hw7 - STAT 428 Spring 2010 Homework 7 Apr 21 Homework 7 Due...

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Unformatted text preview: STAT 428 Spring 2010 Homework 7 Apr 21 Homework 7 Due in class on Friday, Apr 30. 1. Consider the simulation of bivariate normal random variables. Let X = ( X 1 ,X 2 ) and the target distribution is ( x 1 ,x 2 ) N parenleftbiggparenleftbigg parenrightbigg , parenleftbigg 1 . 5 . 5 1 parenrightbiggparenrightbigg . Design a Gibbs sampler to generate samples approximately from the target distribution. Give your estimate of the mean of X 2 . Plot the histogram of X 2 based on your samples, and compare it with the histogram based on i.i.d. samples from N (0 , 1) (which is the true marginal distribution of X 2 ). 2. Consider the one-dimensional Ising model that we discussed in class. Let x = ( x 1 ,... ,x d ), where x i is either +1 or- 1. The target distribution is ( x ) exp braceleftBigg d- 1 summationdisplay i =1 x i x i +1 bracerightBigg . Let = 2, d = 40. (a) Design a Gibbs sampling algorithm to generate samples approximately from the target dis- tribution ( x ), and implement your algorithm in R. Attach the R code.), and implement your algorithm in R....
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