Requirements: $$ We recommend that you typeset your homework using appropriate software such as LATEX. If you submit your handwritten version, please make sure it is cleanly written up and legible. The TAs will not invest undue effort to decrypt bad handwritings. $.$ Please finish your homework independently. However, feel free to discuss with others, but never directly copy$-$paste the answers. In addition, you should write in your homework the set of people with whom you collaborated. $=1$ MLE for Gaussian distribution $[5$pt$]$ Derive the maximum$-$likelihood estimator for the parameter ji $(13)$:$=$and $=(13).$j$=1$ of a gaussian distribution from the observed data $\{2$; E Rd: i$=1,...,$m$\}$: $1$ P$(\backslash$alpha $\backslash$mu $,\backslash$Sigma $)=$exp$(-$; $(2$ u$)?2-12)=(1)$ V$(2\backslash$pi $)\backslash$Iota $\backslash$Sigma $|-1$ Hint: You shall be maximizing the log$-$likelihood with regard to u and $.$ If you think that the matrix computation is difficult, you might first solve it for scaler case where IjH, O ER$.2$ EM for mixture of Bernoulli distributions $[5$pt$]$ We have covered mixture of Gaussians in class, let's now consider mixtures of discrete binary variables described by Bernoulli distributions. Consider a set of D binary variables Ii$,$ where i $=1,...,$D$,$ which is governed by a mixture of Bernoulli distributions K p$($x$|\backslash$mu $,\backslash$pi $)=\backslash$Sigma $\backslash$pi $\backslash$iota $\backslash$Rho $($x$|\backslash$mu $\backslash$kappa $),(2)$ Tap k$=1$ where x $=(11,...,$ID$),$ u $=\{$x$1,...,$K$,7=\{$T$1,...,$ TK$\},$ and p$($x$|$px$)=111$: $(1$ wri$)(2-21).(3)$ i$=1$ Given the dataset x $=\{$x$(1),$x$(2),...,$x$($N$)\},$ please design an EM algorithm to learn the parameters $\{,$ u$\}$ of this mixture model. Hint: The equations in the algorithm can be a little bit complicated, but don't be afraid! You may follow the steps in the lecture slide of the EM algorithm for mixture of Guassians.