Electronic HomeWork 8 University of California, Berkeley COMPSCI 188

Q1 HMMs, Part I
20 Points

Consider the HMM shown below.




The prior probability P (X0 ), dynamics model P (Xt+1 ∣ Xt ), and sensor model P (Et ∣ Xt )
are as follows:




We perform a first dynamics update, and fill in the resulting belief distribution B ′ (X1 ).




We incorporate the evidence E1 = c. We fill in the evidence-weighted distribution
P (E1 = c ∣ X1 )B ′ (X1 ), and the (normalized) belief distribution B(X1 ).




You get to perform the second dynamics update. Fill in the resulting belief distribution B ′ (X2 ).

B ′ (X2 = 0)

.80


B ′ (X2 = 1)

.20

, Now incorporate the evidence E2 = c.
Fill in the evidence-weighted distribution P (E2 = c ∣ X2 )B ′ (X2 ), and the (normalized) belief
distribution B(X2 ).


P (E2 = c ∣ X2 )B ′ (X2 ) when X2 = 0

.04


P (E2 = c ∣ X2 )B ′ (X2 ) when X2 = 1

.12


B(X2 = 0)

.25


B(X2 = 1)

.75




Q2 HMMs, Part II
20 Points

Consider the same HMM (but with different probabilities).




The prior probability P (X0 ), dynamics model P (Xt+1 ∣ Xt ), and sensor model P (Et ∣ Xt )
are as follows:




In this question we'll assume the sensor is broken and we get no more evidence readings after
E2 . We are forced to rely on dynamics updates only going forward. In the limit as t → ∞, our
~
belief about Xt should converge to a stationary distribution B (X∞ ) defined as follows:

~
B (X∞ ) := lim P (Xt ∣ E1 , E2 )
t→∞