Machine Learning Techniques Quiz 2: 23 November 2025 (September 2025 term)

Based on the above data, answer the given subquestions.

Which of the following equations correctly represents the maximum likelihood problem for

estimating
?

Based on the above data, answer the given subquestions.

Derive the maximum likelihood estimate of the parameter
in terms of the training examples

.

Consider the following dataset with 4 features $\{f_1, f_2, f_3, f_4\}$ and three labels $y = 0, -1, +1$. Assume the features to be binary. Answer the given subquestions under the Naive Bayes assumptions.

Find the maximum likelihood estimate
and
of
,
, and , respectively.

Consider the following dataset with 4 features $\{f_1, f_2, f_3, f_4\}$ and three labels $y = 0, -1, +1$. Assume the features to be binary. Answer the given subquestions under the Naive Bayes assumptions.

What class would
belong to? Assume that no Laplacian smoothing has been done.

Consider the following dataset:

Suppose we fit a linear regression model of the form $\hat{y} = w_1x_1 + w_2x_2$. Compute the mean squared error for the training data. Enter the answer correct to two decimal places.

Consider the following trained decision tree learned for the dataset in Figure 1.

Find the training error of the decision tree for the given dataset. Enter the answer correct to one decimal place.

Consider the email filtering problem. Suppose an email can be either spam or ham with probabilities $p_\Theta(1)$ and $p_\Theta(2)$ respectively. Let $\{u_1, u_2, \ldots, u_n\}$ denote a collection of special words whose appearance suggests that the message is likely to be spam. Define the random variable $X_i$ as

$X_i = \begin{cases} 1, & \text{if the word } u_i \text{ appears in the message,} \\ 0, & \text{otherwise.} \end{cases}$

Assume that the features $X_1, X_2, \ldots, X_n$ are conditionally independent given the class label $\Theta$. Assume that the probability of an email being spam is 0.4. Suppose there are three special words with the following conditional probabilities:

A new email represented as $X = (X_1, X_2, X_3)$ contains the words $u_1$ and $u_3$, but not $u_2$, then compute $P(\Theta = 1 \mid X)$. Enter the answer correct to two decimal places.

Consider a dataset of 1000 points for a classification problem using k-NN algorithm. Select the correct statements from the following:

Suppose we observe data sampled from the model
. Since
is unknown, we

would like to estimate it using linear regression. Adding priors to the parameter
, i.e.

is equivalent to adding a
penalty on the parameters in the objective function defined by the log-likelihood.

Consider a training dataset of $n$ points for a regression problem. Suppose that the model is linear. Let $w_1$ and $w_2$ be the optimal weight obtained from solving the following optimization problems:

$w_1 = \text{arg min}_w \sum_{i=1}^{n} (w^Tx_i - y_i)^2$

$w_2 = \text{arg min}_w \sum_{i=1}^{n} (w^Tx_i - y_i)^3$

Which among the following will be true?

Consider four regression models (A, B, C, and D) fitted on the same dataset using Ridge or Lasso regression with different regularization parameters.
Match each model with its corresponding regularization description given below.
Models:

A. Ridge regression with

B. Ridge regression with

C. Lasso regression with

D. Ridge regression with
Regularization Effects:
1. No regularization — same as Ordinary Least Squares (OLS)
2. Strong shrinkage — coefficients are small but non-zero
3. Moderate shrinkage — coefficients are reduced but not zero
4. Some coefficients are exactly zero due to sparsity.
Which of the following correctly matches the models with their effects?

Consider a linear regression problem for the dataset
, where and

for all
. Let
be the least square solution for this problem. Then, which of the following will hold?