CS839: Probabilistic Graphical Models. Lecture 7: Learning Fully Observed BNs. Theo Rekatsinas

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1 CS839: Probabilistic Graphical Models Lecture 7: Learning Fully Observed BNs Theo Rekatsinas 1

2 Exponential family: a basic building block For a numeric random variable X p(x ) =h(x)exp T T (x) A( ) = 1 Z( ) h(x)exp( T T (x)) is an exponential family distribution with natural (canonical) parameter η Function T(x) is a sufficient statistic. Function A(η) = logz(η) is the log normalizer Examples: Bernoulli, multinomial, Gaussian, Poisson, Gamma, Categorical 2

3 Why exponential family? Moment generating property We can easily compute moments of any exponential family distribution by taking the derivatives of the log normalizer A(η) 3

4 MLE for Exponential Family For iid data the log-likelihood is We take the derivatives and set them to zero We perform moment matching We can infer the canonical parameters using 4

5 Generalized Linear Models The graphical model: Linear regression Discriminative linear classification E p (T )=µ = f( T X) Generalized Linear Model The observed input x is assumed to enter into the model via a linear combination of its elements = T x. The conditional mean μ is represented as a function f(ξ) of ξ, where f is known as the response function. The observed output y is assumed to be characterized by an exponential family distribution with conditional mean μ. 5

6 Learning in Graphical Models Goal: Given a set of independent samples (assignments to random variables), find the best Bayesian Network (both DAG and CPDs) (B,E,A,C,R) = (T,F,F,T,F) (B,E,A,C,R) = (T,F,T,T,F) (B,E,A,C,R) = (F,T,T,T,F) 6

7 Learning in Graphical Models Goal: Given a set of independent samples (assignments to random variables), find the best Bayesian Network (both DAG and CPDs) Structure learning (B,E,A,C,R) = (T,F,F,T,F) (B,E,A,C,R) = (T,F,T,T,F) (B,E,A,C,R) = (F,T,T,T,F) 7

8 Learning in Graphical Models Goal: Given a set of independent samples (assignments to random variables), find the best Bayesian Network (both DAG and CPDs) Structure learning (B,E,A,C,R) = (T,F,F,T,F) (B,E,A,C,R) = (T,F,T,T,F) (B,E,A,C,R) = (F,T,T,T,F) Parameter learning 8

9 Learning in Graphical Models Goal: Given a set of independent samples (assignments to random variables), find the best Bayesian Network (both DAG and CPDs) Structure learning Later in the class (B,E,A,C,R) = (T,F,F,T,F) (B,E,A,C,R) = (T,F,T,T,F) (B,E,A,C,R) = (F,T,T,T,F) Parameter learning 9

10 Parameter Estimation for Fully Observed GMs The data: D = (x1, x2, x3,, xn) Assume the graph G is known and fixed Expert design or structure learning Goal: estimate from a dataset of N independent, identically distributed (iid) training examples D Each training example corresponds to a vector of M values one per node random variable Model should be completely observable: no missing values, no hidden variables 10

11 Density estimation A construction of an estimate, based on observed data, of an unobservable underlying probability density function Can be viewed as single-node graphical models Instances of exponential family distribution Building blocks of general GM MLE and Bayesian estimate 11

12 Discrete Distributions Bernoulli distribution: P (x) =p x (1 p) 1 x Multinomial distribution: Mult(1, θ) X =[X 1,X 2,X 3,X 4,X 5,X 6 ] X j =[0, 1], X X j = 1 with probability j, j2[1,,6] X X j =1 j2[1,,6] theta j =1 P (X j = 1) = j 12

13 Discrete Distributions Multinomial distribution: Mult(n, θ) n =[n 1,n 2,...,n k ]where X j n j = N p(n) = N! n 1!n 2! n k! n 1 1 n 2 2 n K K 13

14 Example: multinomial model Data: We observed N iid die rolls (K-sided): D = {5, 1, K, 3}} x n =[x n,1,x n,2,,x n,k ]wherex n,k =0, 1 Model: Likelihood of an observation: Likelihood of D: KX k=1 X n,k = 1 with probability k and P (x 1,x 2,...,x N ) = x n,k =1 X k2{1,,k} k =1 P (x i )=P({x n,k =1, where k is the index of the n-th roll}) KY = k = x n,1 1 x n,2 2 x n,k k = x n,k k NY n=1 P (x n ) =Y k n k k k=1 14

15 MLE: constrained optimization Objective function: l( ; D) = log P (D ) = log Y k n k k = X k n k log k We need to maximize this subject to the constraint: Lagrange multipliers: k = n k k =0 n k = l( ; D) = X k ) X k n k = X k k n k log k + (1 k ) N = X k k ) ˆ k,mle = 1 N X n x n,k Sufficient statistics? 15

16 Bayesian estimation I need a prior over parameters θ Dirichlet distribution Posterior of θ Isomorphism of the posterior with the prior (conjugate prior) Posterior mean estimation P ( ) = Q (P k k) k ( k ) Y k k 1 k P ( x 1,...,x N )= p(x 1,...,x N )p( ) p(x 1,...,x N ) k = Z k p( D)d = C Z = C( ) Y k / Y k k Y k n k k a k 1 Y k k 1 k = Y k k+n k 1 k d = n k + k N + k+n k 1 k 16

17 Continuous Distributions Uniform Gaussian Multivariate Gaussian 17

18 MLE for a multivariate Gaussian You can show that the MLE for μ and Σ is What are the sufficient statistics? 18

19 MLE for a multivariate Gaussian You can show that the MLE for μ and Σ is What are the sufficient statistics? Rewrite Sufficient statistics are: Similar for Bayesian estimation: Normal prior 19

20 MLE for general BNs If we assume the parameters for each CPD are globally independent, and all nodes are fully observed, then the log-likelihood function decomposes into a sum of local terms, one per node MLE-based parameter estimation of GM reduces to local est. of each GLIM. 20

21 Decomposable likelihood of a BN Consider the GM: This is the same as learning four separate smaller BNs each of which consists of a node an its parents. 21

22 MLE for BNs with tabular CPDs Each CPD is represented as a table (multinomial) with In case of multiple parents the CPD is a high-dimensional table The sufficient statistics are counts of variable configurations The log-likelihood is And using a Lagrange multiplier to enforce that conditionals sum up to 1 we have: 22

23 What about parameter priors? In a BN we have a collection of local distributions How can we define priors over the whole BN? We could write P(x1,x2, xn;g,θ)p(θ α) Symbolically the same as before but θ is defined over a vector of random variables that follow different distributions. We need θ to decompose to use local rules. Otherwise we cannot decompose the likelihood any more. We need certain rules on θ Complete Model Equivalence Global Parameter Independence Local Parameter Independence Likelihood and Prior Modularity 23

24 Global and Local Parameter Independence Global Parameter Independence For every DAG model Local Parameter Independence For every node 24

25 Global and Local Parameter Independence 25

26 Which PDFs satisfy these assumptions? Discrete DAG Models Gaussian DAG Models 26

27 Parameter sharing Transition probabilities P(X t X t-1 ) can be different: What is the parameterization cost? 27

28 Parameter sharing Transition probabilities P(X t X t-1 ) can be different: What is the parameterization cost? I need a different local conditional distribution. How can I learn the transition probabilities? I need iid data. So my training examples should correspond multiple rolls of the dice. I need to do alignment etc. What do we do? 28

29 Parameter sharing We will make the assumption that for every transition we follow the same conditional Consider a time-invariant (stationary) 1 st -order Markov model Initial state probability vector State transition probability matrix Now: π (multinomial) What about A? optimize separately 29

30 Learning a Markov chain transition matrix A is a stochastic matrix with X A ij =1 Each row of A is a multinomial distribution MLE of A ij is the fraction of transitions from i to j j Sparse data problem: If i to j did not occur in the data then A ij is 0. Any future sequence with pair i to j will have zero probability Backoff smoothing 30

31 Example: HMM supervised ML estimation Given x = x1, x2,, xn for which the true state path y1, y2,, yn is known: Define The maximum likelihood parameters θ are: If x is continuous we can apply learning rules for a Gaussian 31

32 Supervised ML estimation Intuition: when we know the underlying states, the best estimate of θ is the average frequency of transitions and emissions that occur in the training data Drawback: Given little data, we may overfit (remember zero probabilities) Example: Given 10 rolls we have Then 32

33 Pseudocounts Solution for small training sets: Add pseudocounts The pseudocounts represent our prior belief Large total pseudocounts => strong prior belief Small total pseudocounts => smoothing to avoid 0 probabilities Equivalent to Bayesian estimation under a uniform prior with parameter strength equal to the pseudocounts. 33

34 Summary For fully observed BN, the log-likelihood function decomposes into a sum of local terms, one per node; thus learning is also factored Learning single-node GM density estimation: exp. Family Typical discrete distribution Typical continuous distribution Conjugate priors Learning BN with more nodes: Local operations 34