Problem 1

Consider the data set "Rat_data.csv" (Read the data using the function read.csv()) in R.

Each data point represents the weight of a rat at a certain age. More specifically, weights of 30 rats (in grams) were measured at five time points of x₁ = 8, x₂ = 15, x₃ = 22, x₄ = 29, x₅ = 36(weeks).

For example, weight of rat 3 in week 15 was 214 grams. Let y_ij denote the weight of rat i at age x_j, and consider the following model for the y_ij

??_ij~??(α_i + β_i(x_j - x¯), σ²) for i = 1, . . . , 30 and j = 1, . . . , 5 and x¯ = 22

(a) Obtain the maximum likelihood estimate of α_i , β_i and σ². Show your work, and compute the final estimates

(b) Now consider a Bayesian analysis of this model, where we assume the following priors

α_i~??(α, 100) with hyperprior α~??(0,10000)

β_i~??(β, 1) with hyperprior β~??(0,10000)

Τ = 1/σ² ~ gamma(10^-3, 10^-3)

where α_i , β_i and Τ are assumed to be independent. Our aim here is to use the Gibbs sampling method to generate values from the posterior distribution of(α₁, . . . , α₃₀, β₁, . . . , β₃₀, Τ).

(i) State all the required conditionals. Show your derivation very briefly.

(ii) Write a complete algorithm for this Gibbs sampling algorithm, specific to this problem and code your algorithm, using R.

(iii) Run the chain for 50,000 iterations, and burn-in the first 10,000 generated values. Plot the chain (trace) after the burn-in for α₁, β₁, α₁₅, β₁₅, Τ against the iteration number, and comment on the mixing.

(iv) Draw the histograms for the posterior distribution of α₁, β₁, α₁₅, β₁₅, Τ and provide a point estimate for each parameter

(c) Obtain a 95% credible interval for each of the parameters α₁, β₁. Interpret your intervals in the context of the problem.

(d) In the prior, we assumed that α_i , β_i and Τ are independent. Does the posterior support this assumption? Justify your answer.

Problem 2

Consider simulation of the bivariate normal distribution with μ = (1,2)^?? and covariance matrix

Σ = (1    0.9)

      (0.9   1)

using Metropolis-Hastings algorithm with random walk generating density. Specifically, let x* = x^(t) + ∈ where ∈~??²(0, D) with D being a diagonal matrix with diagonal elements δ₁ and δ₂. Note that δ₁ and δ₂ control the spread of values generated along the first and second coordinate axis, respectively.

(a) Give the M-H algorithm algorithm, and code it in R.

(b) Choose δ₁ = 0.001 and δ₂ = 0.001, and generate a chain, starting at x(0) = (0, 0)t , of size 10,000 using the M-H algorithm.

(i) Compute the acceptance rate, and comment on the acceptance rate.

(ii) Burn 1000, and draw a trace plot and the density plot for the chain for each of the variables. Comment on mixing and the density plots.

(iii) Draw 9000 pairs generated from multivariate normal with μ = (1,2)^?? and covariance matrix

Σ = (1    0.9)

      (0.9   1)

Plot these pairs on a scatterplot overlaying it with the last 9000 points obtained from the M-H chain; use different color for those generated from MH and those directly generated from the target (choosing an alpha-level for color transparency is helpful). Do the M-H generated values match the target? Explain why or why not.

(c) Choose appropriate δ₁ and δ₂ so that the M-H algorithm would generate values that meet the target requirement. Draw a similar plots as in part b(ii) and b(iii) for this case.

(d) How do you expect the acceptance rate would change, if you choose δ₁ and δ₂ to be very large? Briefly explain why?

Attachment:- rats_data.csv