### COSC380/580 Algorithms in Machine Learning Assignment,

Algorithms in Machine Learning Assignment -

Question 1 - A certain type of electronic component has a life time X (in hours) with probability density function given by

Suppose that three such components, tested independently, had lifetimes of x1 = 120, x2 = 130, and x3 = 128 hours.

(a) Plot the likelihood function P(x|θ) as a function of θ for a single data point x1. From the plot, comment about the most likely value of θ.

(b) On the same plot in part (a), add the likelihood function for each data point x2 and x3. For better visalization, you should use a different colour for each likelihood function.

(c) Write down the likelihood function for N data points P(x1, x2, . . . , xN|θ), then derive the maximum likelihood estimate (MLE) of θ, denoted θ^.

(d) Substitute the values of the given data points x1, x2, x3 into θ^ and compare this result with the plot in (b).

Show and justify all working.

Question 2 - To validate the EM algorithm, we'll test it on a synthetic data set where the parameters πk, μk, ∑k with k = 1, 2 are known. Here we only work with bivariate (or two-dimensional) Gaussians.

Step 1: Generate a sample of size N from a mixture of two bivariate Gaussians. To generate a data point, you pick one of the components with probability πk, then draw a sample xi from that component using mvnrnd(). Repeat these steps for each new data point.

Step 2: Implement the EM algorithm given in the lectures for this synthetic data set. Label your script file my_em.m. Plot your resulting mixture of Gaussians. Are your results closed to the true parameter values?

Step 3: Repeat the Steps 1 and 2 for B = 100 times but keep the same parameter values πk, μk, ∑k. Calculate the means of the estimates for each parameter πk, μk, ∑k and present them in a table. Comment about your results.

Question 3 - Revisit the Question 3 in Assignment 1, you toss a bent coin N times, obtaining a sequence of heads and tails. The coin has an unknown bias f of coming up heads.

We assume the prior for f is a beta distribution (Mackay section 23.5 or Bishop 2.1.1), Beta(α, β). The probability density function of the beta distribution is given by the following:

P(f|α, β) = (fα-1(1-f)β-1)/B(α, β)

Where the term in the denominator, B(α, β) is present to act as a normalising constant so that the area under the PDF actually sums to 1.

(a) Show that the posterior distribution of f, P(f|nH), is also a beta distribution. Show and justify all working.

(b) We carried N = 50 tosses and observed nH = 12 heads. Plot the prior Beta(12, 12) and the resulting posterior distribution about P(f|nH) on the same figure.

(c) Implement a Markov chain using the Metropolis method given in the lectures to simulate from the posterior distribution. Label your script file my_metropolis.m. Then compare the results from a closed-form solution in (b) and one calculated by numerical approximation. Plot the analytic and MCMC-sampled posterior distributions about f, overlaid with the prior belief.

Note - Use MATLAB to solve the questions for plotting.

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