Machine Learning Techniques Quiz 1: 27 October 2024 (September 2024 term)

Let $X_1, X_2, \ldots, X_n$ be $n$ i.i.d. samples with parameter $\theta$, which follows one of the following PDFs:

For $\theta = -1$, we have

$f(x \mid \theta) = \begin{cases} 5x^4, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}.$

For $\theta = 1$, we have

$f(x \mid \theta) = \begin{cases} 1, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}.$

Suppose we wish to find the maximum likelihood estimate of $\theta$, then which among the following are true?

Consider a GMM for 5 points:

$x_1 = 1, x_2 = 1.2, x_3 = 2, x_4 = 1.5, x_5 = 0.5$

At some time-step in the EM algorithm, following are the values of $\lambda_k^i$ for the $k$-th mixture after the E-step:

$\lambda_k^1 = 0.3, \lambda_k^2 = 0.1, \lambda_k^3 = 2.5, \lambda_k^4 = 0.6, \lambda_k^5 = 0.8$

What is the estimate of $\mu_k$ after the M-step? Enter your answer correct to two decimal places.

Consider a dataset with 100 total data points. Each data point is classified as either type A or type B. We model this using a Bernoulli distribution, where p is the probability of a data point being type A. If the maximum likelihood estimate (MLE) of p based on the dataset is 0.4, how many data points of type B are there in this dataset?

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Which point has the lowest probability of being chosen as the 3rd cluster mean? Use squared distance to calculate the scores.