[SOLVED] Isye6420 homework 4

ISyE 6420
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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,) distribution, with covariance matrix
.
The density of (X,Y ) MVN2(0,) is
,
with as the only parameter. Take prior on by assuming Jeffreys prior on as () = , since the determinant of is 1 2. Thus
.
(a) If (Xi,Yi),i = 1,,n are observed, write down the likelihood for . 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 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 : At step i generate a candidate 0 from the uniform U(i1 0.1,i1 + 0.1) distribution. Why the proposal distribution cancels in the acceptance ratio expression?
(c) Simulate 51000 samples from the posterior of and discard the first 1000 samples (burn in). Plot two figures: the histogram of s and the realizations of the chain for the last 1000 simulations. What is the Bayes estimator of 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 . When is a constant hazard rate, a simple way to model heterogeneity of hazards is to introduce a multiplicative frailty parameter , so that lifetimes Ti have distribution
Ti f(ti|,) = exp{ti}, ti > 0, , > 0.
The prior on (,) is
(,) c1d1 exp{ },
that is, and are apriori independent with distributions Ga(c,) and Ga(d,), respectively.
The hyperparameters c,d, and are known (elicited) and positive.
Assume that lifetimes t1,t2,,tn are observed.
(a) Show that full conditionals for and are gamma,
,
and by symmetry,
.
(b) Using the result from (a) develop Gibbs Sampler algorithm that will sample 51000 pairs (,) 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, = 100, and = 5. Start the chain with = 0.1.
(c) From the produced chain, plot the scatterplot of (,) as well as histograms of individual components, , and . Estimate posterior means and variances for and . Find 95% equitailed credible sets for and .
(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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