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parameter risk




I believe that how few we consider the "parameter risk".

Suppose that the probability of occurring a accident is P (0<P<1).
For N polices, the probability of occurring n accidents is
N_C_n * P^n * (1-P)^(N-n)
(where N_C_n means the combinatorial number which is chosen n things form N things)

Suppose that the p.d.f.(probability density function) of the prior distribution is g(p),
f(p),the p.d.f. of the posterior distribution by Bayes' theorem
will be proportional to
P^n * (1-P)^(N-n) *g(p)
(because N_C_n is a constant which has no relations with p)

Now if the prior distribution is a uniform distribution, that is
g(p)=1(0<p<1)
f(p) is proportional to P^n*(1-P)^(N-n)
so
P follows to B(n+1,N+1)(beta distribution).



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