Description

ISyE 6420
1. Metropolis for Correlation Coefficient. Pairs (Xi,Yi),i = 1,…,n consist of correlated standard normal random variables (mean 0, variance 1) forming a sample from a bivariate normal MVN2(0,Σ) distribution, with covariance matrix
.
The density of (X,Y ) ∼ MVN2(0,Σ) is
,
with ρ as the only parameter. Take prior on ρ by assuming Jeffreys’ prior on Σ as π(Σ) = , since the determinant of Σ is 1 − ρ2. Thus
.
(a) If (Xi,Yi),i = 1,…,n are observed, write down the likelihood for ρ. Write down the expression for the posterior, up to the proportionality constant (that is, un-normalized posterior as the product of likelihood and prior).
(b) Since the posterior for ρ is complicated, develop a Metropolis-Hastings algorithm to sample from the posterior. Assume that n = 100 observed pairs (Xi,Yi) gave the following summaries:
, and .
In forming a Metropolis-Hastings chain take the following proposal distribution for ρ: At step i generate a candidate ρ0 from the uniform U(ρi−1 − 0.1,ρi−1 + 0.1) distribution. Why the proposal distribution cancels in the acceptance ratio expression?
(c) Simulate 51000 samples from the posterior of ρ and discard the first 1000 samples (burn in). Plot two figures: the histogram of ρs and the realizations of the chain for the last 1000 simulations. What is the Bayes estimator of ρ based on the simulated chain?
(d) Replace the proposal distribution from (b) by the uniform U(−1,1) (independence proposal). Comment on the results of MCMC.
2. Gibbs Sampler and Lifetimes with Multiplicative Frailty. Exponentially distributed lifetimes have constant hazard rate equal to the rate parameter λ. When λ is a constant hazard rate, a simple way to model heterogeneity of hazards is to introduce a multiplicative frailty parameter µ, so that lifetimes Ti have distribution
Ti ∼ f(ti|λ,µ) = λµexp{−λµti}, ti > 0, λ,µ > 0.
The prior on (λ,µ) is
π(λ,µ) ∝ λc−1µd−1 exp{−αλ − βµ},
that is, λ and µ are apriori independent with distributions Ga(c,α) and Ga(d,β), respectively.
The hyperparameters c,d,α and β are known (elicited) and positive.
Assume that lifetimes t1,t2,…,tn are observed.
(a) Show that full conditionals for λ and µ are gamma,
,
and by symmetry,
.
(b) Using the result from (a) develop Gibbs Sampler algorithm that will sample 51000 pairs (λ,µ) from the posterior and burn-in the first 1000 simulations. Assume that n = 20 and that the sum of observed lifetimes is .
Assume further that the priors are specified by hyperparameters c = 3,d = 1,α = 100, and β = 5. Start the chain with µ = 0.1.
(c) From the produced chain, plot the scatterplot of (λ,µ) as well as histograms of individual components, λ, and µ. Estimate posterior means and variances for λ and µ. Find 95% equitailed credible sets for λ and µ.
(d) A frequentist statistician estimates the product .
What is the Bayes estimator of this product? (Hint: It is not the product of averages, it is the average of products, so you will need to save products in the MCMC loop).
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