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Scott
Posts:
66
Registered:
2/2/07
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Laplace's rule of succession
Posted:
Feb 3, 2007 5:45 PM
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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.
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