Question

4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely. In particular: pix- 1 |z= 1) p(X= 2| Z4= 1)= p(&=3 |z= 1)= p(&= 4|z= 1) = 0.25, pix- 1 | z= 2)0.7, p(&- 2| z= 2)= p(X= 3 |z2) = p(=4 122) 0.1 Assume that the casino has an equal probability of starting in either state at time t that p(zi 1 p(zı = 2) = 0.5. The casino usually uses the same die for multiple iterations, but occasionally switches states according to the following probabilities: 1, so p(z+ 1 1 0.8, p(zt1= 2| 2) 0.9. The other transition probabilities you will need are the complements of these. Hint: For all of the questions below, you should not need to enumerate all possible state sequences [4 Points] Under the HMM generative model, what is p(zi = z2 = z3), the probability that the same die is used for the first three rolls? b. [4 Points] Suppose that we observe the first two rolls. What is p(zı 1 |x1 2, x2 4), the probability that the casino used the fair die in the first roll? [12 Points] Using the Viterbi algorithm, decode the path for the following observed sequence x 2, x2= 3, x3 3. x4 3 and xs 1. Show your work (i.e., show each of your belief for based on time). You can also use log2(P) for faster calculation. a. C.

Answer 1

Soliction Pog No - 0 049 Staly a) P2, - 2,) - ).P(2,) Pl223212)P(22-2) O.25 b) Pl, ,-2 2,-4) P (2, u 2, 1). pl2, 21 ) Plx, 2, ,-4) NOL PL 05 P 2, u/ 2 P(,2, -1 ). pix,- u/2,21) (o.15)Pl 2,-1 |2,21)+Plx2u, 2,2/2,) 0 OSS P 2, 4) Now Pl2, xu 2, 2. P(z,= 2)

Pae NO - 2) P(4> 2, LLzu| z, -2) ro S KO S O 6275 P(,222) Pt,-u/2,2) Pu,2221 -2) +Pl1,2u, zz-1|2,-2) кон] P( 2, y-u) O27 t0.0115 KO 5 O 03275 Henle, pL -1 |4, 2, &Ļou) D8397 O 0325 Given, 2t€C,2 > 2 t NOW die stal af st Зиpрае Stat

Pge No-3 P ) 0.5 ptabt bab? Sana die uged exampe Cosino юoшd be P222 -08 TRen NOW tha Suppee P2- 2/ 2-2) =o.9 P(22)0s ard Can be WAi Hes ay, OtRa tanslation pralnbility 2 t +! 1-p[24= 2{2402|] l Pl2i 22 O 66 J P3,2PC22 P22| 2,2+ PC222122. 2/21 P[212,2 Hene Lame die s uged for o Total stot] be Samo t1ye ls [ie, Casino, Ould

Pags NO -(