STAT3011 Graphical Data Analysis Insights Assignment 2026 | ANU

STAT3011 Graphical Data Analysis Insights Assignment 

Introduction to Bayesian Data Analysis

Tutorial 9

(1) Problem 10.4 (Hoff) Gibbs sampling: Consider the general Gibbs sampler for a vector of parameters φ. Suppose φ^(s) is sampled from the target distribution p(φ) and then φ^(s+1) is generated using the Gibbs sampler by iteratively updating each component of the parameter vector. Show that the marginal probability Pr(φ^(s+1) ∈ A) equals the target distribution ∫_A p(φ)dφ.

(2) Problem 10.5 (Hoff) Logistic regression variable selection: Consider a logistic regression model for predicting diabetes as a function of x₁=number of pregnancies, x₂=blood pressure, x₃=body mass index, x₄=diabetes pedigree and x₅=age. Using the data in azdiabetes.dat, centre and scale each of the x-variables by subtracting the sample average and dividing by the sample standard deviation for each variable. Consider a logistic regression model of the form Pr(Yᵢ = 1|xᵢ, γ, β) = exp(θᵢ)/(1 + exp(θᵢ)) where

θᵢ = β₀ + β₁γ₁xᵢ,₁ + β₂γ₂xᵢ,₂ + β₃γ₃xᵢ,₃ + β₄γ₄xᵢ,₄ + β₅γ₅xᵢ,₅

In this model, each γⱼ is either 0 or 1, indicating whether or not variable j is a predictor of diabetes. For example, if it were the case that γ = (1,1,0,0,0), then θᵢ = β₀ + β₁xᵢ,₁ + β₂xᵢ,₂. Obtain the posterior distributions for β and γ, using independent prior distributions for the parameters, such that γⱼ ~ Bern(1/2), β₀ ~ normal(0,16) and βⱼ ~ normal(0,4) for each j > 0.

(a) Implement a Metropolis-Hastings algorithm for approximating the posterior distribution of β and γ. Examine the sequences βⱼ^(s) and βⱼ^(s) × γⱼ^(s) for each j and discuss the mixing of the chain.

(b) Obtain Pr(γⱼ = 1|x,y) for each j. How good do you think the MCMC estimates of these posterior probabilities are?

(c) For each j, plot posterior densities and obtain posterior means for βⱼγⱼ.