GATE 2025 DA Q.35: solution and answer

GATE 2025 · DAQ.35 · 1 mark · NAT

The naive Bayes classifier is used to solve a two-class classification problem with class-labels $y_{1}$ , $y_{2}$ . Suppose the prior probabilities are $P (y_{1}) = \frac{1}{3}$ and $P (y_{2}) = \frac{2}{3}$ . Assuming a discrete feature space with

P (x ∣ y_{1}) = \frac{3}{4} and P (x ∣ y_{2}) = \frac{1}{4}

for a specific feature vector $x$ . The probability of misclassifying $x$ is ______ (Round off to two decimal places)

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Answer and solution

ANSWER

The probability of misclassifying $x$ is $\frac{2}{5} = 0.40$ .

Official answer: 0.39 to 0.41Source: Official answer key, GATE 2025 DA (IIT Roorkee)

Naive Bayes assigns $x$ to the class with the larger posterior, $P (y_{k} ∣ x) \propto P (y_{k}) P (x ∣ y_{k})$ .
Unnormalised posterior for $y_{1}$ : $P (y_{1}) P (x ∣ y_{1}) = \frac{1}{3} \cdot \frac{3}{4} = \frac{1}{4}$ .
Unnormalised posterior for $y_{2}$ : $P (y_{2}) P (x ∣ y_{2}) = \frac{2}{3} \cdot \frac{1}{4} = \frac{1}{6}$ .
Evidence by total probability: $P (x) = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ .
Posteriors by Bayes' theorem: $P (y_{1} ∣ x) = \frac{1/4}{5/12} = \frac{3}{5}$ and $P (y_{2} ∣ x) = \frac{1/6}{5/12} = \frac{2}{5}$ . They sum to 1, as they should.
Since $\frac{3}{5} > \frac{2}{5}$ , the classifier predicts $y_{1}$ .
The probability of misclassifying $x$ is the posterior of the class not chosen: $P (y_{2} ∣ x) = \frac{2}{5} = 0.40$ .

KEY IDEA

The error probability of a Bayes classifier at a given $x$ is $1 - max_{k} P (y_{k} ∣ x)$ . Compute the posteriors by normalising the products of prior and likelihood, then take the smaller one.

WATCH OUT FOR

Using the likelihood $P (x ∣ y_{2}) = \frac{1}{4}$ (or the unnormalised value $\frac{1}{6}$ ) as the error probability. The likelihoods and the unnormalised posteriors do not sum to 1, so the evidence $P (x)$ must be divided out.

From the GATE 2025 DA paper. Every question of this paper · Download the full paper · All 4 Logistic, k-NN, naive Bayes questions, with notes

GATE 2025 DA Q.35: solution and answer

Answer and solution

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