Machine Learning Techniques Quiz 1: 23 February 2025 (January 2025 term)

For the Lloyd's algorithm, what would be the correct relationship among the following three quantities?

(1) $\sum_{i=1}^{n} ||x_i - \mu^t_{z_i^{t+1}}||^2$,
(2) $\sum_{i=1}^{n} ||x_i - \mu^t_{z_i^t}||^2$,
(3) $\sum_{i=1}^{n} ||x_i - \mu^{t+1}_{z_i^{t+1}}||^2$

where $\mu^t_{z_i^t}$ and $\mu^{t+1}_{z_i^{t+1}}$ refer to means of cluster $z_i$ in iterations, $t$ and $t+1$ respectively, and $\mu^t_{z_i^{t+1}}$ refer to the mean of the cluster $z_i$ where $x_i$ is going to move in the next iteration, i.e., $(t+1)$-th iteration.

Which of the following statements is/are true?

Let $X$ be a data matrix of shape $(d, n)$ for a centered dataset. The first principal component of the dataset is given as

$\begin{bmatrix} \frac{\sqrt{2}}{2}, & \frac{\sqrt{2}}{2} \end{bmatrix}^T.$

What will be the scalar proxy of the point $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ onto the first principal component?

Enter the answer correct to two decimal places.

Suppose you toss a coin four times and observe no heads. You then give the coin to your friend, who tosses it until the first head occurs. Your friend tosses the coin a total of five times. Let p denote the probability that the coin comes up heads. Find the maximum likelihood estimate of p.

Suppose we fit a Gaussian Mixture Model with $K = 2$ for the following dataset:

$D = \{x_1 = 1, x_2 = 3, x_3 = 2, x_4 = 5\}.$

At the beginning of the $t$-th time step of the EM algorithm, we have $\theta^{(t)}$ as follows:

$\pi_1 = 0.5, \quad \pi_2 = 0.5$

$\mu_1 = 1, \quad \mu_2 = 3$

$\sigma_1^2 = 1, \quad \sigma_2^2 = 1$

What is the probability that $x_1$ belongs to the first component? Enter the answer correct to two decimal places.

Based on the above data, answer the given subquestions.

Among the vectors given below, choose a representative that has unit length.

Based on the above data, answer the given subquestions.

If standard PCA is performed on this dataset, what is the variance along the first principal component? Enter the answer correct to two decimal places.

Suppose that 100 items are sampled from a manufacturing process and 10 are found to be defective. Let p denote the proportion of defective items. Assume that the prior distribution for p is Beta(1, 1).
Based on the above data, answer the given subquestions.

Select the correct options from the following:

Suppose that 100 items are sampled from a manufacturing process and 10 are found to be defective. Let p denote the proportion of defective items. Assume that the prior distribution for p is Beta(1, 1).
Based on the above data, answer the given subquestions.

Find the posterior mean. Enter the answer correct to two decimal places.