Lecture 6

How hard is inference in practice exact inference in

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Unformatted text preview: lative error ρ is NP-hard. • Thm: Computing P (Xi|e) with absolute error for any ∈ (0, 0.5) is NP-hard. • 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 tree-width 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 NP-hardness 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(Xi|Xπi ) = 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.

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