In this question we consider clustering $1$D data with a mixture of $2$ Gaussians using the EM

algorithm. A GMM with $1$D data represents a distribution as

p$($x$)=$ X

K

k$=1$

$\backslash$pi kN $($x $|$k$,\backslash$sigma k$)$

with $\backslash$pi k the mixing coefficients, where: PK

k$=1\backslash$pi k $=1$ and $\backslash$pi k $>=0,$k$.$ And,

N $($x $|$k$,\backslash$sigma k$)=1$

q

$2\backslash$pi $\backslash$sigma $2$

k

exp$($

$($x $$k$)$

$2$

$2\backslash$sigma

$2$

k

$)$

$2$

You are given the l$-$D data points X $=\{1,10,20\}.$ Suppose the output of the E step is the following

matrix:

R $=$

$$

$$

$10$

$0\mathrm{.}40\mathrm{.}6$

$01$

$$

$$

where entry ri$,$c is the probability of obervation Xi belonging to cluster c $($the responsibility of

cluster c for data point i $).$ You just have to compute the M step. You may state the equations for

maximum likelihood estimates of these quantities $($which you should know from the lecture$)$ and

apply to this data set.

$(1)$ Write down the likelihood function you are trying to optimize.

$(2)$ After performing the M step for the mixing weights $\backslash$pi $1,\backslash$pi $2,$ what are the new values?

$(3)$ After performing the M step for the means $1$ and $2,$ what are the new values?