Machine Learning for Data Science (CS4786) Lecture 24

Transcription

1 Machine Learning for Data Science (CS4786) Lecture 24 Graphical Models: Approximate Inference Course Webpage :

2 BELIEF PROPAGATION OR MESSAGE PASSING Each node passes messages to its parents and children Guaranteed to work on a tree To get marginal of node X v (given evidence) Parents(X v ) messages to X v P(X v Parent(X v )) For general graphs, belief propagation need not work Inference for general graphs can be computationally hard Can we perform inference approximately?

3 WHAT IS APPROXIMATE INFERENCE? Obtain ˆP(X v Observation) that is close to P(X v Observation) Additive approximation: ˆP(Xv Observation) P(X v Observation) Multiplicative approximation: (1 ) ˆP(X v Observation) P(X v Observation) (1 + )

4 APPROXIMATE INFERENCE Two approaches: Inference via sampling: generate instances from the model, compute marginals Use exact inference but move to a close enough simplified model

5 INFERENCE VIA SAMPLING Law of large numbers: empirical distribution using large samples approximates the true distribution Some approaches: Rejection sampling: sample all the variables, retain only ones that match evidence Importance sampling: Sample from a different distribution but then apply correction while computing empirical marginals Gibbs sampling: iteratively sample from distributions closer and closer to the true one

6 REJECTION SAMPLING Example: X 1 X 2 X 3 X 4 ˆP(X v = 1) P(X v = 1) 1 # of samples

7 REJECTION SAMPLING Algorithm: Topologically sort variables (parents first children later) For t = 1 to n For i = 1 to N End For End For Sample x t i P(X i X 1 = x t 1,...,X i 1 = x t i 1) Throw away x t s that do not match observations

8 REJECTION SAMPLING Example: X 1 X 2 X 3 X 4 What about ˆP(Xv = 1observation) P(X v = 1observation)?

9 IMPORTANCE SAMPLING Example: Likelihood weighting X 1 X 2

10 IMPORTANCE SAMPLING Likelihood weighting: Topologically sort variables (parents first children later) For t = 1 to n Set w t = 1 For i = 1 to N End For End For If X i is observed, set w t w t P(X i = x i X 1 = x t 1,...,X i 1 = x t i 1) Else, sample x t i P(X i X 1 = x t 1,...,X i 1 = x t i 1) To compute P(VariableObservation) set, P(Variable = valueobservation) = n t=1 w t1{variable = value} n t=1 w t

11 IMPORTANCE SAMPLING More generally importance sampling is given by: Draw x 1,...,x n Q Notice that E X P [f (X)] = E X Q P(X) Q(X) f (X) Hence, ˆP(X = x) 1 n n P(X = x) 1{x t = x} t=1 Q(X = x) Idea draw samples from Q but re-weight them

12 GIBBS SAMPLING Fix values of observed variables v to the observations (x 1 v = x v ) Randomly initialize all other variables u by randomly sampling x 1 u For t = 2 to n For i = 1 to N If X i is observed set x t i = xt 1 i Else sample x t+1 i from x t+1 i P(X i X 1 = x t+1 1,...,X i 1 = x t+1 i 1, X i+1 = x t i+1,...,x N = x t N) End For End For Take (x n 1,...,xn N ) as one sample and repeat

13 MCMC SAMPLING IN GENERAL Gibbs sampling belongs o a class of methods called Markov Chain Monte Carlo methods We start by sampling from some simple distribution Set up a markov chain whose stationary distribution is the target distribution That is, based on previous sample (state) we transit to the next state, and then to the next state and so on If the transition probabilities are set up right, after multiple transitions, our sample looks like one from target distribution