Hidden Markov Model Homework: State Space S={1,2,3}, Alphabet A={a,b,c}

[daneault23]

Homework Statement

Define a Hidden Markov Model with the following parameters: State space S={1,2,3}, alphabet A = {a,b,c,}, P = 0 .5 .5
1 0 0
0 1 0

initial probability vector, ∏ = 1 0 0

b1(a) = 1/2 ; b1(b) = 1/2 ; b1(c) = 0
b2(a) = 1/2 ; b2(b) = 0; b2(c) = 1/2
b3(a) = 0; b3(b) = 1/2 ; b3(c) = 1/2

List all possible state sequences Q = (q1; q2; q3). Consider the observations
O = (a; b; c). What is the probability to observe this sequence in the model, i.e.
find P(O|λ)? Given these observations, which state sequence has most likely
generated it, i.e., find P(Q|O)?

Homework Equations

Remember the collection of parameters of a Hidden Markov Model is denoted λ = (P, B, ∏)

The Attempt at a Solution

Since ∏ has a 1 in the first entry, we always start in state 1. Thus, the probabilities of starting in other states, (P(O=abc|Q=312)*P(312) for example = 0), so we only need to find P(O=abc|Q=123)*P(Q=123) + P(O=abc|Q=132)*P(132). Well, right off the bat, the probability of going from state 2 to observation b is 0 so P(O=abc|Q=123)*P(Q=123)=0 already. Therefore, the only thing left is P(O=abc|Q=132)*P(132)=.5^4=1/16, so the probability to observe the sequence (a,b,c) in this model is 1/16.

Am I doing something wrong here?

 

[daneault23]

I just realized that the sequence could also be given by Q=111, 112, 113, 121, 122, and 133. It wasn't as easy as I originally thought.