CSE342 Assignment 1 Solution

Assignment
Q1. Consider two Cauchy distributions in one dimension

Assume P(ω1) = P(ω2).
Find the total probability of error. Note you need to first obtain decision boundary using p(ω1|x) = p(ω2|x). Then determine the regions where error occurs and then use p(error) = Rx p(error|x)p(x)dx. Plot the the conditional likelihoods, p(x|ωi)p(ωi), and mark the regions where error will occur. This shall be rough hand-drawn sketch. As p(x) is same when equating posteriors,
we can simply use p(x|ωi)p(ωi). [1]
Q2. Compute the unbiased covariance matrix: [0.5]

Here, X ∈ Rd×N form.
Q3.a. In multi-category case, probability of error p(error) is given as 1p(correct), where p(correct) is the probability of being correct. Consider a case of 3 classes or categories. Draw a rough sketch of p(x|ωi)p(ωi) ∀i = 1,2,3. Give an expression for p(error). Assume equi-probable priors for simplicity. [1]
b. Mark the regions if the three conditional likelihoods are Gaussians p(x|ωi) N(µi,1), µ1 = −1,µ2 = 0,µ3 = 1. Find the p(error) in terms of CDF of standard distribution. [1]
Q4. Find the likelihood ratio test for following Cauchy pdf:

Assume P(ω1) = P(ω2) and 0-1 loss. [1]
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