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Unformatted text preview: lative error ρ is NPhard.
• Thm: Computing P (Xie) with absolute error for any ∈ (0, 0.5) is
NPhard.
• But: special cases may have error bounds. $ & ¨ ¨
... ©
£ $ £ ¡ $ & § & ¦ & & ; • And: heuristics often work well. How hard is inference in practice? Exact inference in Gaussian models takes O (N 3) time • We will show later that exact inference can always be done in time
O (N v W ), where WG is the treewidth of the graph (to be deﬁned
later) and v = maxi Xi is the max number of values (states) each
node can take. • For Gaussian graphical models, exact inference is O (N 3) no matter
what the graph structure is! • The NPhardness proof shows that, in the worst case, we have
WG ∼ N .
• But for many models used in practice, we have WG ∼ constant.
• Also, for Gaussian graphical models, exact inference is O (N 3) no
matter what the graph structure is! • c.f., linear programming easier than integer programming.
• Lecture 3: Any undirected graphical model in which potentials have
the form
ψij = exp(Xi − µi)Σ−1(Xj − µj )
ij
can be converted to a joint Gaussian distribution.
• Book chap 4: any directed graphical model in which CPDs have the
form
p(XiXπi ) = N (Xi; W Xπi + µi, Σi)
can be converted to a joint Gaussian distribution.
• Exact inference in a Gaussian graphical model = matrix inversion. Variable elimination algorithm Working right to left (peeling) Coherence
Difficulty Intelligence
Grade P (J ) = Job L G H I D L S G H I D L S G H I D φJ (J, L, S ) τ3 (G,S ) L L S G H G S τ4 (G,J ) φJ (J, L, S )
L S φL (L, G)τ4(G, J )τ3(G, S )
G
τ5 (J,L,S ) = φJ (J, L, S )τ5(J, L, S )
L S
τ6 (J,L) C φH (H, G, J )
H φS (S, I )φI (I )
I φ( G, I,...
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This document was uploaded on 03/28/2014 for the course CS 532 at UBC.
 Fall '04
 KevinMurphy.

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