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Learning Bayesian Networks

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Dimensions of Learning ModelBayes netMarkov net DataCompleteIncomplete StructureKnownUnknown ObjectiveGenerativeDiscriminative

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Bayes net(s) data X 1 true false true X21532X21532 X 3 0.7 -1.6 5.9 6.3........... Learning Bayes nets from data X1X1 X4X4 X9X9 X3X3 X2X2 X5X5 X6X6 X7X7 X8X8 Bayes-net learner + prior/expert information

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From thumbtacks to Bayes nets Thumbtack problem can be viewed as learning the probability for a very simple BN: X heads/tails X1X1 X2X2 XNXN... toss 1 toss 2toss N

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The next simplest Bayes net X heads/tails Y tails heads “heads”“tails”

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The next simplest Bayes net X heads/tails Y XX X1X1 X2X2 XNXN YY Y1Y1 Y2Y2 YNYN case 1 case 2 case N ?

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The next simplest Bayes net X heads/tails Y XX X1X1 X2X2 XNXN YY Y1Y1 Y2Y2 YNYN case 1 case 2 case N "parameter independence"

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The next simplest Bayes net X heads/tails Y XX X1X1 X2X2 XNXN YY Y1Y1 Y2Y2 YNYN case 1 case 2 case N "parameter independence" two separate thumbtack-like learning problems

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A bit more difficult... X heads/tails Y Three probabilities to learn: X=heads Y=heads|X=heads Y=heads|X=tails

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A bit more difficult... X heads/tails Y XX X1X1 X2X2 Y|X=heads Y1Y1 Y2Y2 case 1 case 2 Y|X=tails heads tails

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A bit more difficult... X heads/tails Y XX X1X1 X2X2 Y|X=heads Y1Y1 Y2Y2 case 1 case 2 Y|X=tails

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A bit more difficult... X heads/tails Y XX X1X1 X2X2 Y|X=heads Y1Y1 Y2Y2 case 1 case 2 Y|X=tails ? ? ?

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A bit more difficult... X heads/tails Y XX X1X1 X2X2 Y|X=heads Y1Y1 Y2Y2 case 1 case 2 Y|X=tails 3 separate thumbtack-like problems

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In general … Learning probabilities in a Bayes net is straightforward if Complete data Local distributions from the exponential family (binomial, Poisson, gamma,...) Parameter independence Conjugate priors

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Incomplete data makes parameters dependent X heads/tails Y XX X1X1 X2X2 Y|X=heads Y1Y1 Y2Y2 case 1 case 2 Y|X=tails

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Solution: Use EM Initialize parameters ignoring missing data E step: Infer missing values using current parameters M step: Estimate parameters using completed data Can also use gradient descent

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Learning Bayes-net structure Given data, which model is correct? XY model 1: XY model 2:

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Bayesian approach Given data, which model is correct? more likely? XY model 1: XY model 2: Data d

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Bayesian approach: Model averaging Given data, which model is correct? more likely? XY model 1: XY model 2: Data d average predictions

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Bayesian approach: Model selection Given data, which model is correct? more likely? XY model 1: XY model 2: Data d Keep the best model: - Explanation - Understanding - Tractability

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To score a model, use Bayes’ theorem Given data d: "marginal likelihood" model score likelihood

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Thumbtack example conjugate prior X heads/tails

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More complicated graphs X heads/tails Y 3 separate thumbtack-like learning problems X Y|X=heads Y|X=tails

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Model score for a discrete Bayes net

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Computation of marginal likelihood Efficient closed form if Local distributions from the exponential family (binomial, poisson, gamma,...) Parameter independence Conjugate priors No missing data (including no hidden variables)

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Structure search Finding the BN structure with the highest score among those structures with at most k parents is NP hard for k>1 (Chickering, 1995) Heuristic methods –Greedy –Greedy with restarts –MCMC methods score all possible single changes any changes better? perform best change yes no return saved structure initialize structure

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Structure priors 1. All possible structures equally likely 2. Partial ordering, required / prohibited arcs 3. Prior(m) Similarity(m, prior BN)

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Parameter priors All uniform: Beta(1,1) Use a prior Bayes net

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Parameter priors Recall the intuition behind the Beta prior for the thumbtack: The hyperparameters h and t can be thought of as imaginary counts from our prior experience, starting from "pure ignorance" Equivalent sample size = h + t The larger the equivalent sample size, the more confident we are about the long-run fraction

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Parameter priors x1x1 x4x4 x9x9 x3x3 x2x2 x5x5 x6x6 x7x7 x8x8 + equivalent sample size imaginary count for any variable configuration parameter priors for any Bayes net structure for X 1 …X n parameter modularity

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x1x1 x4x4 x9x9 x3x3 x2x2 x5x5 x6x6 x7x7 x8x8 prior network+equivalent sample size data improved network(s) x 1 true false true x 2 false true x 3 true false........... Combining knowledge & data x1x1 x4x4 x9x9 x3x3 x2x2 x5x5 x6x6 x7x7 x8x8

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Example: College Plans Data (Heckerman et. Al 1997) Data on 5 variables that might influence high school students’ decision to attend college: –Sex: Male or Female –SES: Socio economic status (low, lower-middle, middle, upper- middle, high) –IQ: discritized into low, lower middle, upper middle, high –PE: Parental Encouragement (low or high) –CP: College plans (yes or no) 128 possible joint configurations Heckerman et. al. computed the exact posterior over all 29,281 possible 5 node DAGs –Except those in which Sex or SAS have parents and/or CP have children (prior knowledge)

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