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. PGM: Tirgul 10 Parameter Learning and Priors

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2 Why learning? Knowledge acquisition bottleneck u Knowledge acquisition is an expensive process u Often we don’t have an expert Data is cheap u Vast amounts of data becomes available to us u Learning allows us to build systems based on the data

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3 Learning Bayesian networks Inducer Data + Prior information E R B A C.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B)

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4 Known Structure -- Complete Data E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) ?? e b e ?? ? ? ?? be b b e BE E B A u Network structure is specified l Inducer needs to estimate parameters u Data does not contain missing values

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5 Unknown Structure -- Complete Data E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) ?? e b e ?? ? ? ?? be b b e BE E B A u Network structure is not specified l Inducer needs to select arcs & estimate parameters u Data does not contain missing values

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6 Known Structure -- Incomplete Data Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) ?? e b e ?? ? ? ?? be b b e BE E B A u Network structure is specified u Data contains missing values l We consider assignments to missing values E, B, A.

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7 Known Structure / Complete Data u Given a network structure G And choice of parametric family for P(X i |Pa i ) u Learn parameters for network Goal u Construct a network that is “closest” to probability that generated the data

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8 Example: Binomial Experiment (Statistics 101) u When tossed, it can land in one of two positions: Head or Tail We denote by the (unknown) probability P(H). Estimation task: Given a sequence of toss samples x[1], x[2], …, x[M] we want to estimate the probabilities P(H)= and P(T) = 1 - HeadTail

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9 Statistical Parameter Fitting Consider instances x[1], x[2], …, x[M] such that l The set of values that x can take is known l Each is sampled from the same distribution l Each sampled independently of the rest The task is to find a parameter so that the data can be summarized by a probability P(x[j]| ). l Depends on the given family of probability distributions: multinomial, Gaussian, Poisson, etc. l For now, focus on multinomial distributions i.i.d. samples

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10 The Likelihood Function u How good is a particular ? It depends on how likely it is to generate the observed data u The likelihood for the sequence H,T, T, H, H is 00.20.40.60.81 L( :D)

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11 Sufficient Statistics To compute the likelihood in the thumbtack example we only require N H and N T (the number of heads and the number of tails) N H and N T are sufficient statistics for the binomial distribution

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12 Sufficient Statistics u A sufficient statistic is a function of the data that summarizes the relevant information for the likelihood Formally, s(D) is a sufficient statistics if for any two datasets D and D’ s(D) = s(D’ ) L( |D) = L( |D’) Datasets Statistics

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13 Maximum Likelihood Estimation MLE Principle: Choose parameters that maximize the likelihood function u This is one of the most commonly used estimators in statistics u Intuitively appealing

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14 Example: MLE in Binomial Data u Applying the MLE principle we get (Which coincides with what one would expect) 00.20.40.60.81 L( :D)L( :D) Example: (N H,N T ) = (3,2) MLE estimate is 3/5 = 0.6

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15 Learning Parameters for a Bayesian Network E B A C u Training data has the form:

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16 Learning Parameters for a Bayesian Network E B A C u Since we assume i.i.d. samples, likelihood function is

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17 Learning Parameters for a Bayesian Network E B A C u By definition of network, we get

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18 Learning Parameters for a Bayesian Network E B A C u Rewriting terms, we get

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19 General Bayesian Networks Generalizing for any Bayesian network: u The likelihood decomposes according to the structure of the network. i.i.d. samples Network factorization

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20 General Bayesian Networks (Cont.) Decomposition Independent Estimation Problems If the parameters for each family are not related, then they can be estimated independently of each other.

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21 From Binomial to Multinomial For example, suppose X can have the values 1,2,…,K We want to learn the parameters 1, 2. …, K Sufficient statistics: N 1, N 2, …, N K - the number of times each outcome is observed Likelihood function: MLE:

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22 Likelihood for Multinomial Networks When we assume that P(X i | Pa i ) is multinomial, we get further decomposition:

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23 Likelihood for Multinomial Networks When we assume that P(X i | Pa i ) is multinomial, we get further decomposition: For each value pa i of the parents of X i we get an independent multinomial problem u The MLE is

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24 Maximum Likelihood Estimation Consistency u Estimate converges to best possible value as the number of examples grow u To make this formal, we need to introduce some definitions

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25 KL-Divergence Let P and Q be two distributions over X u A measure of distance between P and Q is the Kullback-Leibler Divergence KL(P||Q) = 1 (when logs are in base 2) = l The probability P assigns to an instance is, on average, half the probability Q assigns to it u KL(P||Q) 0 KL(P||Q) = 0 iff are P and Q equal

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26 Consistency Let P(X| ) be a parametric family l We need to make various regularity condition we won’t go into now Let P * (X) be the distribution that generates the data u Let be the MLE estimate given a dataset D Thm As N , where with probability 1

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27 Consistency -- Geometric Interpretation P*P* P(X| * ) Space of probability distribution Distributions that can represented by P(X| )

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28 Is MLE all we need? u Suppose that after 10 observations, ML estimates P(H) = 0.7 for the thumbtack l Would you bet on heads for the next toss? Suppose now that after 10 observations, ML estimates P(H) = 0.7 for a coin Would you place the same bet?

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29 Bayesian Inference Frequentist Approach: u Assumes there is an unknown but fixed parameter u Estimates with some confidence u Prediction by using the estimated parameter value Bayesian Approach: u Represents uncertainty about the unknown parameter u Uses probability to quantify this uncertainty: l Unknown parameters as random variables u Prediction follows from the rules of probability: l Expectation over the unknown parameters

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30 Bayesian Inference (cont.) u We can represent our uncertainty about the sampling process using a Bayesian network The values of X are independent given The conditional probabilities, P(x[m] | ), are the parameters in the model u Prediction is now inference in this network X[1]X[2]X[m] X[m+1] Observed dataQuery

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31 Bayesian Inference (cont.) Prediction as inference in this network where Posterior Likelihood Prior Probability of data X[1]X[2]X[m] X[m+1]

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32 Example: Binomial Data Revisited Prior: uniform for in [0,1] P( ) = 1 Then P( |D) is proportional to the likelihood L( :D) (N H,N T ) = (4,1) MLE for P(X = H ) is 4/5 = 0.8 Bayesian prediction is 00.20.40.60.81

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33 Bayesian Inference and MLE u In our example, MLE and Bayesian prediction differ But… If prior is well-behaved u Does not assign 0 density to any “feasible” parameter value Then: both MLE and Bayesian prediction converge to the same value u Both are consistent

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34 Dirichlet Priors u Recall that the likelihood function is A Dirichlet prior with hyperparameters 1,…, K is defined as for legal 1,…, K Then the posterior has the same form, with hyperparameters 1 +N 1,…, K +N K

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35 Dirichlet Priors (cont.) u We can compute the prediction on a new event in closed form: If P( ) is Dirichlet with hyperparameters 1,…, K then u Since the posterior is also Dirichlet, we get

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36 Dirichlet Priors -- Example 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 00.20.40.60.81 Dirichlet(1,1) Dirichlet(2,2) Dirichlet(0.5,0.5) Dirichlet(5,5)

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37 Prior Knowledge The hyperparameters 1,…, K can be thought of as “imaginary” counts from our prior experience Equivalent sample size = 1 +…+ K u The larger the equivalent sample size the more confident we are in our prior

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38 Effect of Priors Prediction of P(X=H ) after seeing data with N H = 0.25N T for different sample sizes 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 020406080100 0 0.1 0.2 0.3 0.4 0.5 0.6 020406080100 Different strength H + T Fixed ratio H / T Fixed strength H + T Different ratio H / T

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39 Effect of Priors (cont.) u In real data, Bayesian estimates are less sensitive to noise in the data 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5101520253035404550 P(X = 1|D) N MLE Dirichlet(.5,.5) Dirichlet(1,1) Dirichlet(5,5) Dirichlet(10,10) N 0 1 Toss Result

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40 Conjugate Families u The property that the posterior distribution follows the same parametric form as the prior distribution is called conjugacy l Dirichlet prior is a conjugate family for the multinomial likelihood u Conjugate families are useful since: l For many distributions we can represent them with hyperparameters l They allow for sequential update within the same representation l In many cases we have closed-form solution for prediction

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41 Bayesian Networks and Bayesian Prediction u Priors for each parameter group are independent u Data instances are independent given the unknown parameters XX X[1]X[2] X[M] X[M+1] Observed data Plate notation Y[1]Y[2] Y[M] Y[M+1] Y|X XX m X[m] Y[m] Query

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42 Bayesian Networks and Bayesian Prediction (Cont.) u We can also “read” from the network: Complete data posteriors on parameters are independent XX X[1]X[2] X[M] X[M+1] Observed data Plate notation Y[1]Y[2] Y[M] Y[M+1] Y|X XX m X[m] Y[m] Query

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43 Bayesian Prediction(cont.) u Since posteriors on parameters for each family are independent, we can compute them separately u Posteriors for parameters within families are also independent: Complete data independent posteriors on Y|X=0 and Y|X=1 XX Y|X m X[m] Y[m] Refined model XX Y|X=0 m X[m] Y[m] Y|X=1

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44 Bayesian Prediction(cont.) Given these observations, we can compute the posterior for each multinomial X i | pa i independently l The posterior is Dirichlet with parameters (X i =1|pa i )+N (X i =1|pa i ),…, (X i =k|pa i )+N (X i =k|pa i ) The predictive distribution is then represented by the parameters

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45 Assessing Priors for Bayesian Networks We need the (x i,pa i ) for each node x j We can use initial parameters 0 as prior information Need also an equivalent sample size parameter M 0 Then, we let (x i,pa i ) = M 0 P(x i,pa i | 0 ) u This allows to update a network using new data

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46 Learning Parameters: Case Study (cont.) Experiment: u Sample a stream of instances from the alarm network u Learn parameters using l MLE estimator l Bayesian estimator with uniform prior with different strengths

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47 Learning Parameters: Case Study (cont.) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0500100015002000250030003500400045005000 KL Divergence M MLE Bayes w/ Uniform Prior, M'=5 Bayes w/ Uniform Prior, M'=10 Bayes w/ Uniform Prior, M'=20 Bayes w/ Uniform Prior, M'=50

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48 Learning Parameters: Summary u Estimation relies on sufficient statistics For multinomial these are of the form N (x i,pa i ) l Parameter estimation u Bayesian methods also require choice of priors u Both MLE and Bayesian are asymptotically equivalent and consistent u Both can be implemented in an on-line manner by accumulating sufficient statistics MLE Bayesian (Dirichlet)

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