Let X; € {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 fx\p(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. Using the data above, plot the Beta(10, 10) prior density, the likelihood function (as a function of p), and the posterior density all on the same plot. Interpret the figure.
Let X; € {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 fx\p(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. Using the data above, plot the Beta(10, 10) prior density, the likelihood function (as a function of p), and the posterior density all on the same plot. Interpret the figure.
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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![**Title: Modeling Relapse in Multiple Sclerosis Patients Using Geometric Distribution**
**Text:**
**Problem Statement:**
Consider \( X_i \in \{1, 2, 3, \ldots \} \) as the number of days until relapse for patient \( i \), who is diagnosed with multiple sclerosis and is currently in remission. We represent this data with a geometric distribution using the probability mass function (pmf):
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} f_{X|p}(x \mid p) = (1-p)^{x-1}p \]
This is defined for \( 0 < p < 1 \) and \( x \in \{1, 2, 3, \ldots \} \), and 0 elsewhere. In this context, \( p \) represents the risk of relapse on each day.
**Objective:**
Using the specified data, generate a plot which includes the following components on the same graph:
- The prior density represented by \( \text{Beta}(10, 10) \)
- The likelihood function considered as a function of \( p \)
- The posterior density
Additionally, analyze and interpret the resulting figure.
**Note:**
The graphical depiction will aid in understanding how the risk of daily relapse is perceived before observing the data (prior), how the data influences the perception of relapse risk (likelihood), and the updated perception after considering the data (posterior).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F361779f5-7517-4850-b433-f5d42195c84d%2Fc41cd5a5-58f4-44f1-af07-f0dac9db1af8%2Fm1demq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Modeling Relapse in Multiple Sclerosis Patients Using Geometric Distribution**
**Text:**
**Problem Statement:**
Consider \( X_i \in \{1, 2, 3, \ldots \} \) as the number of days until relapse for patient \( i \), who is diagnosed with multiple sclerosis and is currently in remission. We represent this data with a geometric distribution using the probability mass function (pmf):
\[ X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} f_{X|p}(x \mid p) = (1-p)^{x-1}p \]
This is defined for \( 0 < p < 1 \) and \( x \in \{1, 2, 3, \ldots \} \), and 0 elsewhere. In this context, \( p \) represents the risk of relapse on each day.
**Objective:**
Using the specified data, generate a plot which includes the following components on the same graph:
- The prior density represented by \( \text{Beta}(10, 10) \)
- The likelihood function considered as a function of \( p \)
- The posterior density
Additionally, analyze and interpret the resulting figure.
**Note:**
The graphical depiction will aid in understanding how the risk of daily relapse is perceived before observing the data (prior), how the data influences the perception of relapse risk (likelihood), and the updated perception after considering the data (posterior).
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