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Let X; e {1, 2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2, ..., X, fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1, 2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X (p | X,). 2. Find the posterior mean of p. Find the MLE of p - you do not need to verify that it is a max.

Let X; e {1, 2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2, ..., X, fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1, 2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X (p | X,). 2. Find the posterior mean of p. Find the MLE of p - you do not need to verify that it is a max.

MATLAB: An Introduction with Applications
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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Answer parts 1 and 2 of the problem below.

 

Let X; e {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2, ..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, } and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fpx„(p| X„). 2. Find the posterior mean of p. Find the MLE of p - you do not need to verify that it is a max.
Transcribed Image Text:Let X; e {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2, ..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, } and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fpx„(p| X„). 2. Find the posterior mean of p. Find the MLE of p - you do not need to verify that it is a max.
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