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Problem 1: Suppose X1,., X are multinomial counts, that is the vector (X1,..., X) follows a multinomial distribution Mult(n; p1,..., Pk) with joint mass function k k n! f(r1,., Tk) IIP, EI; = n. EPj = 1, (0.1) j=1 j=1 (a) Consider the Dirichlet prior for (p1,.., Pk): I(B1 +.. + Bk) B;-1 g(P1, .... Pk) II Bj > 0, Pj = 1, r(B1) x ... x I(BL) j=1 j=1 where I(a) = So wa le"dw is the gamma function. Under this prior, derive the posterior distribution of (P1,.., Pk). (b) Explain why the Dirichlet prior is a conjugate prior in this situation. (c) Note that the mean vector density in (1) is where a' denotes the transpose of a vector a. A common Bayes estimator of a parameter is its posterior mean. Give the Bayes estimator of an arbitrary p; in this situation (j E {1,..., k}). (d) If k = 2, write the form of the Dirichlet prior. What is the familiar name of the Dirichlet distribution with k = 2 categories?

Problem 1: Suppose X1,., X are multinomial counts, that is the vector (X1,..., X) follows a multinomial distribution Mult(n; p1,..., Pk) with joint mass function k k n! f(r1,., Tk) IIP, EI; = n. EPj = 1, (0.1) j=1 j=1 (a) Consider the Dirichlet prior for (p1,.., Pk): I(B1 +.. + Bk) B;-1 g(P1, .... Pk) II Bj > 0, Pj = 1, r(B1) x ... x I(BL) j=1 j=1 where I(a) = So wa le"dw is the gamma function. Under this prior, derive the posterior distribution of (P1,.., Pk). (b) Explain why the Dirichlet prior is a conjugate prior in this situation. (c) Note that the mean vector density in (1) is where a' denotes the transpose of a vector a. A common Bayes estimator of a parameter is its posterior mean. Give the Bayes estimator of an arbitrary p; in this situation (j E {1,..., k}). (d) If k = 2, write the form of the Dirichlet prior. What is the familiar name of the Dirichlet distribution with k = 2 categories?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Problem 1: Suppose X1,., X are multinomial counts, that is the vector (X1,..., X) follows a multinomial distribution Mult(n; p1,..., P) with joint mass function k k n! f(r1,., Tk) II |P, Pi = 1, r; = n. (0.1) j=1 j=1 (a) Consider the Dirichlet prior for (p1, ..., Pk): T(B1 + ... + Br) g(P1, .., Pk) = II B; > 0, EPj = 1, I(B1) x ... x I(B) j=1 j=1 where I(a) = wa-le-wdw is the gamma function. %3D Under this prior, derive the posterior distribution of (p1,., Pk). (b) Explain why the Dirichlet prior is a conjugate prior in this situation. (c) Note that the mean vector density in (1) is ( ) where a' denotes the transpose of a vector a. A common Bayes estimator of a parameter is its posterior mean. Give the Bayes estimator of an arbitrary p; in this situation (j e {1, ..., k}). (d) If k = 2, write the form of the Dirichlet prior. What is the familiar name of the Dirichlet distribution with k = 2 categories? |3D
Transcribed Image Text:Problem 1: Suppose X1,., X are multinomial counts, that is the vector (X1,..., X) follows a multinomial distribution Mult(n; p1,..., P) with joint mass function k k n! f(r1,., Tk) II |P, Pi = 1, r; = n. (0.1) j=1 j=1 (a) Consider the Dirichlet prior for (p1, ..., Pk): T(B1 + ... + Br) g(P1, .., Pk) = II B; > 0, EPj = 1, I(B1) x ... x I(B) j=1 j=1 where I(a) = wa-le-wdw is the gamma function. %3D Under this prior, derive the posterior distribution of (p1,., Pk). (b) Explain why the Dirichlet prior is a conjugate prior in this situation. (c) Note that the mean vector density in (1) is ( ) where a' denotes the transpose of a vector a. A common Bayes estimator of a parameter is its posterior mean. Give the Bayes estimator of an arbitrary p; in this situation (j e {1, ..., k}). (d) If k = 2, write the form of the Dirichlet prior. What is the familiar name of the Dirichlet distribution with k = 2 categories? |3D
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