COSC 578 Final Exam (S2022)
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COSC 578 Final Exam (S2022)
March 21, 2022
Rule: It is an open book and open internet exam. You are allowed to use the internet and the teaching materials. However, you are not allowed to discuss the questions and answers with any human in any manner.
Duration: 75 Minutes.
Q1. Maximum Likelihood Estimation (MLE). [70 points]
1a) Based on the lecture notes, write down the detailed steps of MLE. Give example for each step.
1b) What is a likelihood function? Justify your answer in both English and in math formulas. 1c) Why do we usually use log-likelihood function instead of likelihood function?
1d) Suppose we have a Gaussian Distribution with parameters
The joint probability density of n i.i.d. data points is
What is the optimal θ that maximizes this probability density obtained by MLE? Show the detailed step-by-step derivation. Make sure your answer is thorough.
1e) Suppose we have a Bernoulli Distribution with parameters e = p, where p is unknown. This unknown parameter p is the success rate. Our task is to find out an estimator for p.
The joint probability density of n i.i.d. data points is
What is the optimal p that maximizes this probability density obtained by MLE? Show the detailed step-by-step derivation. Make sure your answer is thorough.
Q2. Bayesian Estimator. [30 points]
2a) What is prior? What is Posterior? Explain in words and in mathematical formulation. 2b) What is prior and posterior’s relationship? Explain in words and in mathematical formulation.
2c) What are the differences between MLE and Bayesian Estimator? What are their pros and cons?
2023-02-24