```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 solvethe 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 successis given by the *expected value of a normalized likelihood function*.The likelihood function isp^s * (1 - p)^(n - s).Normalized with the integral S_{0 to 1}(p^s * (1 - p)^(n - s)) dp, oneobtains as the expected value(s + 1)/(n + 2)for the probability of the next success. Thus, if all we know is thatthe sun has risen 2000 times, the probability of its rising again is2001/2002.Now, I have a question. What's so special about this likelihoodfunction? It seems to be formulated completely ad hoc. If the samplespace were all possible successions, the probability of the nextsuccess would simply be 1/2. So what gives?The figure p^s * (1 - p)^(n - s) is the probability that there will bes successes, with *fixed probability p* for each success, aprobability independent of the trial number. But how can we imposethis property on a sequence? How do we know that there are fixedprobabilities 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 thechoice and justification of the likelihood function. Thank you foryour help.
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