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# 机器学习考试代考 CS5487代写 Machine Learning代写

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## Online Midterm

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.

• 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.

### Problem 1 MLE for Exponential distribution [30 marks]机器学习考试代考

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.

### Problem 2 MAP for Exponential distribution [35 marks]

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?

### Problem 3 Bayesian estimation for Exponential distribution [35 marks]机器学习考试代考

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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