Now compute these probabilities:

$P(O,x=HHH)=P(M|H)P(S|H)P(L|H)=0\mathrm{.}40\mathrm{.}10\mathrm{.}1=0\mathrm{.}004$

$P(O,x=HHC)=P(M|H)P(S|H)P(L|C)=0\mathrm{.}40\mathrm{.}10\mathrm{.}5=0\mathrm{.}02$

$P(O,x=HCH)=P(M|H)P(S|C)P(L|H)=0\mathrm{.}40\mathrm{.}70\mathrm{.}1=0\mathrm{.}028$

$P(O,x=HCC)=P(M|H)P(S|C)P(L|C)=0\mathrm{.}40\mathrm{.}70\mathrm{.}5=0\mathrm{.}14$

$P(O,x=CHH)=P(M|C)P(S|H)P(L|H)=0\mathrm{.}20\mathrm{.}10\mathrm{.}1=0\mathrm{.}002$

$P(O,x=CHC)=P(M|C)P(S|H)P(L|C)=0\mathrm{.}20\mathrm{.}10\mathrm{.}5=0\mathrm{.}01$

$P(O,x=CCH)=P(M|C)P(S|C)P(L|H)=0\mathrm{.}20\mathrm{.}70\mathrm{.}1=0\mathrm{.}014$

$P(O,x=CCC)=P(M|C)P(S|C)P(L|C)=0\mathrm{.}20\mathrm{.}70\mathrm{.}5=0\mathrm{.}07$

Now, summing up these probabilities:

$P(O|)=0\mathrm{.}004+0\mathrm{.}02+0\mathrm{.}028+0\mathrm{.}14+0\mathrm{.}002+0\mathrm{.}01+0\mathrm{.}014+0\mathrm{.}07=0\mathrm{.}288$

So$,P(O|)=0\mathrm{.}288.$

Now compute ${}_i(t)$ using the provided formulas:

Initialize:

${}_0(0)={}_0{b}_0(O_0)=1\mathrm{.}00\mathrm{.}4=0\mathrm{.}4$

${}_0(1)={}_1{b}_1(O_0)=0\mathrm{.}00\mathrm{.}2=0\mathrm{.}0$

Recursion:

${}_1(0)=({}_0(0)a_{00}+{}_0(1)a_{10})b_0(O_1)=(0\mathrm{.}40\mathrm{.}7+0\mathrm{.}00\mathrm{.}4)0\mathrm{.}1=0\mathrm{.}028$

${}_1(1)=({}_0(0)a_{01}+{}_0(1)a_{11})b_1(O_1)=(0\mathrm{.}40\mathrm{.}3+0\mathrm{.}00\mathrm{.}6)0\mathrm{.}4=0\mathrm{.}048$

Recursion:

${}_2(0)=({}_1(0)a_{00}+{}_1(1)a_{10})b_0(O_2)=(0\mathrm{.}0280\mathrm{.}7+0\mathrm{.}0480\mathrm{.}4)0\mathrm{.}5=0\mathrm{.}0194$

${}_2(1)=({}_1(0)a_{01}+{}_1(1)a_{11})b_1(O_2)=(0\mathrm{.}0280\mathrm{.}3+0\mathrm{.}0480\mathrm{.}6)0\mathrm{.}1=0\mathrm{.}00372$

Finally, we sum over the final values of ${}_i(T-1)$ to getP$(O|)$ :

$P(O|)={\displaystyle \underset{i=0}{\overset{N-1}{}}}{}_2(i)={}_2(0)+{}_2(1)=0\mathrm{.}0194+0\mathrm{.}00372=0\mathrm{.}02312$

So$,P(O|)=0\mathrm{.}02312.$

For this problem, use the same model $$ and observation sequence $O$

given in Problem $1.$

a$)$ Determine the best hidden state sequence $(x_0,x_1,x_2)$ in the dy$-$

namic programming sense.

b$)$ Determine the best hidden state sequence $(x_0,x_1,x_2)$ in the HMM

sense.

CAN YOU ANSWER QUESTION $2$ PART A AND B$.$ QUESTION $1$ ANSWERS ARE PROVIDED