Date: Feb 3, 2007 5:45 PM
Author: Scott
Subject: Laplace's rule of succession
In Bayesian statistics, Laplace's rule of succession attempts to solve

the problem of how we can predict that the sun will rise tomorrow,

given its past frequency of rising.

Definitions:

1. Let p be the long-run frequency, as observed.

2. Let n be the total number of trials.

3. Let s be the number of *successes* among these trials, so that n -

s is the number of failures.

The rule of succession states that the probability of the next success

is given by the *expected value of a normalized likelihood function*.

The likelihood function is

p^s * (1 - p)^(n - s).

Normalized with the integral S_{0 to 1}(p^s * (1 - p)^(n - s)) dp, one

obtains as the expected value

(s + 1)/(n + 2)

for the probability of the next success. Thus, if all we know is that

the sun has risen 2000 times, the probability of its rising again is

2001/2002.

Now, I have a question. What's so special about this likelihood

function? It seems to be formulated completely ad hoc. If the sample

space were all possible successions, the probability of the next

success would simply be 1/2. So what gives?

The figure p^s * (1 - p)^(n - s) is the probability that there will be

s successes, with *fixed probability p* for each success, a

probability independent of the trial number. But how can we impose

this property on a sequence? How do we know that there are fixed

probabilities of success and failure on each trial?

Is Laplace's rule even accepted nowadays?

I would like to understand more of the philosophical theory behind the

choice and justification of the likelihood function. Thank you for

your help.