1 / 8

I-maps and perfect maps

This document explores the concepts of I-maps and perfect maps within the framework of probabilistic graphical models. It explains how a graph G can serve as an I-map for a distribution P by factoring its independencies, while noting that not all I-maps capture all independencies present in P. The aim is to design a sparse graph that encodes essential independencies, leading to more informative models. It further discusses the relationship between minimal I-maps, perfect maps, and the challenge of exact independence capture within Bayesian networks (BN) and Markov networks (MN).

betrys
Télécharger la présentation

I-maps and perfect maps

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Representation Probabilistic Graphical Models Independencies I-maps andperfect maps

  2. Capturing Independencies in P • P factorizes over G  G is an I-map for P: • But not always vice versa: there can be independencies in I(P) that are not in I(G)

  3. Want a Sparse Graph • If the graph encodes more independencies • it is sparser (has fewer parameters) • and more informative • Want a graph that captures as much of the structure in P as possible

  4. Minimal I-map • I-map without redundant edges D D I I G G

  5. MN as a perfect map • Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) D D I I G G

  6. BN as a perfect map A A A A • Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) D D D D B B B B C C C C

  7. More imperfect maps • Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) X1 X2 Y XOR

  8. Summary • Graphs that capture more of I(P) are more compact and provide more insight • But it may be impossible to capture I(P) perfectly (as a BN, as an MN, or at all) • BN to MN: lose the v-structures • MN to BN: add edges to undirected loops

More Related