MLT End Term: 3 September 2023, Set QPD1-S1 (May 2023 term)

Suppose you have been given a task of estimating the conversion rate of google click of new online advertisement campaign. You have collected data from a limited sample of 1000 users, where 150 of them have converted. You want to use Bayesian estimation with a prior distribution to provide a more robust estimate of the conversion rate.

Assume you have prior information suggesting that conversion rates of click typically fall within the range of 0.1 to 0.3, and you decide to use a beta distribution as your prior. You choose a beta distribution with parameters $\alpha = 5$ and $\beta = 20$ to capture your prior beliefs.

Calculate the posterior mean?

You are working on a text classification problem using a Naive Bayes classifier to determine whether an email is "Spam" or "Not Spam" . You have trained your model using a dataset of 1000 emails, with 600 of them labeled as "Spam" and 400 labeled as "Not Spam." You've collected statistics on the occurrence of two words, "Won" and "Money" in these emails:

You receive a new email containing both the "Won" and "Money" keywords and want to classify it using Naive Bayes.

Use the Naive Bayes formula to calculate the probability that the new email is classified as "Spam."

Hint: Assume that these are only two possible words (that is there are only two features)

Which of the following statements is/are true?

Consider the following dataset with two features.

Which of the following property does the solution $w$ satisfy that fits the linear regression model $y = w^T x$?

While training a perceptron model, the weight vector at some iteration $t$ is $\mathbf{w}^t$. The next two data-points picked up by the perceptron algorithm in the course of its execution is $(\mathbf{x}_1, y_1)$, and $(\mathbf{x}_2, y_2)$ where $y$ is the true label:

$\mathbf{w}^t = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \quad \mathbf{x}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 1 \end{bmatrix}, \quad y_1 = 1, \quad \mathbf{x}_2 = \begin{bmatrix} -2 \\ 1 \\ 1 \\ -1 \end{bmatrix}, \quad y_2 = -1$

What are the value of $\mathbf{w}^{t+1}$ and $\mathbf{w}^{t+2}$?

Consider the following optimization problem involving the variable $\mathbf{w} = \begin{bmatrix} w_1 & w_2 \end{bmatrix}^T$:

$\min_{\mathbf{w}} \ w_1^2 - 3w_2$

$\text{sub. to} \quad w_1 + 7w_2 \le 5$

What is the output of the following expression for $\mathbf{w}' = \begin{bmatrix} 1 & 1 \end{bmatrix}^T$? Here, $L(\mathbf{w}', \alpha)$ is the Lagrangian function for this optimization problem evaluated at $\mathbf{w}'$.

$\max_{\alpha \ge 0} \ L(\mathbf{w}', \alpha)$

You are working on a spam email classification problem. You have collected a dataset of some 1000 emails, labeled as either "Spam" or "Not Spam," and you want to build a decision tree to classify them based on certain features. One of the features is the presence of a specific keyword: "Won."

Here's the breakdown of emails containing or not containing the keyword: "Won."

"Won" keyword present: 300 Spam, 50 Not Spam
"Won" keyword not present: 100 Spam, 550 Not Spam

Calculate the information gain for selecting the "Won" keyword feature for splitting.

Which of the following statements is/are true about the K-NN algorithm?

You are working on a binary classification problem where you need to predict whether an email is spam (1) or not (0). You have trained a machine learning model that produces predicted probabilities for each email being spam. You have two different loss functions to consider: the 0-1 loss and the squared loss.

You have a test dataset with the following true labels and predicted probabilities for a set of emails:

For the given data set if Predicted probability is greater than 0.5, the predicted label will be 1 and 0 otherwise.

Consider a simple neural network with one hidden layer. The network has the following architecture:

Input layer with 3 neurons.
Hidden layer with 2 neurons, using the sigmoid activation function.
Output layer with 1 neuron, using the linear activation function.

The weights and biases for the network are as follows:

Hidden Layer:

Neuron 1:
Weights: [0.5, -0.2, 0.8]
Bias: 0.1

Neuron 2:
Weights: [-0.3, 0.6, -0.7]
Bias: -0.4

Output Layer:

Neuron 1:
Weights: [0.2, 0.4]
Bias: -0.3

Assume that the input values are [0.6, 0.3, 0.8].

Calculate output of Neuron 1 in hidden layer