described in part 3.1.
4. Using the intuition in part 3.3, design a dynamic programming algorithm (caching
partial results) which computes
P
(
X
i
, X
j
) for all
n
choose 2 choices of
i
and
j
in time
asymptotically much lower than the complexity you described in part 3.2. What is the
asymptotic running time of your algorithm?
4
Belief Propagation
Two graduate students in University of Tehran have gotten into an argument over the
weather. One thinks summer is over and Autumn has already come, while the other thinks
it is still summer. In Tehran, there are four seasons – Spring (S), Summer (M), Autumn (A),
and Winter (W). Given the season the previous day, the season on a day is conditionally
3
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independent of the season on all previous days. The weather is either Hot (H), Rainy (R) or
Freezing (F). Given the season on any given day, the weather that day is independent of all
other variables. More formally, if we let
C
i
denote the season on the i–th day (taking values
S, M, A, W) and
O
i
denote the observed weather pattern (one of H, R, F).
We have
∀
j < i

1
, C
i
⊥
C
j

C
i

1
and
∀
X, O
i
⊥
X

C
i
where
X
is any random variable other
than
C
i
,
O
i
.
1. Draw a graphical model over
C
1
. . . C
N
, O
1
. . . O
N
that satisfies the conditional inde
pendencies listed above.
2. Implement sumproduct and maxproduct algorithms in MATLAB for this graphical
model.
3. We has made 20 observation of the weather over the last few months (i.e.,
O
1
. . . O
N
) :
{
R, F, F, H, F, H, H, H, H, H, H, H, H, R, H, H, H, R, H, H
}
Some of the values for the conditional probability table (CPT) are as follows.
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 Spring '13
 Dr. Zare
 Dynamic Programming, Probability, Probability theory, Bayesian network, graphical model, Chordal graph

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