a probabilistic event
in the sense of frequentism.
>It is true, we can never eliminate the subjective element from
>public concerns completely, nor from science or math.
>But we ought to be seeking ways to DETECT and
>MINIMIZE the subjective element. NOT building it into the system!!!
It seems to me that Bayesianism does a *better* job at this than
frequentism.
The way I would put it is this: if we are trying to reason in
the presence of uncertainty, then we should try to eliminate or
reduce the uncertainty. Subjectivity is *inevitable* when there
is uncertainty. We should try to eliminate the *need* for subjectivity.
Eliminating subjectivity without reducing uncertainty doesn't
make sense to me.
--
Daryl McCullough
Ithaca, NY
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