: There are four candidates in an election. The election comprises of two voting stages, morning and afternoon, and there are 100 votes for each stage. Denote yij as the vote number of Candidate j at stage i, i = 1, 2 and j = 1, 2, 3, 4. However, an accident occurs during the vote counting such that we only know that the vote number of some candidates are larger or equal to 10 as the following table: Candidate 1 Candidate 2 Candidate 3 Candidate 4 Stage I 41 ? 10 ? 10 13 Stage II 38 32 ? 10 ? 10 If we assume that Y i = (yi1, yi2, yi3, yi4), i = 1, 2 satisfies the same multinomial distribution 1 with 100 trials and event probabilities p = (p1, p2, p3, p4). Based on the available information, please carry out a hybrid Gibbs sampler to estimate p. Note: assign the Dirichlet prior ?(p) ? p1p2p3p4 to p. Regard y13 and y23 as latent variables and update y13 by the proposal: 1. if 10

STAT 3006 Assignment 3 Due date: 5:00 pm on 26 April (30%)Q1: There are 100 samples {X1 , X2 , . . . , X100 }. For each sample i, Xi = (Xi1 , Xi2 ). You know that these samples are from three clusters. For sample i, we use Zi to denote the cluster number to which sample i belongs. The proportion of the three clusters is denoted by 1 , 2 , 3 . Specifically, for each sample i, P (Zi = k) = k , 1 k 3. Given Zi = k, Xi1 N (1k , 1) and Xi2 N (2k , 1). Use Q1 dataset to estimate parameters (1k , 2k ), k = 1, 2, 3, (1 , 2 , 3 ) and Zi (1 i 100). Note: assign Dirichlet(2, 2, 2) prior to (1 , 2 , 3 ), and assign uniform prior p(jk ) 1 to jk , j = 1, 2; k = 1, 2, 3; implement Gibbs sampler 10,000 iterations, and only samples in the last 8,000 iterations are kept; use posterior mean to estimate , and use posterior mode to estimate Z . (35%)Q2: There are T = 1, 500 independent normal distributed random variables {X1 , X2 , . . . , XT }. Xt (1 t T ) denotes the value we observed at time t. We know that there are two mean-shift change points in this data stream. More specifically, there are two different change points at time k and l (1