Do part c,d,e Suppose that1X\µ ~ Bernoulli(1+e-(u+2)and a standard normal N(0, 1) prior is specified for u. In a trial, the valuex = 1 is observed.(a) Show that the posterior for u when X is observed to be r = 1 isf(4x = 1) x (1+e¯(u+2)-le¬w²/2 on R.%3DThis posterior distribution cannot be sampled from directly. Suppose thatthe Metropolis-Hastings algorithm is to be used to simulate samples fromthe target posterior density f(u|r = 1).In order to simulate samples from the posterior distribution of parameter u,consider the sequence of random variables u1, #2, - such that µi is thevariable for iteration t of the Metropolis-Hastings algorithm. Let be itssimulated value. Suppose that the proposal distribution for 4+1H=4,which has density denoted by q(µ+1\4), is N(H, 2).(b) Let u =0 be the starting value for the algorithm. What distributionwill the Metropolis-Hastings algorithm use to generate the candidatevalue u* for 12?(c) Explain why, in this case, the acceptance probability for u can bewritten as%3Da(u"|14) = min%3D(d) Suppose that the candidate value generated from the distribution inpart (b) is u* = 1.2. Calculate the acceptance probability for thiscandidate value.(e) Suppose that u is simulated from U(0,1) so that u = 0.81. What is thevalue of u,? From what distribution will the candidate value u** for 43be generated?

c) In the Metropolis-Hastings algorithm, the acceptance probability is defined as:Let's break down…

Suppose that Xu~ Bernoulli (1+(+2)) and a standard normal N(0, 1) prior is specified for u. In a trial, the value x= 1 is observed. (a) Show that the posterior for when X is observed to be z = 1 is ƒ(μ|x = 1) x (1 + e−(n+2))-¹₁-4²/2 on R. This posterior distribution cannot be sampled from directly. Suppose that the Metropolis-Hastings algorithm is to be used to simulate samples from the target posterior density f(ulx = 1). In order to simulate samples from the posterior distribution of parameter , consider the sequence of random variables #1, #2,... such that is the variable for iteration t of the Metropolis-Hastings algorithm. Let u be its simulated value. Suppose that the proposal distribution for 4+1 = ₂ which has density denoted by q(+1), is N(μ, 2). (b) Let = 0 be the starting value for the algorithm. What distribution will the Metropolis-Hastings algorithm use to generate the candidate value for ₂? (e) Explain why, in this case, the acceptance probability for u can be written as a(μμ₁) = min (f(µ|x = 1) \ƒ(µ₁₂₁|x = 1)' =B₁¹). (d) Suppose that the candidate value generated from the distribution in part (b) is u = 1.2. Calculate the acceptance probability for this candidate value. (e) Suppose that u is simulated from U(0, 1) so that u= 0.81. What is the value of 2? From what distribution will the candidate value μ** for µ3 be generated?

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.3: Special Probability Density Functions

Problem 30E