1In the context of supervised learning, what is the primary goal of Classification?
A.To group similar data points together without predefined labels.
B.To map input variables to discrete output categories or classes.
C.To reduce the dimensionality of the dataset.
D.To predict a continuous numerical value based on input features.
Correct Answer: To map input variables to discrete output categories or classes.
Explanation:
Classification involves identifying which of a set of categories (sub-populations) a new observation belongs to, on the basis of a training set of data containing observations (or instances) whose category membership is known. This distinguishes it from regression, which predicts continuous values.
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2What does a Decision Boundary represent in a classification problem?
A.The boundary where the training data ends and the testing data begins.
B.The maximum error rate acceptable for the model.
C.The limit of the computational power required to train the model.
D.A hypersurface that partitions the underlying vector space into two or more sets, one for each class.
Correct Answer: A hypersurface that partitions the underlying vector space into two or more sets, one for each class.
Explanation:
A decision boundary is the hypersurface (line in 2D, plane in 3D, etc.) that separates different classes in the feature space. On one side of the boundary, data points are classified as one class, and on the other side, a different class.
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3Consider a linear classifier in a 2-dimensional feature space defined by . What geometric shape is the decision boundary?
A.A circle
B.A parabola
C.A hyperbola
D.A straight line
Correct Answer: A straight line
Explanation:
In 2D space, a linear equation of the form represents a straight line. In 3D it is a plane, and in higher dimensions, it is a hyperplane.
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4For a Linear Classifier, the decision rule is often given by . What is the role of the bias term ?
A.It translates the decision boundary away from the origin.
B.It rotates the decision boundary around the origin.
C.It scales the length of the weight vector .
D.It creates non-linear curves in the boundary.
Correct Answer: It translates the decision boundary away from the origin.
Explanation:
The weight vector determines the orientation of the decision boundary, while the bias term allows the decision boundary to be offset (translated) from the origin. Without , the boundary would always pass through the origin.
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5In the One-vs-All (One-vs-Rest) strategy for multi-class classification with classes, how many binary classifiers are trained?
A.
B.
C.
D.
Correct Answer:
Explanation:
In One-vs-All, we train separate binary classifiers. The -th classifier distinguishes class from all other classes combined.
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6In the One-vs-One strategy for multi-class classification with classes, how many binary classifiers are trained?
A.
B.
C.
D.
Correct Answer:
Explanation:
One-vs-One builds a classifier for every pair of classes. The number of unique pairs in classes is given by the combination formula .
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7When using the One-vs-One strategy, how is the final classification decision typically made for a new data point?
A.By a voting scheme where the class with the most 'wins' is selected.
B.By averaging the regression outputs of all classifiers.
C.By choosing the class that was trained last.
D.By selecting the class with the highest probability from a single classifier.
Correct Answer: By a voting scheme where the class with the most 'wins' is selected.
Explanation:
In OvO, each of the classifiers casts a vote for one of the two classes it distinguishes. The class that receives the maximum number of votes is predicted.
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8Which of the following is a potential disadvantage of the One-vs-All strategy when classes are imbalanced?
A.The datasets for the binary classifiers become heavily skewed (e.g., 1 vs 99 others).
B.It requires too many classifiers to be trained.
C.It cannot handle linear decision boundaries.
D.It is computationally more expensive than One-vs-One during inference.
Correct Answer: The datasets for the binary classifiers become heavily skewed (e.g., 1 vs 99 others).
Explanation:
In OvA, the negative class includes samples from all other classes. If there are many classes, the negative set will vary largely outnumber the positive set for a specific class classifier, leading to class imbalance problems during training.
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9In Bayes Theorem, given by , what is the term called?
A.Evidence
B.Likelihood
C.Posterior
D.Prior
Correct Answer: Likelihood
Explanation:
is the Likelihood (or Class-Conditional Probability), representing the probability of observing feature given that the class is .
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10In Bayes Theorem, , what does represent?
A.The probability of the data occurring regardless of the class.
B.The probability of the class after observing the data (Posterior).
C.The probability of the class before observing any data (Prior).
D.The conditional dependence of on .
Correct Answer: The probability of the class before observing any data (Prior).
Explanation:
is the Prior probability, reflecting our initial belief about the probability of class occurring before seeing any specific features.
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11What is the fundamental assumption of the Naïve Bayes classifier?
A.The prior probabilities of all classes are equal.
B.Features are conditionally independent given the class label.
C.All features contribute equally to the decision boundary regardless of the class.
D.Features are dependent on each other given the class label.
Correct Answer: Features are conditionally independent given the class label.
Explanation:
The 'Naïve' assumption is that the value of a particular feature is independent of the value of any other feature, given the class variable. Mathematically: .
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12Which of the following equations represents the decision rule for a Naïve Bayes classifier (ignoring the evidence as it is constant for all classes)?
A.
B.
C.
D.
Correct Answer:
Explanation:
According to the MAP (Maximum A Posteriori) estimation and the independence assumption, we select the class that maximizes the product of the prior and the individual likelihoods .
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13In a Naïve Bayes classifier, what is the Zero Frequency Problem?
A.When the entire dataset has zero variance.
B.When a feature value appears in the test set but was never observed with a specific class in the training set, resulting in a zero likelihood.
C.When the prior probability of a class is zero.
D.When the computation results in a divide-by-zero error during normalization.
Correct Answer: When a feature value appears in the test set but was never observed with a specific class in the training set, resulting in a zero likelihood.
Explanation:
If a categorical variable has a category in the test data that was not observed in the training data for a specific class, the probability becomes 0. Since probabilities are multiplied, this makes the entire posterior probability 0.
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14What technique is commonly used to solve the Zero Frequency Problem in Naïve Bayes?
Laplace smoothing adds a small count (usually 1 or ) to each variable count in the training data to ensure that no probability is ever exactly zero.
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15Which variation of Naïve Bayes is most appropriate when feature values are continuous and assumed to follow a normal distribution?
A.Multinomial Naïve Bayes
B.Gaussian Naïve Bayes
C.Bernoulli Naïve Bayes
D.Poisson Naïve Bayes
Correct Answer: Gaussian Naïve Bayes
Explanation:
Gaussian Naïve Bayes assumes that the continuous values associated with each class are distributed according to a Gaussian (Normal) distribution.
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16In Bayesian Decision Theory, the concept of Risk is defined as:
A.The computational complexity of the algorithm.
B.The probability of choosing the wrong class.
C.The expected loss associated with a decision rule.
D.The inverse of the likelihood function.
Correct Answer: The expected loss associated with a decision rule.
Explanation:
Risk is the expected value of the loss function. Ideally, a decision rule is chosen to minimize this total risk.
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17If we use the Zero-One Loss function (loss is 0 for correct classification, 1 for incorrect), minimizing the Risk is equivalent to:
A.Maximizing the entropy.
B.Minimizing the probability of error.
C.Maximizing the likelihood.
D.Minimizing the squared error.
Correct Answer: Minimizing the probability of error.
Explanation:
With Zero-One loss, the risk corresponds exactly to the probability of misclassification. Therefore, the optimal strategy (Bayes decision rule) is to select the class with the highest posterior probability to minimize error.
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18Given the formula for Gaussian Naïve Bayes likelihood: , what parameters need to be estimated from the training data?
A.The median and mode.
B.The min and max values of .
C.The weights and bias .
D.The mean and variance for each class and feature.
Correct Answer: The mean and variance for each class and feature.
Explanation:
Gaussian Naïve Bayes parametrizes the likelihood using the mean and variance of the feature specific to class .
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19Why is the Naïve Bayes classifier considered a Generative Model?
A.Because it directly learns the decision boundary without modeling densities.
B.Because it generates new training data to balance classes.
C.Because it models the joint probability (via ) and captures how the data is generated.
D.Because it uses genetic algorithms for optimization.
Correct Answer: Because it models the joint probability (via ) and captures how the data is generated.
Explanation:
Generative models learn the distribution of individual classes (the likelihood and prior) and can essentially 'generate' samples from the learned distribution. Discriminative models (like Logistic Regression) model directly.
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20In the context of multi-class classification, if the decision regions are separated by linear boundaries, the classifier is known as:
A.A Linear Classifier
B.A Quadratic Classifier
C.A Non-parametric Classifier
D.A Decision Tree
Correct Answer: A Linear Classifier
Explanation:
A linear classifier makes decisions based on the value of a linear combination of the characteristics. This results in decision boundaries that are hyperplanes (linear).
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21Which version of Naïve Bayes is best suited for binary feature vectors (e.g., word presence/absence in text classification)?
A.Bernoulli Naïve Bayes
B.Linear Naïve Bayes
C.Multinomial Naïve Bayes
D.Gaussian Naïve Bayes
Correct Answer: Bernoulli Naïve Bayes
Explanation:
Bernoulli Naïve Bayes is designed for binary/boolean features. It explicitly models the absence of terms as well as their presence.
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22What is MAP estimation in the context of Bayesian classification?
A.Maximum Average Precision
B.Minimum Absolute Posteriori
C.Mean Average Probability
D.Maximum A Posteriori
Correct Answer: Maximum A Posteriori
Explanation:
MAP stands for Maximum A Posteriori. It is the method of estimating the variable (class) that maximizes the posterior probability .
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23How does Maximum Likelihood (ML) estimation differ from MAP estimation?
A.There is no difference; they are identical.
B.MAP assumes a uniform likelihood, ML calculates likelihood.
C.ML is for regression, MAP is for classification.
D.ML assumes a uniform prior (or ignores the prior), while MAP accounts for the prior .
Correct Answer: ML assumes a uniform prior (or ignores the prior), while MAP accounts for the prior .
Explanation:
MAP maximizes , whereas ML maximizes only . If the prior is uniform (all classes equally likely), MAP reduces to ML.
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24What is the computational complexity of predicting a class for a single instance using Naïve Bayes with features and classes?
A.
B.
C.
D.
Correct Answer:
Explanation:
For each of the classes, we multiply probabilities (one for each feature). Thus, the complexity is linear with respect to both the number of classes and features.
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25In Bayesian Decision Theory, typically denotes:
A.The likelihood of feature .
B.The learning rate.
C.The loss incurred by taking action when the true state of nature is .
D.The probability of class .
Correct Answer: The loss incurred by taking action when the true state of nature is .
Explanation:
This notation represents the Loss Function matrix, quantifying the cost of making a specific decision (action) given the actual ground truth.
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26Which of the following is true regarding the Decision Boundary of a Gaussian Naïve Bayes classifier if all classes share the same covariance matrix?
A.The boundary is linear.
B.The boundary is quadratic.
C.There is no decision boundary.
D.The boundary is circular.
Correct Answer: The boundary is linear.
Explanation:
If the covariance matrices are identical for all classes, the quadratic terms in the discriminant functions cancel out, leaving a linear function of . Thus, it behaves like a linear classifier (similar to Linear Discriminant Analysis).
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27Why do we often work with Log-Probabilities (sums of logs) instead of direct probabilities (products) in Naïve Bayes?
A.Because log probabilities are required by the Bayes theorem definition.
B.To prevent numerical underflow when multiplying many small probabilities.
C.Because logs are always positive.
D.To make the math harder.
Correct Answer: To prevent numerical underflow when multiplying many small probabilities.
Explanation:
Probabilities are . Multiplying many of them results in extremely small numbers that can vanish (underflow) in floating-point arithmetic. Summing logs () avoids this and preserves the ranking order (since log is monotonic).
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28Which of the following text classification scenarios is Multinomial Naïve Bayes typically used for?
A.When the text length is infinite.
B.When features represent the presence/absence of words (binary).
C.When features are continuous word embeddings.
D.When features represent word counts or term frequencies.
Correct Answer: When features represent word counts or term frequencies.
Explanation:
Multinomial NB models the distribution of counts (frequencies) of events (words) generated from a multinomial distribution.
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29A classifier that distinguishes between 'Spam' and 'Not Spam' is an example of:
A.Binary Classification
B.Reinforcement Learning
C.Clustering
D.Regression
Correct Answer: Binary Classification
Explanation:
There are exactly two possible output classes, making it a binary classification problem.
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30In the formulation , if and , which side of the boundary does the point fall on?
A.Positive side ()
B.Negative side ()
C.On the boundary ()
D.Undefined
Correct Answer: Positive side ()
Explanation:
Substitute the values: . Since , it falls on the positive side.
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31In Bayesian Decision Theory, the Evidence acts as a:
A.Prior belief about the feature distribution.
B.Loss function.
C.Weighting factor for the likelihood.
D.Normalization constant to ensure probabilities sum to 1.
Correct Answer: Normalization constant to ensure probabilities sum to 1.
Explanation:
The evidence scales the numerator () so that the posterior probabilities over all classes sum to 1. It does not affect the ranking of classes.
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32Which of the following statements about Decision Regions is correct?
A.Decision regions are only defined for training data.
B.Decision regions must always be convex.
C.The union of all decision regions must cover the entire feature space.
D.Decision regions can never be disjoint.
Correct Answer: The union of all decision regions must cover the entire feature space.
Explanation:
A classifier assigns every point in the feature space to a class (or a reject option), so the regions partition the space.
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33The One-vs-One strategy generally requires more space to store models than One-vs-All ( vs ). Why might it still be preferred?
A.It guarantees 100% accuracy.
B.Each individual classifier is trained on a smaller subset of data (only two classes), potentially making training faster.
C.It is the only method that supports Neural Networks.
D.It does not require labels.
Correct Answer: Each individual classifier is trained on a smaller subset of data (only two classes), potentially making training faster.
Explanation:
Although there are more classifiers, each one only looks at data from two classes. For algorithms that scale super-linearly with data size (like SVMs), training many small models is often faster than training fewer large models (OvA).
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34In a probabilistic classifier, if and , and the loss for misclassifying class 1 is much higher than misclassifying class 2, Bayesian Decision Theory might suggest:
A.Choosing class 1 regardless of cost.
B.Choosing the class with the highest probability always.
C.Refusing to classify.
D.Choosing class 2 if the expected risk is lower, even if the probability is lower.
Correct Answer: Choosing class 2 if the expected risk is lower, even if the probability is lower.
Explanation:
Bayesian Decision Theory minimizes Risk, not just error rate. If the cost (loss) of a specific error is very high, the optimal decision might be to select a less probable class to avoid that high cost.
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35What is the vector in a linear classifier geometrically orthogonal to?
A.The data points.
B.The decision boundary (hyperplane).
C.The y-axis.
D.The x-axis.
Correct Answer: The decision boundary (hyperplane).
Explanation:
The weight vector is the normal vector to the separating hyperplane defined by .
This is the formula for calculating the probability of a word given a class in Multinomial Naïve Bayes with Laplace smoothing (Add-one smoothing).
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37Why does the Naïve Bayes independence assumption often work well in practice even when features are somewhat dependent?
A.Because the algorithm corrects the dependencies during training.
B.Because classification relies on the correct sign/ranking of the posterior, not the exact probability value.
C.Because real-world data is always independent.
D.Because dependencies cancel each other out.
Correct Answer: Because classification relies on the correct sign/ranking of the posterior, not the exact probability value.
Explanation:
Even if the probability estimates are biased due to the independence assumption, the classifier will still classify correctly as long as the correct class remains the one with the highest (albeit inaccurate) probability score.
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38If a classifier produces a probability , it is known as a:
A.Hard Classifier
B.Deterministic Classifier
C.Regressive Classifier
D.Soft (Probabilistic) Classifier
Correct Answer: Soft (Probabilistic) Classifier
Explanation:
Soft classifiers output the probabilities of class membership, whereas hard classifiers output just the label.
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39In the context of Bayes Theorem, if the Prior is uniform for all classes, the MAP estimate is equivalent to maximizing:
A.The Variance
B.The Likelihood
C.The Evidence
D.The Loss Function
Correct Answer: The Likelihood
Explanation:
If is constant, maximizing is the same as maximizing .
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40What is the dimension of the decision boundary for a binary classification problem with 10 input features?
A.1
B.9
C.2
D.10
Correct Answer: 9
Explanation:
The decision boundary is a hyperplane of dimension , where is the dimension of the feature space. Here .
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41Which of the following is NOT a property of a Linear Classifier?
A.Simple to interpret (weights indicate feature importance).
B.Computationally efficient (fast inference).
C.Can learn complex XOR relationships directly without feature engineering.
D.Less prone to overfitting compared to high-degree polynomial classifiers.
Correct Answer: Can learn complex XOR relationships directly without feature engineering.
Explanation:
Linear classifiers cannot solve the XOR problem because the XOR classes are not linearly separable.
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42When applying Naïve Bayes, how is the Prior usually estimated from training data?
A.Correlation coefficient of class .
B.It is always set to 0.5.
C.Fraction of training samples belonging to class .
D.Average value of features for class .
Correct Answer: Fraction of training samples belonging to class .
Explanation:
, where is the count of samples in class and is total samples.
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43Which statement best describes the Bayes Error Rate?
A.The rate at which the algorithm converges.
B.The error rate when is ignored.
C.The error rate of a Naïve Bayes classifier.
D.The lowest possible error rate for any classifier on a given distribution.
Correct Answer: The lowest possible error rate for any classifier on a given distribution.
Explanation:
The Bayes Error Rate is the irreducible error inherent in the data distribution (due to noise/overlap of classes). No classifier can outperform the theoretical Bayes Optimal Classifier.
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44In a 3-class problem using One-vs-Rest, if the outputs of the three classifiers for a point are , , , which class is predicted?
A.Class 3
B.None
C.Class 1
D.Class 2
Correct Answer: Class 3
Explanation:
In One-vs-Rest, we choose the class corresponding to the classifier with the highest confidence score (signed distance). $1.5$ is the maximum.
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45If features and are duplicates (), how does this affect Naïve Bayes?
A.It causes a division by zero.
B.It has no effect.
C.It violates the independence assumption and 'double counts' the importance of that feature.
D.It improves accuracy by reinforcing the signal.
Correct Answer: It violates the independence assumption and 'double counts' the importance of that feature.
Explanation:
Naïve Bayes multiplies probabilities. If a feature is duplicated, its probability is multiplied twice, effectively squaring it and giving that feature undue weight (double counting), which skews the posterior.
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46What is a Reject Option in classification?
A.Deleting features that are not useful.
B.Rejecting the null hypothesis.
C.Refraining from making a prediction if the posterior probability is below a certain threshold.
D.Removing outliers from the training set.
Correct Answer: Refraining from making a prediction if the posterior probability is below a certain threshold.
Explanation:
In ambiguity (low confidence in all classes), a classifier may choose to 'reject' the input (e.g., send it for human review) rather than making a likely incorrect guess.
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47Geometrically, what does the likelihood in a Gaussian Naïve Bayes represent?
A.The probability of the class .
B.The volume of the dataset.
C.The density of the data point within the cluster of class .
D.The distance of from the decision boundary.
Correct Answer: The density of the data point within the cluster of class .
Explanation:
For a Gaussian distribution, the likelihood indicates how close the point is to the mean of the class distribution, scaled by the variance.
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48Which of the following is an example of a Discriminative approach to classification?
A.Gaussian Mixture Models
B.Hidden Markov Models
C.Logistic Regression
D.Naïve Bayes
Correct Answer: Logistic Regression
Explanation:
Logistic Regression models the posterior directly (Discriminative), whereas the others listed are Generative models (modeling ).
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49In the context of the One-vs-One strategy, if there is a tie in the voting (e.g., class A and class B both get same number of votes), how is it typically resolved?
A.The process is restarted.
B.Random selection or based on highest aggregate confidence score.
C.Both classes are returned.
D.The model crashes.
Correct Answer: Random selection or based on highest aggregate confidence score.
Explanation:
Ties are usually broken by either summing the raw decision function values (confidence) for the tied classes or simply picking one randomly/lexicographically.
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50What happens to the decision boundary in a Linear Classifier if we multiply all weights and bias by a positive constant ?
A.The boundary shifts.
B.The boundary remains unchanged.
C.The boundary becomes non-linear.
D.The boundary rotates.
Correct Answer: The boundary remains unchanged.
Explanation:
The equation is equivalent to (since $c
eq 0$). The hyperplane geometry is identical, though the magnitude of the output scores changes.