Recall that for a standard HMM the Elapse Time update and the Observation update are of the respective forms:

P(X_t | e_1:t−1) = Σ_xt−1 P(X_t | x_t−1)P(x_t−1 | e_1:t−1)

P(X_t | e_1:t) ∝ P(X_t | e_1:t−1)P(e_t | x_t)

We now consider the following two HMM-like models:

Derive the modified Elapse Time update and the modified Observation update that correctly compute the beliefs from the quantities that are available in the Bayes’ Net.

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