[SOLVED] SML Assignment 1

Description

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

Decide ω₁ if x > θ; otherwise decide ω₂.

• 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 θ satisfy p(θ|ω₁)P(ω₁) = p(θ|ω₂)P(ω₂)

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

2                                      (2)

Assuming P(ω₁) = P(ω₂), show that P(ω₁|x) = P(ω₂|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(µ,Σ) in d dimensions.

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

normal distribution with Σ = σ²I and prior probability P(ω_i).

1. Compare the discriminant function’s values for two different distributions

) and    = 2 dimensions.

Assume the test sample to be and P(ω₁) = 1/3 and P(ω₂) = 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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