Question 5 The observed data y = {y₁j, i = 1,...,n, j = 1,...,m;} are the average results in an exam for school j within county i. The following hierarchical model is considered reasonable: Yij- Normal (₁,3), j = 1,...,mi μ₁~Normal (μc, o), i = 1,...,n. where μc, σs and σoc are unknown parameters which are each assigned a prior distribution. Suppose that we have generated a sample of size M from the joint posterior distribution pμc,σs,σc,μ1,...,pn | y). (a) Explain how to use the posterior sample to estimate the following: (i) the posterior mean for μc; (ii) a 95% credible interval for σs/oci (iii) the posterior probability that μ1 <μ2. (b) Explain how to generate a sample from the posterior predictive distribution of the result for a school not in our dataset, in each of the following two cases: (i) if the county containing the school is in our dataset; (ii) or if the county is not in our dataset.
Question 5 The observed data y = {y₁j, i = 1,...,n, j = 1,...,m;} are the average results in an exam for school j within county i. The following hierarchical model is considered reasonable: Yij- Normal (₁,3), j = 1,...,mi μ₁~Normal (μc, o), i = 1,...,n. where μc, σs and σoc are unknown parameters which are each assigned a prior distribution. Suppose that we have generated a sample of size M from the joint posterior distribution pμc,σs,σc,μ1,...,pn | y). (a) Explain how to use the posterior sample to estimate the following: (i) the posterior mean for μc; (ii) a 95% credible interval for σs/oci (iii) the posterior probability that μ1 <μ2. (b) Explain how to generate a sample from the posterior predictive distribution of the result for a school not in our dataset, in each of the following two cases: (i) if the county containing the school is in our dataset; (ii) or if the county is not in our dataset.
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
The observed data y = {yij, i = 1,...,n, j = 1,...,m;} are the average results in an exam for
school j within county i. The following hierarchical model is considered reasonable:
yij ~ Normal (ui, o), j = 1,...,m¡
μ¡~ Normal (μc, oc), i = 1,...,n.
where μc, σs and σc are unknown parameters which are each assigned a prior distribution.
Suppose that we have generated a sample of size M from the joint posterior distribution
pluc,σs,σc,,...,ny).
(a) Explain how to use the posterior sample to estimate the following:
(i) the posterior mean for μc;
(ii) a 95% credible interval for σs/c;
(iii) the posterior probability that μ₁ <μ2.
(b) Explain how to generate a sample from the posterior predictive distribution of the result
for a school not in our dataset, in each of the following two cases:
(i) if the county containing the school is in our dataset;
(ii) or if the county is not in our dataset.
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