## NPTEL Introduction to Machine Learning Week 3 Assignment Answers 2024

1. Which of the following statement(s) about decision boundaries and discriminant functions of classifiers is/are true?

- In a binary classification problem, all points x on the decision boundary satisfy δ
_{1}(x)=δ_{2}(x) - In a three-class classification problem, all points on the decision boundary satisfy δ
_{1}(x)=δ_{2}(x)=δ_{3}(x) - In a three-class classification problem, all points on the decision boundary satisfy at least one of δ
_{1}(x)=δ_{2}(x),δ_{2}(x)=δ_{3}(x)orδ_{3}(x)=δ_{1}(x). - Let the input space be R
^{n}. If x does not lie on the decision boundary, there exists an ϵ>0 such that all inputs y satisfying ||y−x||<ϵ belong to the same class.

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2. The following table gives the binary ground truth labels y_{i} for four input points xi

(not given). We have a logistic regression model with some parameter values that computes the probability p(x_{i}) that the label is 1. Compute the likelihood of observing the data given these model parameters.

- 0.346
- 0.230
- 0.058
- 0.086

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3. Which of the following statement(s) about logistic regression is/are true?

- It learns a model for the probability distribution of the data points in each class.
- The output of a linear model is transformed to the range (0, 1) by a sigmoid function.
- The parameters are learned by optimizing the mean-squared loss.
- The loss function is optimized by using an iterative numerical algorithm.

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4. Consider a modified form of logistic regression given below where k is a positive constant and β_{0}andβ_{1} are parameters.

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5. Consider a Bayesian classifier for a 3-class classification problem. The following tables give the class-conditioned density f_{k}(x) for three classes k=1,2,3 at some point x

in the input space.

Note that π_{k }denotes the prior probability of class k. Which of the following statement(s) about the predicted label at x is/are true?

- If the three classes have equal priors, the prediction must be class 2
- If π
_{3}<π_{2}andπ_{1}<π_{2,}the prediction may not necessarily be class 2 - If π
_{1}>2π_{2,}the prediction could be class 1 or class 3 - If π
_{1}>π_{2}>π_{3}, the prediction must be class 1

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7. Which of the following statement(s) about a two-class LDA model is/are true?

- It is assumed that the class-conditioned probability density of each class is a Gaussian
- A different covariance matrix is estimated for each class
- At a given point on the decision boundary, the class-conditioned probability densities corresponding to both classes must be equal
- At a given point on the decision boundary, the class-conditioned probability densities corresponding to both classes may or may not be equal

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9. Which of the following statement(s) about LDA is/are true?

- It minimizes the between-class variance relative to the within-class variance
- It maximizes the between-class variance relative to the within-class variance
- Maximizing the Fisher information results in the same direction of the separating hyperplane as the one obtained by equating the posterior probabilities of classes
- Maximizing the Fisher information results in a different direction of the separating hyperplane from the one obtained by equating the posterior probabilities of classes

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10. Which of the following statement(s) regarding logistic regression and LDA is/are true for a binary classification problem?

- For any classification dataset, both algorithms learn the same decision boundary
- Adding a few outliers to the dataset is likely to cause a larger change in the decision boundary of LDA compared to that of logistic regression
- Adding a few outliers to the dataset is likely to cause a similar change in the decision boundaries of both classifiers
- If the within-class distributions deviate significantly from the Gaussian distribution, logistic regression is likely to perform better than LDA

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