We will be working with the following data set for Problems $1$ and $2.$ There are two trinary features

$x_1$ and $x_2,$ and a class variable $y$ with values $0$ and $1.$

Problem $1$: Na$$ve Bayes $(20$ points$)$

The maximum likelihood class CPT is $Pr(Y)=(0\mathrm{.}5,0\mathrm{.}5).$ Estimate the feature CPTs

$Pr(x_1|Y)$ and $Pr(x_2|Y)$ without Laplace smoothing.

Which value$($s$)$ of $x_1$ give a direct prediction of $y$ regardless of the value of $x_2,$ and what are

the predictions of $y$ for each? Give the same analysis for values of $x_2.$ Are there any feature

combinations $(x_1,x_2)$ for which a prediction is not well defined?

Suppose that we find out that the class values $y$ in the data set may not be correct. We can

use the estimated probabilities in Part $1$ as a starting point for expectation$-$maximization.

Compute the expected counts $Pr(Y|x_1,x_2)$ for each sample $(x_1,x_2)$ in the data set.

Perform the maximization step and find the new CPTs $Pr(Y),Pr(x_1|Y),$ and $Pr(x_2|Y).$