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Please Explain and Show How - It is OK to Use Python for Part B to Calculate Answers

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), f ( 0 / Mb, T ) = 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, 2), 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 ( " , ? ) , [ ]~ ga ( 2, 2 ) . that is, T T ( T 2 + ( 0 - 7 ) 2 ) 0 2TT2 exp - 27 2 ( 0 - 14 ) 2 1 . 13-1 exp { - ax . The full conditionals can be derived from the product of the densities for the likelihood and priors, [y 10 , 02 ] ~ N ( Q , 0- ) 101 X] ~ N (14 , ? ) [A] ~ ga (?'2) (a) Show that full conditionals are normal and exponential, T2 do2 T2 . 02 / n [ely, X] ~ N T2 + 102 / ny + 72 + 102/ n T2 + 102) [x /y, 0] ~ E T + (0 - 14) 2 272 (b) Jeremy models the score on his IQ tests as N(0, o2) with o? = 90. He places Cauchy Ca(110, V120) prior on 0. 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 0. Using Gibbs sampler described in (a) approximate the posterior mean and variance. Approximate 95% equi-tailed credible set by sample quantiles