CS228 Final
1
CS 228, Winter 2007
Final
Handout #17
You have 24 hours to complete this exam.
The exam is given out at noon, and due at noon
(12:00 pm) one day after you pick it up. The exam will be handed out and collected in Gates
120 (the Fishbowl).
This exam is long and difficult, and we do not expect everyone to finish
all of the questions. Be sure to use good test taking skills and attack the easier problems first,
spend more time on questions worth more points, and generally pay attention to how you spend
your time. Also, you are welcome (and we expect you) to use or refer algorithms from the reader
when appropriate, without having to rederive or explain them. Furthermore, please use standard
notation from the reader (when possible) and clearly define any terms you introduce. Algorithm
answers should be provided in the form of pseudocode (with explanations where necessary), and
your answers will be easier to grade (read: you will get higher grades) if you use proper spacing
and layout of your answers on the page. We have provided approximate times with each of the
problems to give you a rough estimate of how long we think it might take (in time and pages
not including diagrams).
1.
[10 points] Representation
(a)
[5 points]
Let
G
be a Bayesian network with no immoralities. Let
H
be a Markov
network with the same skeleton as
G
. Show that
H
is an Imap of
I
(
G
). That is, prove
Corollary 5.7.4 — Corollary 5.5.4 in the version of Chapter 5 posted on Coursework
— for the case in which
G
has no immoralities.
(You may not simply assume this
corollary or use theorems that follow it in the readings.)
(b)
[5 points]
Let
X
and
Y
be variables in a Bayesian network
G
, and suppose that
neither is a parent of the other. Let
W
=
Pa
X
∪
Pa
Y
. Prove that (
X
⊥
Y

W
)
∈
I
(
G
).
Estimate:
1 page, 1 hour
2.
[20 points] Sampling on a Tree
Suppose we have a distribution
P
(
X
,
E
) over two sets of variables
X
and
E
. Our distri
bution is represented by a nasty Bayes Net with very dense connectivity, and our sets of
variables
X
and
E
are spread arbitrarily throughout the network. In this problem our goal
is to use the sampling methods we learned in class to estimate the posterior probability
P
(
X
=
x

E
=
e
).
More specifically, we will use a treestructured Bayes Net as the
proposal distribution for use in the importance sampling algorithm.
(a)
[1 points]
For a particular value of
x
and
e
, can we compute
P
(
x

e
) exactly, in
a tractable way? Can we sample directly from the distribution
P
(
X

e
)? Can we
compute
P
(
x

e
) =
P
(
x
,
e
) exactly, in a tractable way? For each question, provide
a Yes/No answer and a single sentence explanation or description.
(b)
[13 points]
Now, suppose your friendly TAs have given you a tree network (each
variable besides the root has exactly one parent) that defines a distribution
Q
. They
tell you that
Q
(
X
,
E
) is “close” to the distribution
P
(
X
,
E
) of the nasty network.
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 Winter '09
 belief state, template clique tree, approximate belief state

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