MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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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.
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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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