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The Gibbs sampler proceeds as follows: . How to find full conditionals? Start0 0 = (01, 02,...,0p) [initials] Form a kernel of joint distribution of all parameters and data Sample on+1 from n (0, 105, 05, .., op, x) To find the full conditional for component 0;, select Sample On+1 from n ( Ozler+1, 05, ..., op, x) only the parts of the kernel that contain 0;, all other 9's and data are considered constant Sample 03+1 from n (03lenti, on+1, or. .., op, x ) Normalize the selected part as a distribution. Often, you can recognize what distribution it is from the form of kernel Sample op+1 from n (Oplenti, .., op+ ], Important: You should be able to sample from all conditionals Increase n GTX GTX For example: Example 1. X1, ...,Xn N (0,z); t precision, = = 02 X|0~N(0, 1) u~N(0,1) T~ Ga(2,1) 0~Ca(0,1) ntin - Er,( xi-H ) e-zu2 Te-T Find S(2) by Gibbs sampling oint o (21 ) 2 tze 2 2 + 00 TT (0 ) 72 + (0 - 14) 2 0 2nTz exp (- 272 (0 - 1)2) x2-e-zda MIT , X ~ N TZXi 1 (1 + no'1 + nT) = 0~ Ca(M, T) -012 -N (M. ). -Ga (2 2) IT (TIM, X ) & Tze 2 1=1(21-") Textile-7/1+ 21=1 (21-1)2] If X|0~N(0, 62) 012 , x ~N (72+202 X + 102 + 72+202 /, 72 +262) TIM, X ~ Ga (" + 2, 1 + - [ ( x1-41)2) 210, x ~Exp (- +(0-H) 2 ) 272 GTX norcaugibbs.m GTX2. 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 ( 01 1, T ) = T' 72 + (0-14) 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) = 101 x] ~ N ( 1, 7 ) , [ ]~ ga ( 3, 2 ) , that is, T TT ( T 2 + ( 0 - T ) 2 ) x 60 Vantrz exp ( 272 10 -10)2 ). 13 -" exp { -?} ax. The full conditionals can be derived from the product of the densities for the likelihood and priors, [y10 , 02 ] ~ N ( 0, - ) (a) Show that full conditionals are normal and exponential, [oly, X] ~ N T2 102 T2 . 02 T2 + 10 2 / n 7 72 + 102 / n ' T2 + 102 / n / [ly, 0] ~ E T + ( 0 - 14)2 ) 272 (b) Jeremy models the score on his IQ tests as N(0, 02) with o2 = 90. He places Cauchy Ca(110, V120) prior on e. 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