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?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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