[SOLVED] Biostat276 Project 1

Please describe all your work in clear terms, before implementing R code. Each question should include a description of your approach with clear indication of where I can find the associated source code. Your code should be attached to your assignment or uploaded online to a repository I can freely access.

Consider the implementation of a posterior simulation algorithm for Bayesian adaptive LASSO. More precisely consider the following model:

Y | β,σ² ∼ N(Xβ,σ²I_n) Let β = (β^[1],…,β_p)⁰, the model is completed with priors:

β_j | τ_j² ∼ N(0,τ_j²); j = 1,…,p

h = 1/σ² ∼ Gamma(0.1,rate = 10)

• Consider p = 1. Simulate 5,000 Monte Carlo samples from the conditional prior β | τ² = 1 and obtain a plot of the density using the R function density.

1. Consider p = 1. Simulate 5,000 Monte Carlo samples from the marginal prior β, considering λ² = 2, so that E(τ² | λ) = 1. Obtain a plot of the density as in
2. Consider p = 1. Add a hyper prior on γ = 1/λ ∼ Gamma(a,rate = b). Assess how the marginal prior of β changes for a = 1 and values of b ≥ 1.
3. Considering the hyper prior in c., describe a Markov Chain Monte Carlo algorithm to sample from the posterior distribution of β and σ².
4. Implement such algorithm in R and compare your results with estimates obtained using glmnet(). In particular, you should test your results on the diabetes data available from lars, (use the matrix of predictors x).
5. For the diabetes data, fix λ and produce a regularization path for adaptive Bayesian Lasso obtained on a grid of values for the tuning parameter λ. Describe your approach and compare your result with the path obtained using glmnet().
6. Free λ and carry out a sensitivity analysis assessing the behavior of the posterior distribution of β and σ², as hyper parameters a and b are changed. Explain clearly the rationale you use to assess sensitivity and provide recommendations for the analysis of the diabetes data.
7. [Extra credit] Using rcpp attempt and quantify acceleration of your code in (e.).