Answer Happy

A random sample y = (y₁,..., yn) of size n has been collected from a Log Normal distribution. A Log Normal distribution has the density function: T p(y | μ, T) = - (-7(log(y) — μ)²) exp y > 0, 2πy² 2 where -∞< < x and > 0 are unknown parameters. A Bayesian analysis will be conducted to inferu and T. Assume an independent prior on the parameters is defined as ~ N(0, 1) T~ Exp(a) where a is some given fixed value. Answer the following: (a) Show that the posterior of u and 7 up to a constant of proportionality is given by Ti exp (log(y) - ) - M³²] - 1²) i=1 (b) Derive the full conditional densities for and 7 up to a constant of proportionality. Identify the probability distributions if you recognise them. (c) Write the pseudocode for implementing a Gibbs algorithm for sampling the joint posterior (μ, Ty) the pseudocode should include all inputs and outputs to the algorithm and all of the steps. (d) Write an expression for the deviance of this model D(μ, T) for a fixed dataset y. (e) Assume that an MCMC sample {₁,1 has been generated from the joint posterior. Assume also that a suitable burn-in has been chosen and MCMC convergence diagnostics are satisfied. Write down an expression for the (estimated) DIC of this model based on the MCMC sample and previous expressions (using the first definition for effective number of parameters).