Question 4 3]. The observed data is y = (y₁,..., yn), a sample from a negative binomial distribution with parameters q and r, where r is assumed to be known. The prior distribution for q is Betala, p). Suppose that y₁ =.= yn = 0. Take n = = 10 +A, where A is the third-to-last digit of your ID number; a = 5+ B,where B is the second-to-last digit of your ID number; r = 3; and ß = 1. B=6 A = 6 (a) What is the posterior probability density function for q? (b) Find an expression for the quantile function for this posterior distribution, and hence find a 95% credible interval for q. (c) Let x be a new data-point generated by the same negative binomial distribution with parameters q and r. Find P(x = 0|y), the posterior predictive probability that x is 0. Suppose now that we want to compare two models. Model M₁ is the model and prior distribution described above. Model M₂ assumes that the data follow a negative binomial distribution with q known to be qo = 0.9. (d) Find the Bayes factor B12 for comparing the two models. (e) We assign prior probabilities of 1/3 that M₁ is the true model, and 2/3 that M₂ is the true model. Find the posterior probability that M₁ is the true model.
Question 4 3]. The observed data is y = (y₁,..., yn), a sample from a negative binomial distribution with parameters q and r, where r is assumed to be known. The prior distribution for q is Betala, p). Suppose that y₁ =.= yn = 0. Take n = = 10 +A, where A is the third-to-last digit of your ID number; a = 5+ B,where B is the second-to-last digit of your ID number; r = 3; and ß = 1. B=6 A = 6 (a) What is the posterior probability density function for q? (b) Find an expression for the quantile function for this posterior distribution, and hence find a 95% credible interval for q. (c) Let x be a new data-point generated by the same negative binomial distribution with parameters q and r. Find P(x = 0|y), the posterior predictive probability that x is 0. Suppose now that we want to compare two models. Model M₁ is the model and prior distribution described above. Model M₂ assumes that the data follow a negative binomial distribution with q known to be qo = 0.9. (d) Find the Bayes factor B12 for comparing the two models. (e) We assign prior probabilities of 1/3 that M₁ is the true model, and 2/3 that M₂ is the true model. Find the posterior probability that M₁ is the true model.
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 4
The observed data is y = (y1,..., Yn), a sample from a negative binomial distribution with
parameters q and r, where r is assumed to be known.
The prior distribution for q is Beta(a,b). Suppose that y₁ = ... =yn = 0. Take n = 10 + A,
where A is the third-to-last digit of your ID number; a = 5+ B,where B is the second-to-last
digit of your ID number; r = 3; and ß = 1. B=6
A = 6
(a) What is the posterior probability density function for q?
(b) Find an expression for the quantile function for this posterior distribution, and hence
find a 95% credible interval for q.
(c) Let x be a new data-point generated by the same negative binomial distribution with
parameters q and r. Find P(x = 0|y), the posterior predictive probability that x is 0.
Suppose now that we want to compare two models. Model M₁ is the model and prior
distribution described above. Model M₂ assumes that the data follow a negative binomial
distribution with q known to be qo = 0.9.
(d) Find the Bayes factor B12 for comparing the two models.
(e) We assign prior probabilities of 1/3 that M₁ is the true model, and 2/3 that M₂ is the
true model. Find the posterior probability that M₁ is the true model.
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