Re: A simple but confusing question

From: Ian Jermyn (ianjermyn_at_wanadoo.fr)
Date: 10/03/04 

Date: Sun, 3 Oct 2004 18:32:37 +0200

This is only approximately true, for N >> n, where N is the number of balls
in the bucket and n the number of draws. Let w_{n} be the proposition 'the
nth ball is white'. Let W_{n} be the proposition w_{n} & w_{n - 1} & ... &
w_{2} &w_{1}. Assume we know how many balls (N) are in the bucket. Let the
number of white balls in the bucket be M. The probability of the next ball
being white after n white balls is given by

Pr(w_{n + 1} | W_{n}, N) = Pr(W_{n + 1} | N) / Pr(W_{n} | N) .

Then

Pr(W_{n} | N) = \sum_{M = 0}^{N} Pr(W_{n} | M, N) Pr(M | N)

= 2^{-N} N^{-n} \sum_{M = 0}^{N} M^{n} (N choose M) ,

because Pr(M | N) is binomial with p = 1/2 and Pr(W_{n} | M, N) is simply
(M/N)^{n}. Thus

Pr(w_{n + 1} | W_{n}, N) = N^{-1}[ \sum_{M = 0}^{N} M^{n + 1} (N choose M) /
\sum_{M = 0}^{N} M^{n} (N choose M) ] .

For n >> N, the sums will be dominated by the terms with M = N,
and we find that

Pr(w_{n + 1} | W_{n}, N) = N^{-1} N = 1 ,

whereas for N >> n the term with M = N/2 will dominate, and we find

Pr(w_{n + 1} | W_{n}, N) = N^{-1} (N/2) = 1/2 .

This is only natural. When we have much more data than balls in the bucket,
the data dominates our estimate of M. When the reverse is true, the prior
dominates. When both N and n are small, no one term will really dominate and
the sum must actually be evaluated.

I am still thinking about the case in which we do not know the number of
balls in the bucket.

Ian. 

-- 
--------------------------------------------------
Ian Jermyn
ianjermyn@wanadoo.fr
"Henry" <se16@btinternet.com> a écrit dans le message de
news:mnjrl09pi6r05hjdqnchr4j8cvtpqnfgi7@4ax.com...
> On Sat, 02 Oct 2004 20:30:38 +0300, George Kahrimanis
> <anakreon@hol.gr> wrote:
> > I admit that I feel somewhat like Alice in Wonderland
>
> Then use Lewis Carroll's prior in one of his "Pillow Problems", which
> assumes that the balls were originally put in the bucket at random
> with each a probability 1/2 white and 1/2 black.  After 100 whites
> drawn from the bucket, the probability of the next drawn being white
> is still 1/2.
>

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