# Problem 1: Warm-up (a) Suppose we have a sensor reading…..solution

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Problem 1: Warm-up
(a) Suppose we have a sensor reading for the second timestep, D2 = 0. Compute the
posterior distribution P(C2 = 1|D2 = 0). We encourage you to draw out the (factor)
graph.

(b) Suppose a time step has elapsed and we got another sensor reading, D3 = 1, but we are
still interested in C2. Compute the posterior distribution P(C2 = 1|D2 = 0, D3 = 1).
The resulting expression might be moderately complex. We encourage you to draw
out the (factor) graph.

(c) Suppose =0.1 and η=0.2.
i. Compute and compare the probabilities P(C2 = 0|D2 = 1) and P(C2 = 1|D2 =
0, D3 = 1). Give numbers, round your answer to 4 significant digits

ii. How did adding the second sensor reading D3 = 1 change the result? Explain your
intuition in terms of the car positions with respect to the observations.

iii. What would you have to set  while keeping η=0.2 so that P(C2 = 1D2 = 0) =
P(C2 = 1D2 = 0, D3 = 1)? Explain your intuition in terms of the car positions with
respect to the observations.

Problem 5: Which car is it?
(a) Suppose we have K=2 cars and one time step T=1. Write an expression for the
conditional distribution P(C11, C12|E1 = e1) as a function of the PDF of a Gaussian
p(v; µ, σ2) and the prior probability p(c11) and p(c12) over car locations. Your final
answer should not contain variables d11, d12.

(b) Assuming the prior p(c1i) is the same for all i, show that the number of assignments for
all K cars (c11, , c1K) that obtain the maximum value of P(C11 = c11, , C1K = c1K|E1 =
e1) is at least K!.

(c) For general K, what is the treewidth corresponding to the posterior distribution over
all K car locations at all T time steps conditioned on all the sensor readings.