[FOM] 2 problems in the foundations of statistics

[Robert Black]

>  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

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