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1. If $X_1,X_2,\ldots,X_n$ are discrete r.v.'s with joint pmf $f(x_1,\ldots,x_n|\theta)$. Let theta be a discrete random variable with prior pmf $\pi(\theta)$. Let $H(x_1,x_2,\ldots,x_n)$ be a sufficient statistic. Show that $\pi(\theta|x_1=x_1,\ldots,x_n=x_n) = \pi(\theta|H=h)$.

Attempt: Basically the posterior distribution of theta given the sample is equivalent to the posterior of theta given a sufficient statistic of the sample as a sufficient statistic contains all the relevant information about the sample.

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