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a) [2m] Let (X1,..., X) Multin(n......6m) with 0; > 0 and 21-10; = 1. Show that the Dirichlet distribution is the conjugate prior to the Multinomial sampling distribution. b) [Am] If we do not want to assume that the sampling distribution of our data or our prior distribution of the unknown parameter is a member of some parametric family of distribu- tions such as Normal, Gamma etc, the Dirichlet process can be used to define a probability distribution on the space of all probability distributions on R. It is defined as follows: Definition: Let Go be a edf on D CR and a > 0. The Dirichlet Process DP(a, Go) with base edf Go and concentration parameter a generates random probability distribution P on R such that for any finite partition A].....At of D, its joint probability has a Dirichlet distribution (P(A1),...,P(Ax)) Dirichlet (aPG.(A1),...,OPG. (4x)) We can use this definition to simulate realizations (sample paths) from a Dirichlet process in the following way: Let 31 0 and 21-10; = 1. Show that the Dirichlet distribution is the conjugate prior to the Multinomial sampling distribution. b) [Am] If we do not want to assume that the sampling distribution of our data or our prior distribution of the unknown parameter is a member of some parametric family of distribu- tions such as Normal, Gamma etc, the Dirichlet process can be used to define a probability distribution on the space of all probability distributions on R. It is defined as follows: Definition: Let Go be a edf on D CR and a > 0. The Dirichlet Process DP(a, Go) with base edf Go and concentration parameter a generates random probability distribution P on R such that for any finite partition A].....At of D, its joint probability has a Dirichlet distribution (P(A1),...,P(Ax)) Dirichlet (aPG.(A1),...,OPG. (4x)) We can use this definition to simulate realizations (sample paths) from a Dirichlet process in the following way: Let 31