Bayes Nets Representation joint distribution and conditional independence Yi Zhang 10 701 Machine Learning Spring 2011 February 9th 2011 Parts of the slides are from previous 10 701 lectures 1 Outline Conditional independence C I Bayes nets overview Local Markov assumption of BNs Factored joint distribution of BNs Infer C I from factored joint distributions D separation motivation 2 Conditional independence X is conditionally independent of Y given Z In short Equivalent to 3 Bayes nets Bayes nets directed acyclic graphs express sets of conditional independence via graph structure All about the joint distribution of variables Conditional independence assumptions are useful Na ve Bayes model is an extreme example 4 Three key questions for BNs Representation What joint distribution does a BN represent Inference How to answer questions about the joint distribution Conditional independence Marginal distribution Most likely assignment Learning How to learn the graph structure and parameters of a BN from data 5 Local Markov assumptions of BNs A variable X is independent of its nondescendants given only its parents Intuition flu and allergy causes headache only through sinus 6 Local Markov assumptions of BNs A variable X is independent of its nondescendants given only its parents 7 Local Markov assumptions of BNs Local Markov assumptions only express a subset of C I s on a BN Is XM conditionally independent of X1 given X2 But they are sufficient to infer all others 8 Factored joint distribution of a BN A BN can represent the joint distributions of the following form 9 Factored joint distribution of a BN A BN can represent the joint distributions of the following form 10 Factored joint distribution of a BN Local Markov assumptions imply the factored joint distribution 11 Factored joint distribution of a BN Na ve Bayes Local Markov assumptions Factored joint distribution 12 Infer C I from the factored joint distribution We already see local Markov assumptions factored joint distribution Also factored joint distribution all C I in the BN 13 Infer C I from the factored joint distribution Factored Joint distribution Show that 14 Infer C I from the factored joint distribution Factored Joint distribution Do we have In general no Cannot be written into two separate terms of a and b 15 D separation motivation Is XM conditionally independent of X1 given X2 Intuitively yes X1 affects XM only through X2 Method 1 using factored joint distribution to derive Method II D separation not today 16
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