Question 5 We have observed data y = {yij: i= 1,...,n, j = 1,...,m;}. Each yij is the number of times a certain type of machine needs to be repaired during length of time Tij, where j = 1,...,m; are the machines in factory i, for i = 1,...,n, with n > 2. A hierarchical model is used to model the data. We assume that Yij~ Poisson(Tijμi). μ; is the repair rate for factory i, which varies between factories according to a gamma distribution Hi~ Gamma(a,p), i = 1,...,n. The parameters & and 3 are given prior distributions, p(a) and p(3). Suppose that we have generated a sample of size M from the joint posterior distribution P(μ1,...,Hn, a,Bly). (a) Explain how to estimate the following using the joint posterior sample: (i) The posterior mean of a. α (ii) The posterior median of v=- B (iii) A 95% equal tail credible interval for v. (b) Explain how to generate a sample from the posterior predictive distribution of the number of repairs during time U for a machine not in our dataset, in each of the following two cases: (i) If the factory containing this machine is in our dataset. (ii) If the factory is not in our dataset. Also explain how to estimate the posterior predictive probability that such a machine will not need any repairs during time U.
Question 5 We have observed data y = {yij: i= 1,...,n, j = 1,...,m;}. Each yij is the number of times a certain type of machine needs to be repaired during length of time Tij, where j = 1,...,m; are the machines in factory i, for i = 1,...,n, with n > 2. A hierarchical model is used to model the data. We assume that Yij~ Poisson(Tijμi). μ; is the repair rate for factory i, which varies between factories according to a gamma distribution Hi~ Gamma(a,p), i = 1,...,n. The parameters & and 3 are given prior distributions, p(a) and p(3). Suppose that we have generated a sample of size M from the joint posterior distribution P(μ1,...,Hn, a,Bly). (a) Explain how to estimate the following using the joint posterior sample: (i) The posterior mean of a. α (ii) The posterior median of v=- B (iii) A 95% equal tail credible interval for v. (b) Explain how to generate a sample from the posterior predictive distribution of the number of repairs during time U for a machine not in our dataset, in each of the following two cases: (i) If the factory containing this machine is in our dataset. (ii) If the factory is not in our dataset. Also explain how to estimate the posterior predictive probability that such a machine will not need any repairs during time U.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Question 5
We have observed data
y = {yij: i= 1,...,n, j = 1,...,m;}.
Each yij is the number of times a certain type of machine needs to be repaired during length of
time Tij, where j = 1,...,m; are the machines in factory i, for i = 1,...,n, with n > 2.
A hierarchical model is used to model the data. We assume that
Yij~ Poisson(Tijμi).
μ; is the repair rate for factory i, which varies between factories according to a gamma
distribution
Hi~ Gamma(a,p), i = 1,...,n.
The parameters & and 3 are given prior distributions, p(a) and p(3).
Suppose that we have generated a sample of size M from the joint posterior distribution
P(μ1,...,Hn, a,Bly).
(a) Explain how to estimate the following using the joint posterior sample:
(i) The posterior mean of a.
α
(ii) The posterior median of v=-
B
(iii) A 95% equal tail credible interval for v.
(b) Explain how to generate a sample from the posterior predictive distribution of the
number of repairs during time U for a machine not in our dataset, in each of the
following two cases:
(i) If the factory containing this machine is in our dataset.
(ii) If the factory is not in our dataset. Also explain how to estimate the posterior
predictive probability that such a machine will not need any repairs during time U.
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