## Description

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.