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We consider the following hidden Markov model (HMM).

S1		S2
Z1		Z2

We have binary states (labeled s and s) and binary observations (labeled z and z) and the probabilities as in the given table.

S1	P(S1)
s	0.3
s	0.7

Si	Si+1	P(Si+1|Si)
s	s	0.2
s	s	0.8
s	s	0.9
s	s	0.1

Si	Zi	P(Zi|Si)
s	z	0.1
s	z	0.9
s	z	0.8
s	z	0.2

The above is the setup for the following question:

We observed that Z₁=z and then Z₂=z. We used the forward algorithm and obtained P(S₁=s|Z₁=z) = 0.6585. Please use the forward algorithm to obtain the following probabilities (keep only four decimal values for all probabilities):

1. [5 points] P(S₂=s|Z₁=z)
2. [5 points] P(S₂=s|Z₁=z, Z₂=z)

The following is for bonus points. You should leave this to the last in your to-do list.

After observing the two evidences, we can have a better inference of the state S₁. Use either (i) Bayesian inference or (ii) Bayes' rule and conditional independence to calculate the following probability:

3. [6 bonus points] P(S₁=s|Z₁=z, Z₂=z)