Problem $4.$ Expectation Maximization $(25$pts$).$ Consider a data set of binary $($black and white$)$

images. Each image is arranged into a vector of pixels by concatenating the columns of pixels in the

image.

The data set has $N$ images $\{x^{(1)},$dots,$x^{(N)}\}$ and each image has $D$ pixels, where $D$ is $($number of rows $x$

number of columns$)$ in the image. For example, image $x^{(n)}$ is a vector $(x_1^{(n)},$dots,$x_D^{(n)})$ where $x_d^{(n)}in$

$\{0,1\}$ for all nin$\{1,$dots,$N\}$ and din$\{1,$dots,$D\}.$

Write down the likelihood for a model consisting of a mixture of $K$ multivariate Bernoulli distributions.

Use the parameters ${}_1,$dots,${}_K$ to denote the mixing proportions $(\{$:$0{}_{k}1$;${\displaystyle \underset{k}{\overset{?}{}}}{}_k=1)$ and arrange

the $K$ Bernoulli parameter vectors into a matrix $P$ with elements $p_{kd}$ denoting the probability that pixel

$d$ takes value $1$ under mixture component $k.$Expectation Maximization $(25$pts$).$ Consider a data set of binary $($black and white$)$ images. Each image is arranged into a vector of pixels by concatenating the columns of pixels in the image. The data set has N images $\{$x$(1),...,$ x$($N$)\}$ and each image has D pixels, where D is $($number of rows X number of columns$)$ in the image. For example, image x$($n$)$ is a vector $($x$1($n$),...,$ xD $($n$))$ where xd $($n$)$ in $\{0,1\}$ for all n in $\{1,...,$ N$\}$ and d in $\{1,...,$ D$\}.$ Write down the likelihood for a model consisting of a mixture of K multivariate Bernoulli distributions. Use the parameters $\backslash$pi $1,...,\backslash$pi K to denote the mixing proportions pi k ; $\backslash$pi kk $=1)$ and arrange the K Bernoulli parameter vectors into a matrix P with elements pkd denoting the probability that pixel d takes value $1$ under mixture component k$.$ "PLEASE REPLY WITH CORRECT ANSWER".