Re: A simple but confusing question
From: Alex Strashny (alex_strashny_at_yahoo.com)
Date: 10/06/04
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Date: 6 Oct 2004 00:05:08 -0700
szhao@darkwing.uoregon.edu (Shanyu Zhao) wrote in message news:<96a39245.0409270026.63fd76c@posting.google.com>...
> Here is the naive question:
>
> There are a large number of balls in a bucket, the white color balls
> occupy p and the black balls (1-p). If p is unknown, when you pick 100
> balls from the bucket, find that all of them are white. Then you pick
> the 101st ball, what is the probability that the ball still a white
> one?
The answer can be anything from 0 to 1. (See below.) The answer that I
would consider reasonable is 101/102.
> Is this problem a parameter estimation or hypothesis testing? If we
> use parameter estimation, clearly p=1, which means the probability is
> 100%. But this is not true.
This is not right.
There are a couple of different definitions of probability. The
frequentist definition states: if you pick n balls, and x of them are
white, then probability of a white ball is p = (x/n) **as n goes to
infinity**. People sometimes forget this last part, and it often does
not matter; it matters here. 100 is not equal to infinity, so you
cannot apply the frequentist definition to calculate probability.
This is an estimation problem. p, the probability, is a parameter that
you want to estimate. A model consists of two parts: a likelihood,
which expresses the interaction between the parameters and the data;
and a prior, which expresses your prior beliefs about the parameters.
For this particular problem, the likelihood is Binomial(x | n,p). If
the prior is Beta(a,b), then the best estimator of p (under quadratic
loss) is p_hat = (x+a)/(n+a+b).
What should a and b be? In principle, they could be anything. They
express your beliefs about p before you drew any of the balls. That is
why I said that your estimate of p could be anything between 0 and 1.
If a = 1, b -> Infinity, then p_hat -> 0. But is such a prior
reasonable? Probably not.
If you have no prior information about p, then I would say, set a = b
= 1. Beta(1,1) is the Uniform(0,1). This prior says that, a priori, p
is equally likely to be anything between 0 and 1. Under this prior,
p_hat = 101/102.
Some people would say that if you have no prior information, set a = b
= 0.5. (This is the Jeffrey's prior.) I would disagree, but in
practice it doesn't matter. Under Jeffrey's prior, p_hat = 100.5 /
101.
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