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 $=$

$10111011$

$00010011$

$10011010$

$01001101$

$00011011$

$00011001$

xsport $=$

$11000000$

$00100000$

$11010000$

$11010001$

$11011000$

$00010100$

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?