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1027MSCA 37014 Machine Learning Final Project (Replacing Assignment 4) 机器学习作业代写 Airbnb is interested in better understanding data relating to price of listings on their website. They ha...
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机器学习考试代考 Time: 2 hours 1. The following resources are allowed on the midterm: • You are allowed a cheat sheet that is one A4 page (single-sided only)
Time: 2 hours
1. The following resources are allowed on the midterm:
• You are allowed a cheat sheet that is one A4 page (single-sided only) hand-written with pen or pencil.
2. All other resources are not allowed, e.g., internet searches, classmates, textbooks.
3. Answer the questions on physical paper using pen or pencil.
• Answer ALL questions.
• Remember to write your name, EID, and student number at the top of each answer paper.
4. You should stay on Zoom during the entire exam time.
• If you have any questions, use the private chat function in Zoom to message Antoni.
5. Midterm submission
• Take pictures of your answer paper and submit it to the “Midterm Quiz” Canvas assignment. You may submit it as jpg/png/pdf.
• It is the student’s responsibility to make sure that the captured images are legible. Illegible images will be graded as is, similar to illegible handwriting.
In this problem you will consider the maximum likelihood estimate (MLE) of the parameters of a Exponential Distribution.
The exponential distribution is a distribution of the time interval between consecutive events occurring in a Poisson process. A Poisson process is a process where events occur continuously at some constant average rate. For example, if we model a telephone switch as a Poisson process, then the time between incoming calls can be modeled as an exponential
distribution.
Suppose we have a set of N samples of time intervals, D = {x1, · · · , xN } with xi ≥ 0.
(a) [5 marks] Write down the log-likelihood of the data D, i.e., log p(D|γ).
(b) [5 marks] Write down optimization problem for maximum-likelihood estimation of the parameter γ.
(c) [15 marks] Derive the MLE for the parameter γ.
(d) [5 marks] What is the intuitive interpretation of the derived MLE for γ.
In this problem you will compute the MAP estimate for exponential distribution in Problem 1. Let the prior distribution of γ be an Inverse Gamma distribution,
(a) [5 marks] Write down the optimization problem for MAP estimation of γ using an Inverse Gamma prior with known hyperparameters (α, β).
(b) [15 marks] Derive the MAP estimator for the parameter γ using the Inverse Gamma prior.
(c) [10 marks] Compare this MAP estimator with the ML estimator derived in Problem 1. What is the intuitive interpretation of the MAP solution with regards to the ML solution?
(d) [5 marks] What is the intuitive effect on the MAP estimate as the number of samples N increases?
In this problem you will derive the Bayesian estimate for the Exponential distribution with Inverse Gamma prior, using the same setup as Problems 1 and 2.
(a) [5 marks] Write down the posterior distribution of the parameters, p(γ|D), in terms of the prior and likelihood function.
(b) [15 marks] Derive the form of the posterior distribution p(γ|D) in terms of the hyperparameters (α, β) and data D.
(c) [5 marks] What is the intuitive interpretation of the derived Bayesian estimate in Problem 3(b), e.g., the mean of the posterior?
(d) [10 marks] What happens to the posterior distribution as the number of samples N increases?
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