[Solved] SML Assignment 1

Q1. Consider the following decision rule for a two-category one-dimensional problem:

Decide ₁ if x > ; otherwise decide ₂.

• Show that the probability of error for this rule is given by

(1)

• By differentiating, show that a necessary condition to minimize P(error) isthat satisfy p(|₁)P(₁) = p(|₂)P(₂)

Q2. Let the conditional densities for a two-category one-dimensional problem be given by the Cauchy distribution

2 (2)

Assuming P(₁) = P(₂), show that P(₁|x) = P(₂|x) if x = (a₁ +a₂)/2, i.e., the minimum error decision boundary is a point midway between the peaks of the two distributions, regardless of b.

Q3. Suppose we have three equi-probable categories in two dimensions with the following underlying distributions:

By explicit calculation of posterior probabilities, classify the point x =

for minimum probability of error.

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Q4. a. Write a procedure to generate random samples according to a normal distribution N(,) in d dimensions.

1. Write a procedure to calculate the discriminant function for a given

normal distribution with = ²I and prior probability P(_i).

1. Compare the discriminant functions values for two different distributions

) and = 2 dimensions.

Assume the test sample to be and P(₁) = 1/3 and P(₂) = 2/3.

In a general process, you would be given several samples from two (or more) classes. Counting each class frequency will give the priors. With these samples as d dimensional vectors, you can estimate mean and covariance using MLE or other techniques, which is a part of later lecture. This computed info is sufficient for computing discriminants and thereby classifying the sample into one of the classes.

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