$($a$)(3$ marks$)$ A training set consists of one dimensional examples from two classes. The training examples from class $1$ are $\{0\mathrm{.}5,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}3,0\mathrm{.}2,0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}35,0\mathrm{.}25\}$ and from class $2$ are $\{0\mathrm{.}9,0\mathrm{.}8,0\mathrm{.}75,1\mathrm{.}0\}.$ Fit a $($one dimensional$)$ Gaussian using Maximum Likelihood to each of these two classes. You can assume that the variance for class $1$ is $0\mathrm{.}0149,$ and the variance for class $2$ is $0\mathrm{.}0092.$ Also estimate the class probabilities $p_1$ and $p_2$ using Maximum Likelihood. What is the probability that the test point $x=0\mathrm{.}6$ belongs to class $1?$

$($b$)(3$ marks$)$ You are now going to make a text classifier. To begin with, you attempt to classify documents as either sport or politics. You decide to represent each document as a $($row$)$ vector of attributes describing the presence or absence of the following words.

$x=($goal$,$football,golf, defence,offence,wicket,office,strategy$)$

Training data from sport documents and from politics documents is represented below in a matrix in which each row represents the $8$ attributes.

xpolitics=[10111011001001100110100100110100011011

00011001]

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$1$

$0$

$x_{\text{s}\text{p}\text{o}\text{r}\text{t}}=\left[\begin{array}{ccccccccccccccccccccccccccccccccccccccccccc}1& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 0& 1& 0& 1& 0& 0& 0& 1& 1& 0& 1& 1& 0& 0& 0& 0& 0& 1& 0& 1& 0& 0\end{array}\right]$

$0$

$1$

$1$

$1$

$0$

Using a maximum likelihood naive Bayes classifier, what is the probability that the document $x=(1,0,0,1,1,1,1,0)$ is about politics?