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Scott
Posts:
66
Registered:
2/2/07


Laplace's rule of succession
Posted:
Feb 3, 2007 5:45 PM


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 longrun 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.



