[ExI] Bayesian epistemology

[Lee Corbin]

Samantha writes

> If a mathematical method useful in decision making has become an -ism
> then we are surely lost.

Unfortunately, methods do differ between frequentists, Bayesians,
proponents of Popper's propensity view, and the "chance" view.
Well, at least I know that methods do differ between the first two
schools mentioned.  For example, the frequentists (and classical
statisticians) can become embarrassed over applications of the
stopping rule, whereas Bayesians (so I am informed by Peter Lee's
book "Bayesian Statistics") likewise can become embarrassed 
over "Jeffreys' rule for finding reference priors, [which is] 
incompatible wit the likelihood principle" (p. 210).

Has statistics gone irreversibly down the route of dismality in the
wake of economics?   :-)

Now I was taught the use of confidence intervals when I first
took statistics, and it came as a shock that Bayesians never use
them. I did convert to Bayesianism, however, but still can't help
but lapse into non-Bayesian concepts often.  Currently I
am taking more and more seriously what seemed for me---but
I am a total amateur---a possible Achilles' Heel:  we know of
one area in which objective probability is very manifest: QM
and Quantum Field Theory.  What's worse, (in my opinion) is
that if you take MWI very seriously, which I do, then you are
stuck with a definite fraction of worlds in which one thing happens
a certain way.

For a concrete example, let's suppose that George Bush is about
to give a scheduled speech on a certain day at a certain time.
What is the probability that he will read the speech exactly as
written by his speechwriter and as it appears on the teleprompter?
More precisely, what is the distribution of when he'll digress, or
stumble, or deviate from that exact sequence of words?

David Deutsch would remind us that we want to start with a 
particular "group of identical universes" at a particular time (say,
just before the speech begins).  Then that particular group of
identical universes (see Fabric of Reality where the phrase is
used over and over again and is meant quite literally) will begin
to fray at once.  (There will be meteor strikes, insects, or 
whatever that can immediately cause branching to occur, as
well as the harder to conceive of but equally real build-up of
neuronal quantum events---all of which cause George to 
deviate.)

Now on this reading, given a particular group of identical universes,
Barbara Bush's knowledge and his speechwriter's knowledge of
Bush do not---unlike in Bayesianism---play any role!  Jaynes brilliantly
(it seemed to me, though again I am no expert and may have mis-
understood his motivation) strongly emphasized the concept of a
*robot* and what the robot *knows*.  I surmise that he did this
to make the reader aware that he was talking about something
quite objective and real, namely, the state of the robot's
knowledge.  I liked that.  Maybe it even helped explain why
Jaynes belongs to the school of "objective Bayesians", which,
I readily confess, had a pleasing sound to me.

I don't know how to reconcile the preceding two paragraphs.

Here is how the whole mess started for me.  At first, it was
simplicity itself.  At age 18 I took my first course in probability,
and actually felt insulted that somebody's name (Bayes') was
attached to what seemed then to be a very obvious formula
of elementary probability. I was unaware of how some problems
are pretty tricky, but I still would not have been impressed, because
it still was nothing more than relative weights of cases.  What
*would* have impressed me would have been if I had known
exactly what problem Bayes was struggling with, and for which
he obtained a solution:

  "Given that the number of times in which an unknown event has
  happened and failed: *Required* [is] the chance that the
  probability of its happening in a single trial lies somewhere
  between any two degres of probability that can be named."

----Reverend Bayes in 1764, Philosophical Transactions of
the Royal Society (as I quote from my book "Against the Gods,
the Remarkable Story of Risk" by Peter Bernstein).

Now that's a hard problem!  Suppose you remain just outside
a betting parlor and your friend enters and begins to play some 
game (but you have no idea what).  All you hear is him cry out
in his unmistakable voice "Damnation!", "Oh yes!", "Damnation!",
"Damnation!", "Oh yes!", e.g., he's had two successes in five 
tries.  What probability of winning are you to assign to that game?
It's not so simple.  One calculation, for example, yields Laplace's
Rule of Succession, in which the correct probability is not one's
first guess 2/5, but rather is (2 + 1)/(2 + 3 + 1 + 1), or 3/7.
(This assumes a uniform prior distribution---in other words,
as you decide you know nothing about what is going on, you
assign as equally probable the true probability lying anywhere
between 0 and 1, as Laplace, I suppose, would advise, and
then "update" that information with your friend's results. I did
the calculus a few years ago and I think I got Laplace's Rule to
come out.

Anyway, as I understand it, this is only the tip of the iceberg.
I think that I read that one definition of a Bayesian is if he or
she swears by MaxEnt, the principle of Maximum Entropy.
Jaynes certainly does, and makes it a cornerstone of his big
2003 book (unfortunately, Jaynes did not live to see the
publication). 

Believe it or not---and I'm sure you won't have any trouble
believing this---this also relates (in my opinion) to the 
philosophical problem of personal identity, about which
we have spoken so much.  As
one takes MWI more and more seriously, and if one agrees
entirely with the patternist (state) notion of identity, then the
frequencies I mentioned above become correspondingly 
more important. As I contend that many people fail to appreciate
MWI *precisely* because they cannot believe that there are
other versions of themselves in adjacent universes, and that
they themselves already *are* all their duplicates in identical
and nearly identical universes, the horrid thought has occurred
to me that failure to agree with the state view of identity could
even be playing a role in the Great Statistical debate!

So it's high time I reviewed the reasons for my 2003 conversion
to Bayesianism.

Lee