2. Normal-Cauchy by Gibbs. Assume that y1, y2, . .., yn is a sample from N(0, o2) distribution, and that the prior on 0 is Cauchy Ca( M, T), T f(0/ M, T) TT T2 + ( 0 - 1 ) 2 . Even though the likelihood for y1, ..., yn simplifies by sufficiency arguments to a likeli- hood of y ~ N(0, o2), a closed form for the posterior is impossible and numerical integra- tion is required. The approximation of the posterior is possible by Gibbs sampler as well. Cauchy Ca( M, T) distribution can be represented as a scale-mixture of normals: [0] ~ Ca(M, T) = [01X] ~ N ( M, ) , [A] ~ Ga 2 . that is, T TT(T2 + ( 0 - 7) 2) 2 TTT2 exp 2 72 ( 0 - 1) 2 8 . 13 -1 exp ) - , dx.The full conditionals can be derived from the product of the densities for the likelihood and priors, [y10, 02 ] ~ NO,S n [A] ~ ga (2'2 (a) Show that full conditionals are normal and exponential, 72 T2 . 02 [oly, X] ~ N 102 72 + 102 / ny + T2 + 102" T2 + 102) [x /y, 0 ] ~ E T2 + ( 0 - 1 ) 2 272 (b) Jeremy models the score on his IQ tests as N(0, 02) with o? = 90. He places a Cauchy prior on 0: Ca(110, v120). In 10 random IQ tests Jeremy scores y = [100, 112, 110, 95, 104, 112, 120, 95, 98, 109]. The average score is 105.5, which is the frequentist estimator of 0. Using the Gibbs sampler described in (a), approximate the posterior mean and variance. Approximate the 94% equi- tailed credible set by sample quantiles