3. Maximum Entropy Estimation: Shannon's entropy is defined as: where =- H(p) Pilnpi = Σ1 Possibility of each event: pi = p(xi) = [0,1] You are given a six-sided dice numbered from 1 to 6. You are given the information that the average result from 10000 times of rolling dice is E[x]=3.5. What is your estimation of the probabilities associated with different sides (what is the probabilities of having 1, 2, ..., respectively)? The following nonlinear optimization problem max H (p) subject to Σ - Pi = 1, Σi±1 xipi = E[x] gives the least-biased probability distribution (a) Solve this problem by calling an optimization solver. Include your script and result output. Plot how the probabilities change as E[x] varies between 1 and 6. (b) Derive the optimality conditions using Lagrange Multiplier Method. Save this result, as we will come back to it in the next homework assignment.
3. Maximum Entropy Estimation: Shannon's entropy is defined as: where =- H(p) Pilnpi = Σ1 Possibility of each event: pi = p(xi) = [0,1] You are given a six-sided dice numbered from 1 to 6. You are given the information that the average result from 10000 times of rolling dice is E[x]=3.5. What is your estimation of the probabilities associated with different sides (what is the probabilities of having 1, 2, ..., respectively)? The following nonlinear optimization problem max H (p) subject to Σ - Pi = 1, Σi±1 xipi = E[x] gives the least-biased probability distribution (a) Solve this problem by calling an optimization solver. Include your script and result output. Plot how the probabilities change as E[x] varies between 1 and 6. (b) Derive the optimality conditions using Lagrange Multiplier Method. Save this result, as we will come back to it in the next homework assignment.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![3. Maximum Entropy Estimation:
Shannon's entropy is defined as:
where
=-
H(p) Pilnpi
= Σ1
Possibility of each event: pi = p(xi) = [0,1]
You are given a six-sided dice numbered from 1 to 6. You are given the information that
the average result from 10000 times of rolling dice is E[x]=3.5. What is your estimation
of the probabilities associated with different sides (what is the probabilities of having 1,
2, ..., respectively)?
The following nonlinear optimization problem
max H (p) subject to Σ - Pi = 1, Σi±1 xipi = E[x]
gives the least-biased probability distribution
(a) Solve this problem by calling an optimization solver. Include your script and result
output. Plot how the probabilities change as E[x] varies between 1 and 6.
(b) Derive the optimality conditions using Lagrange Multiplier Method. Save this result,
as we will come back to it in the next homework assignment.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e532244-1a69-487f-88f4-10ae0a919295%2F7679e8a3-eb84-4e24-9ffe-0ebc650a6610%2F3t96da_processed.png&w=3840&q=75)
Transcribed Image Text:3. Maximum Entropy Estimation:
Shannon's entropy is defined as:
where
=-
H(p) Pilnpi
= Σ1
Possibility of each event: pi = p(xi) = [0,1]
You are given a six-sided dice numbered from 1 to 6. You are given the information that
the average result from 10000 times of rolling dice is E[x]=3.5. What is your estimation
of the probabilities associated with different sides (what is the probabilities of having 1,
2, ..., respectively)?
The following nonlinear optimization problem
max H (p) subject to Σ - Pi = 1, Σi±1 xipi = E[x]
gives the least-biased probability distribution
(a) Solve this problem by calling an optimization solver. Include your script and result
output. Plot how the probabilities change as E[x] varies between 1 and 6.
(b) Derive the optimality conditions using Lagrange Multiplier Method. Save this result,
as we will come back to it in the next homework assignment.
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