A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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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.](https://content.bartleby.com/qna-images/question/6e532244-1a69-487f-88f4-10ae0a919295/eb5d77e2-176c-4f68-bfbe-84327fc15751/rlnmw7_processed.png)
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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