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Let the observed data be y = (6, 4, 9, 2, 0, 3), a random sample from the Poisson distribution with mean A, where A> 0 is unknown. Suppose that we assume a Gamma(1, 1) prior distribution for A. The posterior density, p(A | y), for λ is Gamma (1+ S, 1 + n), where S = ₁ and n = 6. Suppose that you want to construct a symmetric Metropolis-Hastings on the log-scale to generate a sample from this posterior distribution by using a normal proposal distribution with standard deviation b = 0.2. (a) Write down the steps in this symmetric Metropolis-Hastings (on the log-scale) to simulate realisations from the posterior density p(x|y). (b) Implement the algorithm in R. and plot the observations as a function of the iter- ations. Use M = 5000 for the number of iterations. (c) To assess the accuracy compare the empirical distribution of the sample with the exact posterior density, Gamma(1 + S, 1 + n). (d) Rerun the algorithm in R using a smaller b = 0.01 and a larger b = 20. What are the effects on the behaviour of the algorithm of making b smaller? What are the

Let the observed data be y = (6, 4, 9, 2, 0, 3), a random sample from the Poisson distribution with mean A, where A> 0 is unknown. Suppose that we assume a Gamma(1, 1) prior distribution for A. The posterior density, p(A | y), for λ is Gamma (1+ S, 1 + n), where S = ₁ and n = 6. Suppose that you want to construct a symmetric Metropolis-Hastings on the log-scale to generate a sample from this posterior distribution by using a normal proposal distribution with standard deviation b = 0.2. (a) Write down the steps in this symmetric Metropolis-Hastings (on the log-scale) to simulate realisations from the posterior density p(x|y). (b) Implement the algorithm in R. and plot the observations as a function of the iter- ations. Use M = 5000 for the number of iterations. (c) To assess the accuracy compare the empirical distribution of the sample with the exact posterior density, Gamma(1 + S, 1 + n). (d) Rerun the algorithm in R using a smaller b = 0.01 and a larger b = 20. What are the effects on the behaviour of the algorithm of making b smaller? What are the

Calculus For The Life Sciences
Calculus For The Life Sciences
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.CR: Chapter 13 Review
Problem 40CR
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Let the observed data be y = (6, 4, 9, 2, 0, 3), a random sample from the Poisson distribution with mean A, where A> 0 is unknown. Suppose that we assume a Gamma(1, 1) prior distribution for A. The posterior density, p(A | y), for A is Gamma(1+ S, 1 + n), where S = ₁₁y₁ and n = 6. Suppose that you want to construct a symmetric Metropolis-Hastings on the log-scale to generate a sample from this posterior distribution by using a normal proposal distribution with standard deviation b = 0.2. (a) Write down the steps in this symmetric Metropolis-Hastings (on the log-scale) to simulate realisations from the posterior density p(x|y). (b) Implement the algorithm in R. and plot the observations as a function of the iter- ations. Use M = 5000 for the number of iterations. (c) To assess the accuracy compare the empirical distribution of the sample with the exact posterior density, Gamma(1 + S, 1 + n). (d) Rerun the algorithm in R using a smaller b = 0.01 and a larger b = 20. What are the effects on the behaviour of the algorithm of making b smaller? What are the effects of making it larger? (e) Add code to count how many times the proposed value for À was accepted. Rerun the algorithm using values of b = 0.01, b = 0.2 and b = 20, and each time calculate the proportion of steps that were accepted. Then plot this acceptance probability against b. Examine how the acceptance probability for this algorithm depends b.
Transcribed Image Text:Let the observed data be y = (6, 4, 9, 2, 0, 3), a random sample from the Poisson distribution with mean A, where A> 0 is unknown. Suppose that we assume a Gamma(1, 1) prior distribution for A. The posterior density, p(A | y), for A is Gamma(1+ S, 1 + n), where S = ₁₁y₁ and n = 6. Suppose that you want to construct a symmetric Metropolis-Hastings on the log-scale to generate a sample from this posterior distribution by using a normal proposal distribution with standard deviation b = 0.2. (a) Write down the steps in this symmetric Metropolis-Hastings (on the log-scale) to simulate realisations from the posterior density p(x|y). (b) Implement the algorithm in R. and plot the observations as a function of the iter- ations. Use M = 5000 for the number of iterations. (c) To assess the accuracy compare the empirical distribution of the sample with the exact posterior density, Gamma(1 + S, 1 + n). (d) Rerun the algorithm in R using a smaller b = 0.01 and a larger b = 20. What are the effects on the behaviour of the algorithm of making b smaller? What are the effects of making it larger? (e) Add code to count how many times the proposed value for À was accepted. Rerun the algorithm using values of b = 0.01, b = 0.2 and b = 20, and each time calculate the proportion of steps that were accepted. Then plot this acceptance probability against b. Examine how the acceptance probability for this algorithm depends b.
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