3. Maximum Entropy Estimation: Shannon's entropy is defined as: where H(p)=1 Pi ln pi 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, respectively)? 2,..., The following nonlinear optimization problem 6 max H (p) subject to Σi±1 P₁ = 1, Σ²±1 xipi = = E[x] i=1 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)=1 Pi ln pi 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, respectively)? 2,..., The following nonlinear optimization problem 6 max H (p) subject to Σi±1 P₁ = 1, Σ²±1 xipi = = E[x] i=1 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)=1 Pi ln pi
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,
respectively)?
2,...,
The following nonlinear optimization problem
6
max H (p) subject to Σi±1 P₁ = 1, Σ²±1 xipi = = E[x]
i=1
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%2Feb5d77e2-176c-4f68-bfbe-84327fc15751%2Frlnmw7_processed.png&w=3840&q=75)
Transcribed Image Text:3. Maximum Entropy Estimation:
Shannon's entropy is defined as:
where
H(p)=1 Pi ln pi
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,
respectively)?
2,...,
The following nonlinear optimization problem
6
max H (p) subject to Σi±1 P₁ = 1, Σ²±1 xipi = = E[x]
i=1
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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