[FOM] 2 problems in the foundations of statistics
Robert Black
Mongre at gmx.de
Sat Jul 30 12:39:43 EDT 2005
> we have seen something happen 1 of 2
>times, and ask for the chance it will happen once in the next 2 tries,
>and the formula gives an answer of 2/5.
>But is 2/5 really the right answer? Is the uniform prior a valid
>assumption? Could there be a better prior distribution, or a better
>solution that does not correspond to assuming a prior distribution and
>calculating a la Bayes? Other solutions have been proposed, and there is
>no consensus.
>Suppose you knew nothing about two basketball teams except that the
>first two games they played were split. Is the chance the next two will
>be split really two fifths? It would be nice to have an explanation for
>this number that would satisfy the proverbial "man in the street".
>
You've got to distinguish between credence, i.e. (rational) degree of
belief and chance, i.e. objective tendency to occur. Obviously
they're not the same: consider a coin which we know to be biased, but
we don't know which way. Then it might be reasonable to assign
credence 1/2 to heads on the first toss while knowing that the chance
of heads on the first toss is certainly not 1/2. It would be mad to
conclude that the *chance* of games 3 and 4 being split has to be 2/5
just because games 1 and 2 were split, but that doesn't stop it being
a reasonable credence.
Bayesianism is about credence. A Bayesian can either (like de
Finetti) think that credence is the only sort of probability, or he
can also believe in chance. For simple cases like the repeated
tossing of a 'coin of unknown bias' these turn out to be
mathematically equivalent (de Finetti's theorem shows that starting
with prior credence spread somehow over all the possible 'real
biases' of the coin, using evidence from tosses to update these
credences and moving from credences about chances to credences about
outcomes via Lewis's 'principal principle' gives the same results for
credences about the actual tosses as starting with exchangeable
priors about the tosses themselves and conditionalizing directly).
Note that the tosses are independent from each other as regards
chance (coins don't have memories), but *not* as regards credence (a
head raises our credence that the next toss will give heads). No sane
Bayesian thinks the priors should always be uniform: I don't know
about basketball, but I suspect it's impossible to construct a coin
with a bias greater than about 3/4, so our priors about a coin should
never be flat. What priors we regard as sensible will vary from case
to case, and there's no reason to think this can be captured in a
mathematical formula.
The general bayesian answer to the basketball case goes as follows
(and since we're talking about a probability space with only 16
atomic outcomes, I'll do it de Finetti's way which keeps everything
finite).
There are five possible *frequencies* of wins for the 'first' team,
ranging from all four won to all four lost. To these five
possibilities assign *any* credences you like so long as the five
numbers are non-zero and add up to 1.
Within each frequency, assign equal credence to all arrangements
generating that frequency. So, for example, if you assign c to two
out of four games being wins, then assign c/6 to any one of the 6
possible arrangements giving that result.
That guarantees exchangeability and by de Finetti's theorem means
that your priors correspond to some distribution or other over all
the 'objective chances' (from 0 to 1).
Now just conditionalize on the results as they come in.
2/5 arises from assigning equal prior credence to the five
frequencies (corresponding to a flat distribution from 0 to 1 of the
chances of the first team winning).1/2 arises from assigning the
credences 1/16, 1/4, 3/8, 1/4, 1/16 to the five frequencies
(corresponding to certainly that the two teams are objectively
matched, and thus giving every possible sequence of outcome equal
credence). My claim is that there just is no answer to the question
'what priors should one have?', but of course given priors there is
always an answer to the question of what they lead to after
conditionalization on even split on the first two games.
Also significant of course is that with longer sequences of trials so
long as your priors are sufficiently spread out so as not to lead to
high confidences about long term frequencies (e.g. don't correspond
to certainty that the teams are equally matched, which leads to near
certainty of long-term frequencies near 1/2), your credence for the
next game being a win after n out of m wins will approach n/m as m
becomes large (so-called 'swamping of the priors', which explains why
we don't need to have agreed much on our priors in order to largely
agree on our credences after conditionalizing on the same evidence).
An appealing thing about the 'flat' distribution is that it's
particularly easily swamped, and that perhaps gives it a special
status.
Robert
--
PS I am at present in Berlin, which is why this message is coming
from a gmx address. But you can reply to my usual
<Robert.Black at nottingham.ac.uk>.
Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD
tel. 0115-951 5845
home tel. 0115-947 5468
[in Berlin: 0(049)30-44 05 69 96]
mobile 0(044)7974 675620
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