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Learning: Parameter Estimation

Learning: Parameter Estimation. Eran Segal Weizmann Institute. Learning Introduction. So far, we assumed that the networks were given Where do the networks come from? Knowledge engineering with aid of experts Automated construction of networks Learn by examples or instances.

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Learning: Parameter Estimation

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  1. Learning:Parameter Estimation Eran Segal Weizmann Institute

  2. Learning Introduction • So far, we assumed that the networks were given • Where do the networks come from? • Knowledge engineering with aid of experts • Automated construction of networks • Learn by examples or instances

  3. Learning Introduction • Input: dataset of instances D={d[1],...d[m]} • Output: Bayesian network • Measures of success • How close is the learned network to the original distribution • Use distance measures between distributions • Often hard because we do not have the true underlying distribution • Instead, evaluate performance by how well the network predicts new unseen examples (“test data”) • Classification accuracy • How close is the structure of the network to the true one? • Use distance metric between structures • Hard because we do not know the true structure • Instead, ask whether independencies learned hold in test data

  4. Prior Knowledge • Prespecified structure • Learn only CPDs • Prespecified variables • Learn network structure and CPDs • Hidden variables • Learn hidden variables, structure, and CPDs • Complete/incomplete data • Missing data • Unobserved variables

  5. Learning Bayesian Networks X1 X2 Inducer • Data • Prior information Y

  6. Known Structure, Complete Data • Goal: Parameter estimation • Data does not contain missing values X1 X2 X1 X2 Inducer Initial network Y Y InputData

  7. Unknown Structure, Complete Data • Goal: Structure learning & parameter estimation • Data does not contain missing values X1 X2 X1 X2 Inducer Initial network Y Y InputData

  8. Known Structure, Incomplete Data • Goal: Parameter estimation • Data contains missing values • Example: Naïve Bayes X1 X2 X1 X2 Inducer Initial network Y Y InputData

  9. Unknown Structure, Incomplete Data • Goal: Structure learning & parameter estimation • Data contains missing values X1 X2 X1 X2 Inducer Initial network Y Y InputData

  10. Parameter Estimation • Input • Network structure • Choice of parametric family for each CPD P(Xi|Pa(Xi)) • Goal: Learn CPD parameters • Two main approaches • Maximum likelihood estimation • Bayesian approaches

  11. Biased Coin Toss Example • Coin can land in two positions: Head or Tail • Estimation task • Given toss examples x[1],...x[m] estimateP(H)= and P(T)=1- • Assumption: i.i.d samples • Tosses are controlled by an (unknown) parameter  • Tosses are sampled from the same distribution • Tosses are independent of each other

  12. L(D:)  0 0.2 0.4 0.6 0.8 1 Biased Coin Toss Example • Goal: find [0,1] that predicts the data well • “Predicts the data well” = likelihood of the data given  • Example: probability of sequence H,T,T,H,H

  13. L(D:)  0 0.2 0.4 0.6 0.8 1 Maximum Likelihood Estimator • Parameter  that maximizes L(D:) • In our example, =0.6 maximizes the sequence H,T,T,H,H

  14. Maximum Likelihood Estimator • General case • Observations: MH heads and MT tails • Find  maximizing likelihood • Equivalent to maximizing log-likelihood • Differentiating the log-likelihood and solving for  we get that the maximum likelihood parameter is:

  15. Sufficient Statistics • For computing the parameter  of the coin toss example, we only needed MH and MT since •  MH and MT are sufficient statistics

  16. Datasets Statistics Sufficient Statistics • A function s(D) is a sufficient statistics from instances to a vector in k if for any two datasets D and D’ and any  we have

  17. Sufficient Statistics for Multinomial • A sufficient statistics for a dataset D over a variable Y with k values is the tuple of counts <M1,...Mk> such that Mi is the number of times that the Y=yi in D • Sufficient statistic • Define s(x[i]) as a tuple of dimension k • s(x[i])=(0,...0,1,0,...,0) (1,...,i-1) (i+1,...,k)

  18. Sufficient Statistic for Gaussian • Gaussian distribution: • Rewrite as •  sufficient statistics for Gaussian: s(x[i])=<1,x,x2>

  19. Maximum Likelihood Estimation • MLE Principle: Choose  that maximize L(D:) • Multinomial MLE: • Gaussian MLE:

  20. MLE for Bayesian Networks • Parameters • x0, x1 • y0|x0, y1|x0, y0|x1, y1|x1 • Data instance tuple: <x[m],y[m]> • Likelihood X Y  Likelihood decomposes into two separate terms, one for each variable

  21. MLE for Bayesian Networks • Terms further decompose by CPDs: • By sufficient statistics where M[x0,y0] is the number of data instances in which X takes the value x0 and Y takes the value y0 • MLE

  22. MLE for Bayesian Networks • Likelihood for Bayesian network •  if Xi|Pa(Xi) are disjoint then MLE can be computed by maximizing each local likelihood separately

  23. MLE for Table CPD BayesNets • Multinomial CPD • For each value xX we get an independent multinomial problem where the MLE is

  24. Terms for <y0,z0> and <y0,z1> can be combined Optimization can be done by leaves MLE for Tree CPDs • Assume tree CPD with known tree structure Y Z X Y y0 y1 Z x0|y0,x1|y0 z0 z1 x0|y1,z1,x1|y1,z1 x0|y1,z1,x1|y1,z1

  25. MLE for Tree CPD BayesNets • Tree CPD T, leaves l • For each value lLeaves(T) we get an independent multinomial problem where the MLE is

  26. Limitations of MLE • Two teams play 10 times, and the first wins 7 of the 10 matches  Probability of first team winning = 0.7 • A coin is tosses 10 times, and comes out ‘head’ 7 of the 10 tosses  Probability of head = 0.7 • Would you place the same bet on the next game as you would on the next coin toss? • We need to incorporate prior knowledge • Prior knowledge should only be used as a guide

  27. … X[2] X[M] X[1] Bayesian Inference • Assumptions • Given a fixed  tosses are independent • If  is unknown tosses are not marginally independent – each toss tells us something about  • The following network captures our assumptions

  28. Bayesian Inference • Joint probabilistic model • Posterior probability over   … X[2] X[M] X[1] Likelihood Prior For a uniform prior, posterior is the normalized likelihood Normalizing factor

  29. Bayesian Prediction • Predict the data instance from the previous ones • Solve for uniform prior P()=1 and binomial variable

  30. (MH,MT ) = (4,1) 0 0.2 0.4 0.6 0.8 1 Example: Binomial Data • Prior: uniform for  in [0,1] • P() = 1 •  P(|D) is proportional to the likelihood L(D:) • MLE for P(X=H) is 4/5 = 0.8 • Bayesian prediction is 5/7

  31. Dirichlet Priors • A Dirichlet prior is specified by a set of (non-negative) hyperparameters 1,...k so that  ~ Dirichlet(1,...k) if • where and • Intuitively, hyperparameters correspond to the number of imaginary examples we saw before starting the experiment

  32. Dirichlet Priors – Example 5 Dirichlet(1,1) Dirichlet(2,2) 4.5 Dirichlet(0.5,0.5) Dirichlet(5,5) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1

  33. Dirichlet Priors • Dirichlet priors have the property that the posterior is also Dirichlet • Data counts are M1,...,Mk • Prior is Dir(1,...k) •  Posterior is Dir(1+M1,...k+Mk) • The hyperparameters 1,…,K can be thought of as “imaginary” counts from our prior experience • Equivalent sample size = 1+…+K • The larger the equivalent sample size the more confident we are in our prior

  34. 0.55 0.6 Different strength H + T Fixed ratio H / T 0.5 Fixed strength H + T Different ratio H / T 0.5 0.45 0.4 0.4 0.35 0.3 0.3 0.2 0.25 0.1 0.2 0.15 0 0 20 40 60 80 100 0 20 40 60 80 100 Effect of Priors • Prediction of P(X=H) after seeing data with MH=1/4MT as a function of the sample size

  35. MLE Dirichlet(.5,.5) Dirichlet(1,1) Dirichlet(5,5) Dirichlet(10,10) Effect of Priors (cont.) • In real data, Bayesian estimates are less sensitive to noise in the data 0.7 0.6 0.5 P(X = 1|D) 0.4 0.3 0.2 N 0.1 5 10 15 20 25 30 35 40 45 50 1 Toss Result 0 N

  36. General Formulation • Joint distribution over D, • Posterior distribution over parameters • P(D) is the marginal likelihood of the data • As we saw, likelihood can be described compactly using sufficient statistics • We want conditions in which posterior is also compact

  37. Conjugate Families • A family of priors P(:) is conjugate to a model P(|) if for any possible dataset D of i.i.d samples from P(|) and choice of hyperparameters  for the prior over , there are hyperparameters ’ that describe the posterior, i.e.,P(:’) P(D|)P(:) • Posterior has the same parametric form as the prior • Dirichlet prior is a conjugate family for the multinomial likelihood • Conjugate families are useful since: • Many distributions can be represented with hyperparameters • They allow for sequential update within the same representation • In many cases we have closed-form solutions for prediction

  38. Bayesian Estimation in BayesNets • Instances are independent given the parameters • (x[m’],y[m’]) are d-separated from (x[m],y[m]) given  • Priors for individual variables are a priori independent • Global independence of parameters Bayesian network for parameter estimation Bayesian network X X … X[2] X[M] X[1] … Y Y[2] Y[M] Y[1] Y|X

  39. Bayesian Estimation in BayesNets • Posteriors of  are independent given complete data • Complete data d-separates parameters for different CPDs • As in MLE, we can solve each estimation problem separately Bayesian network for parameter estimation Bayesian network X X … X[2] X[M] X[1] … Y Y[2] Y[M] Y[1] Y|X

  40. Bayesian Estimation in BayesNets • Posteriors of  are independent given complete data • Also holds for parameters within families • Note context specific independence between Y|X=0 and Y|X=1 when given both X and Y Bayesian network for parameter estimation Bayesian network X X … X[2] X[M] X[1] … Y Y[2] Y[M] Y[1] Y|X=0 Y|X=1

  41. Bayesian Estimation in BayesNets • Posteriors of  can be computed independently • For multinomial Xi|paiposterior is Dirichlet with parameters Xi=1|pai+M[Xi=1|pai],..., Xi=k|pai+M[Xi=k|pai] Bayesian network for parameter estimation Bayesian network X X … X[2] X[M] X[1] … Y Y[2] Y[M] Y[1] Y|X=0 Y|X=1

  42. Assessing Priors for BayesNets • We need the (xi,pai) for each node xi • We can use initial parameters 0 as prior information • Need also an equivalent sample size parameter M’ • Then, we let (xi,pai) = M’  P(xi,pai|0) • This allows to update a network using new data • Example network for priors • P(X=0)=P(X=1)=0.5 • P(Y=0)=P(Y=1)=0.5 • M’=1 • Note: (x0)=0.5 (x0,y0)=0.25 X Y

  43. MINVOLSET KINKEDTUBE PULMEMBOLUS INTUBATION VENTMACH DISCONNECT PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL FIO2 VENTALV PVSAT ANAPHYLAXIS ARTCO2 EXPCO2 SAO2 TPR INSUFFANESTH HYPOVOLEMIA LVFAILURE CATECHOL LVEDVOLUME STROEVOLUME ERRCAUTER HR ERRBLOWOUTPUT HISTORY CO CVP PCWP HREKG HRSAT HRBP BP Case Study: ICU Alarm Network • The “Alarm” network • 37 variables • Experiment • Sample instances • Learn parameters • MLE • Bayesian

  44. 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 Case Study: ICU Alarm Network • MLE performs worst • Prior M’=5 provides best smoothing 1.4 1.2 1 0.8 KL Divergence 0.6 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 M

  45. Parameter Estimation Summary • Estimation relies on sufficient statistics • For multinomials these are of the form M[xi,pai] • Parameter estimation • Bayesian methods also require choice of priors • MLE and Bayesian are asymptotically equivalent • Both can be implemented in an online manner by accumulating sufficient statistics Bayesian (Dirichlet) MLE

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