CS228 Problem Set #3
1
CS 228, Winter 2008
Problem Set #3
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1.
Entanglement in DBNs [20 points]
(a)
[6 points]
Prove Proposition 15.2.4:
Let
I
be the influence graph for a 2TBN
B
→
. Then
I
contains a directed path from
X
to
Y
if and only if, in the unrolled DBN, for every
t
, there exists a directed path
from
X
(
t
)
to
Y
(
t
)
for some
t
≥
t
.
(b)
[10 points]
Prove the entanglement theorem, Theorem 15.2.5:
Let
G
0
,
G
→
be a fully persistent DBN structure over
X
=
X
∪
O
, where the state
variables
X
(
t
)
are hidden in every time slice, and the observation variables
O
(
t
)
are
observed in every time slice.
Furthermore, assume that, in the influence graph for
G
→
:
•
there is a trail (not necessarily a directed path) between every pair of nodes, i.e.,
the graph is connected;
•
every state variable
X
has some directed path to some evidence variable in
O
.
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 Winter '09
 Bayesian probability, Dirichlet distribution

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