Machine Learning Techniques End Term: 31 August 2025, Set QDD1 (May 2025 term)

Consider clustering 1D data with a mixture of 2 Gaussians using the EM algorithm. You are given the data points $x_1 = 1, x_2 = 10, x_3 = 20$.

Suppose the output of the E-step is as follows:

$\begin{gathered} \lambda_1^1 = 1, \; \lambda_2^1 = 0 \\ \lambda_1^2 = 0.4, \; \lambda_2^2 = 0.6 \\ \lambda_1^3 = 0.1, \; \lambda_2^3 = 0.9, \end{gathered}$

where $\lambda_C^i$ denotes the probability of the $i$-th data point to be in cluster $C$. After performing the M-step for the means $\mu_1$ and $\mu_2$, what are the updated means?

Which among the following is TRUE for k-NN algorithm? Select all that apply.

With respect to the Lloyd’s algorithm, choose the correct statements:

Consider a naive Bayes classification problem with three binary features $f_1, f_2, f_3$ for a binary classification problem. We have a total of 8 training samples given in the table below:

If a test point has label 0, what is the probability of $x_{test} = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}^T$? Assume that the Laplacian smoothing is done, (i.e., add two pseudo training exmaples $x_{pseudo}^0 = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}^T$ for class 0 and $x_{pseudo}^1 = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}^T$ for class 1). Enter the answer correct to two decimal places.

Consider a binary classification problem with three data points, denoted by $\{(x_i, y_i)\}$, where $x_i$ represents a one-dimensional feature and $y_i$ is the corresponding class label, as given below:

$\{(1, +1), (2, -1), (3, +1)\}$

Each data point gets uniform initial weights $w_1^{(0)} = w_2^{(0)} = w_3^{(0)} = \frac{1}{3}$, where $w_i^{(t)}$ denotes the weight of $i$-th data point for $t^{th}$ iteration. Following are the two decision stumps given to us:

$h_1(x) = \begin{cases} +1, & \text{if } x \leq 2 \\ -1, & \text{if } x > 2 \end{cases}, \quad h_2(x) = \begin{cases} +1, & \text{if } x \leq 1 \\ -1, & \text{if } x > 1 \end{cases}.$

We run the AdaBoost algorithm for one iteration by selecting the decision stump with the lowest training error. Then, find the value of $w_1^{(1)} + w_2^{(1)}$. Enter the answer correct to one decimal place.

Suppose we are solving a binary classification task using a neural network with the following architecture:

• Input: $x = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
• Hidden layer: 2 neurons (ReLU activation)
• Output layer: 1 neuron (sigmoid activation)

The weight parameters of the network are given as:

$W^{(1)} = \begin{bmatrix} w_1^{(1)} & w_2^{(1)} \\ w_1^{(2)} & w_2^{(2)} \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix}, \quad W^{(out)} = \begin{bmatrix} w_1^{(out)} & w_2^{(out)} \end{bmatrix}^T = \begin{bmatrix} 1 & 0 \end{bmatrix}^T,$

where $w_i^{(j)}$ represents the weight associated with the $j$-th neuron for the $i$-th input feature. Compute the cross-entropy loss $L$. Write the answer to correct to three decimal places.

Loss function: $L = -[y\log_e(\hat{y}) + (1 - y)\log_e(1 - \hat{y})]$, where $\hat{y}$ is the network output and $y = 1$ is the true label.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

What is the probability that a 60-year-old patient with a cholesterol level of 200 mg/dL has the disease? Enter the answer correct to two decimal places.

Based on the above data, answer the given subquestions.

How high should the cholesterol level be for the patient mentioned in the previous question to have a 50% chance of being diagnosed with the disease?

Consider the datapoints (−2,−2) with label +1 and (2, 2) with label −1. A hard margin SVM is applied on this dataset. Answer the given subquestions:

Which of the following is the decision boundary for the given dataset?

Consider the datapoints (−2,−2) with label +1 and (2, 2) with label −1. A hard margin SVM is applied on this dataset. Answer the given subquestions:

The solution to the dual problem is given by $\alpha^* = \begin{bmatrix} \frac{1}{16} & \frac{1}{16} \end{bmatrix}^T$. Find the width of the widest margin for this dataset. Enter the answer correct to two decimal places.

Note: Width of the widest margin is given by $\frac{2}{||w||}$.