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The observed data is y = (v₁....yn), a sample from a negative binomial distribution with parameters q andr, where r is assumed to be known. The prior distribution for q is Beta(a,ß). 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 /3 = 1. A=S A=S₂B=1 (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.

The observed data is y = (v₁....yn), a sample from a negative binomial distribution with parameters q andr, where r is assumed to be known. The prior distribution for q is Beta(a,ß). 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 /3 = 1. A=S A=S₂B=1 (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.

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
Section13.3: Special Probability Density Functions
Problem 53E
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The observed data is y = (₁,...,n), 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,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 3 = 1. A = S ·A=S₂B=1 (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 a 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.
Transcribed Image Text:The observed data is y = (₁,...,n), 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,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 3 = 1. A = S ·A=S₂B=1 (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 a 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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