[SOLVED] Cse342 sml assignment 1

Q1. Consider two Cauchy distributions in one dimension
p(x|ωi) = 1
πb
1
1 + ( x−ai
b
)
2
, i = 1, 2
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) = R
x
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]
X =


1 0 0
−1 0 1
0 1 1


Here, X ∈ Rd×N form.
Q3.a. In multi-category case, probability of error p(error) is given as 1-
p(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:
p(x|ωi) = 1
πb
1
1 + ( x−ai
b
)
2
, i = 1, 2
Assume P(ω1) = P(ω2) and 0-1 loss. [1]