Problem $1$:

Consider an HMM model with $5$ active states describing the weather at noon with following states: Sunny$=$S$1,$ Cloudy $=S2,$ Raining $=S3,$ and Sandstorms $=S4.$ The initial state probability as follows: $=[0\mathrm{.}4,0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}3].$ Also, the transition matrix as following:

$A=\left[\begin{array}{cccc}0\mathrm{.}4& 0\mathrm{.}3& 0\mathrm{.}2& 0\mathrm{.}1\\ 0\mathrm{.}5& 0\mathrm{.}3& 0\mathrm{.}1& 0\mathrm{.}1\\ 0\mathrm{.}15& 0\mathrm{.}15& 0\mathrm{.}5& 0\mathrm{.}2\\ 0\mathrm{.}05& 0\mathrm{.}05& 0\mathrm{.}2& 0\mathrm{.}7\end{array}\right]$

In this model, we can consider three system observations: Umbrella$=U,$Coat$=C,$ and Mask $=M.$

$B_{s1}=\left[\begin{array}{c}P(U|S1)\\ P(C|S1)\\ P(M|S1)\end{array}\right]=\left[\begin{array}{c}0\mathrm{.}6\\ 0\mathrm{.}25\\ 0\mathrm{.}15\end{array}\right],B_{s2}=\left[\begin{array}{c}P(U|S2)\\ P(C|S2)\\ P(M|S2)\end{array}\right]=\left[\begin{array}{c}0\mathrm{.}4\\ 0\mathrm{.}3\\ 0\mathrm{.}3\end{array}\right],B_{s3}=\left[\begin{array}{c}P(U|S3)\\ P(C|S3)\\ P(M|S3)\end{array}\right]=\left[\begin{array}{c}0\mathrm{.}5\\ 0\mathrm{.}25\\ 0\mathrm{.}25\end{array}\right],B_{s4}=\left[\begin{array}{c}P(U|S4)\\ P(C|S4)\\ P(M|S4)\end{array}\right]=\left[\begin{array}{c}0\mathrm{.}1\\ 0\mathrm{.}2\\ 0\mathrm{.}7\end{array}\right]$

Answer the following:

a$)$ Draw this HMM model showing all states, transitions and initial states.

b$)$ What is the most probable initial state in this model?

c$)$ Is the model Bakis network? Justify your answer.

d$)$ How likely is the observation sequence CM $?$

e$)$ Compute the following $P(S3S2S4S1S3)?$

f$)$ Compute the following probability $P(\frac{C}{M}UU|S1S3S2S4)?$