You are given the temporal model in the figure below, where all variables are Boolean.
(a) (20 points) Find the probability estimates p(Xi= t| e) for i=1,2,3 using the
appropriate algorithm cover in class. Make sure to show step of your solution, not
just final number.
For the first state (X0 no associated observation and P(e0) = <0.5, 0.5>, we can
know that P (X1 | e0:t-1 ) = = ΣX0 P(X1 | X0) P(X0), which is equivalent as: <0.7,
0.4> * 0.5 + <0.4, 0.7> * 0.5, which is equal to <0.55, 0.55> We can now add
the evidence from t=1 (observation is T) and we can estimate that P(X1 = t | e
= t) = αP(e1 | X1) P(X1), which is equal to: α <0.8, 0.3> < 0.55, 0.55> ~ <0.727,
0.273>
With the information from the first step, we can calculate the second state as:
P (X2 | e0:t-1 ) = = Σ P(X2 | X1) P(X1 | e1), which is equivalent as: <0.7, 0.4> *
0.727 + <0.4, 0.7> * 0.273, which is approximately to <0.618, 0.482> We can
add the evidence from t=1 (observation is F) and we can now estimate that
P(X2 = t | e = f) = αP(e2 | X2) P(X2 | e1), which is equal to: α <0.2, 0.7> < 0.618,
0.482> ~ <0.268, 0.732>
With the information from the second step, we can calculate the third state
as:
P (X3 | e0:t-1 ) = = Σ P(X3 | X2) P(X2 | e2), which is equivalent as: <0.7, 0.4> *
0.268 + <0.4, 0.7> * 0.732, which is approximately to <0.480, 0.620> We can
now add the evidence from t=1 (observation is T) and we can estimate that
P(X3 = t | e = t) = αP(e3 | X3) P(X3 | e2), which is equal to: α <0.8, 0.3> < 0.480,
0.620> ~ <0.674, 0.326>
(b) (20 points) Find the most likely sequence of states (s1, s2, s3 ) of variables X1,
X2, X3 using the appropriate algorithm covered in class. Again, make sure to
show the steps of your solution, not just the final numbers
