Problem 3. In this problem, we are going to do both maximum likelihood estimation and Bayesian estimation. (a) We have one unknown parameter 0. We draw X1, X2,..., Xg independently from a Bernoulli(0) distribution. Suppose X₁ = X2 = X5 = 1 and X3 = X₁ = X6 = X7 = X8 = 0, i.e. we get 3 successes and 5 failures. What is the likelihood function L(0) and what is the log-likelihood function In L(0)? (b) What's the maximum likelihood estimator of given this data? (c) Suppose we have a prior that = 0.75 with probability 0.6 and that = 0.25 with probability 0.4. What is the posterior distribution in this case? What is the maximum a posteriori estimator? (d) Instead of the prior in Part (c), suppose instead we have a prior that is uniformly distributed on [0,1]. What is the posterior distribution in this case? What is the maximum a posteriori estimator?
Problem 3. In this problem, we are going to do both maximum likelihood estimation and Bayesian estimation. (a) We have one unknown parameter 0. We draw X1, X2,..., Xg independently from a Bernoulli(0) distribution. Suppose X₁ = X2 = X5 = 1 and X3 = X₁ = X6 = X7 = X8 = 0, i.e. we get 3 successes and 5 failures. What is the likelihood function L(0) and what is the log-likelihood function In L(0)? (b) What's the maximum likelihood estimator of given this data? (c) Suppose we have a prior that = 0.75 with probability 0.6 and that = 0.25 with probability 0.4. What is the posterior distribution in this case? What is the maximum a posteriori estimator? (d) Instead of the prior in Part (c), suppose instead we have a prior that is uniformly distributed on [0,1]. What is the posterior distribution in this case? What is the maximum a posteriori estimator?
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![Problem 3. In this problem, we are going to do both maximum likelihood estimation and Bayesian
estimation.
(a) We have one unknown parameter 0. We draw X1, X2,..., X8 independently from a Bernoulli (0)
distribution. Suppose X₁ = X2 = X5 = 1 and X3 = X₁ = X6 = X7 = X8 = 0, i.e. we get 3 successes
and 5 failures. What is the likelihood function L(0) and what is the log-likelihood function In L(0)?
(b) What's the maximum likelihood estimator of given this data?
(c) Suppose we have a prior that = 0.75 with probability 0.6 and that = 0.25 with probability 0.4.
What is the posterior distribution in this case? What is the maximum a posteriori estimator?
(d) Instead of the prior in Part (c), suppose instead we have a prior that is uniformly distributed on
[0,1]. What is the posterior distribution in this case? What is the maximum a posteriori estimator?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F423822cb-5d3f-48f4-83a4-f0ba428ae79c%2F4d3891de-e028-422c-914a-02fb777a9c15%2Fqulkdev_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3. In this problem, we are going to do both maximum likelihood estimation and Bayesian
estimation.
(a) We have one unknown parameter 0. We draw X1, X2,..., X8 independently from a Bernoulli (0)
distribution. Suppose X₁ = X2 = X5 = 1 and X3 = X₁ = X6 = X7 = X8 = 0, i.e. we get 3 successes
and 5 failures. What is the likelihood function L(0) and what is the log-likelihood function In L(0)?
(b) What's the maximum likelihood estimator of given this data?
(c) Suppose we have a prior that = 0.75 with probability 0.6 and that = 0.25 with probability 0.4.
What is the posterior distribution in this case? What is the maximum a posteriori estimator?
(d) Instead of the prior in Part (c), suppose instead we have a prior that is uniformly distributed on
[0,1]. What is the posterior distribution in this case? What is the maximum a posteriori estimator?
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VIEWStep 5: Determine posterior distribution when prior is uniform distributed on [0,1].
VIEWStep 6: Determine the maximum posteriori estimator when prior is uniform distributed on [0,1].
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