A DNA sequence is a series of components from {A, C, G, T}. Suppose there is one hidden variable S that controls the generation of DNA sequence. S can have 2 possible states: {cg, at}. Both states are equally likely initially. At each time step, the probability of transitioning to the other state is 0.2; otherwise, the state remains the same. In state cg, the probability of observing each of C or G is 0.4, and the probability of observing each of A and T is 0.1. In state at, the probability of observing each of A or T is 0.3, and the probability of observing each of C and G is 0.2. A) Give a precise formulation of this HMM (initial belief, transition and observation probabilities).
B) Recall that an HMM assumes the Markov assumption St L St-k | St-1 for k > 1. Hence we know that S2 L So. Show that this does not imply that S2 is independent of So, i. e., show that S2 is not equal to So. This therefore shows that conditional independence does not imply independence.
Hint: Using the provided model, write out the joint distribution P(So, Si, S2). Then compute the marginal distributions P(So), P(S2), and P(So, S2), and show that these marginal distributions do not satisfy the definition of independence.
C) Suppose we observed the sequence e = TGC ACA. Using the HMM recursive forward filtering equations, compute P(S6|E = e).