Description

ISyE 6420
1. Simple Metropolis: Normal Precision – Gamma. Suppose X = −2 was observed from the population distributed as and one wishes to estimate the parameter θ.
(Here θ is the reciprocal of the variance σ2 and is called the precision parameter). Suppose the analyst believes that the prior on θ is Ga(1/2,1).
2. Normal-Cauchy by Gibbs. Assume that y1,y2,…,yn is a sample from N(θ,σ2) distribution, and that the prior on θ is Cauchy Ca(µ,τ),
.
Even though the likelihood for y1,…,yn simplifies by sufficiency arguments to a likelihood of ¯y ∼ N(θ,σ2/n), a closed form for the posterior is impossible and numerical integration is required.
The approximation of the posterior is possible by Gibbs sampler as well. Cauchy Ca(µ,τ) distribution can be represented as a scale-mixture of normals:
,
that is,

The full conditionals can be derived from the product of the densities for the likelihood and priors,
,
.
(a) Show that full conditionals are normal and exponential,
,
(b) Jeremy models the score on his IQ tests as√ N(θ,σ2) with σ2 = 90. He places Cauchy Ca(110, 120) prior on θ.
In 10 random IQ tests Jeremy scores y = [100,106,110,97,90,112,120,95,96,109]. The average score is 103.5, which is the frequentist estimator of θ. Using Gibbs sampler described in (a) approximate the posterior mean and variance. Approximate 95% equi-tailed credible set by sample quantiles.
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