ECE521 Lecture 21 HMM cont. Message Passing Algorithms
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1 ECE521 Lecture 21 HMM cont Message Passing Algorithms
2 Outline Hidden Markov models Numerical example of figuring out marginal of the observed sequence Numerical example of figuring out the most probable sequence Message-passing algorithm Factor graph review Sum-product algorithms 2
3 Inference in HMMs z1 z2 z3 zd x1 x2 x3 xd Given the joint distribution over the observations and the latent states: Computing the marginal distribution of the observations requires sum over the latent states: local summation The marginal dist is used for both posterior inference and learning: 3
4 Inference in HMMs Example1: z1 z2 z3 zd x1 x2 x3 xd Consider an HMM with three discrete states: The observations are also discrete random variables: We have the following transition and emission probabilities: For a particular training example: C C observations: The marginal distribution of the observed sequence: 4
5 Inference in HMMs z1 z2 z3 zd Example1: The marginal distribution of the observed sequence: x1 x2 x3 xd observations: C C element-wise product 5
6 Inference in HMMs z1 z2 z3 zd Example2: The posterior distribution over the latent states for a training example x1 x2 x3 xd observations: C C posterior over the latent sequence prior over the latent sequence 6
7 Outline Hidden Markov models Numerical example of figuring out marginal of the observed sequence Numerical example of figuring out the most probable sequence Message-passing algorithm Factor graph review Sum-product algorithms 7
8 Factor graph Why Factor graphs (FGs)? Problem: computing marginals and posteriors are tedious and expensive in a graphical model (such as Ex 1 and Ex 2) We need an algorithm to automate the local summations in analogous to backpropagation Advantage over BNs and MRFs: FG provides a simple and unified way to think about local computations (factors and messages) FG makes it easy to design message-passing algorithms that efficiently exploit local factorization structure in a graph
9 The sum product algorithm on factor graph Our first message-passing algorithm: the sum-product algorithm The sum-product algorithm computes probabilities for a subset of the variables of a factor graph, eg P(a, b, c, d) Marginal distributions, eg P(b) Joint distributions of a subset of variables, eg P(a,b) Conditional distributions ( often the posterior distributions of our interest ), eg P(a,c d) = P(a,c,d) / P(d) We need marginals to normalize the posterior distributions
10 Review: convert BNs, MRFs to factor graph Converting Bayesian Networks to factor graph takes the following steps: Consider all the parents of a child node Pinch all the edges from its parents to the child into one factor Create an additional edge from the factor to the child node Move on the the next child node Last step is to add all the priors as individual dongles to the corresponding variables Sanity check: Let the original BN have N variables and E edges The converted factor graph will have N+E edges in total
11 Review: convert BNs, MRFs to factor graph Converting Markov Random Fields to factor graph takes the following steps: Consider all the maximum cliques of the MRF Create a factor node for each of the maximum cliques Connect all the nodes of the maximum clique to the new factor nodes
12 Review: conditional independence in factor graph The Markov blanket for our factor graphs is very similar to MRFs The Markov blanket of a variable node in a factor graph is given by the variables second-neighbour nodes Consider the nearest-neighbours of variable d: a b f 1 f2 f 3 c d e
13 Review: conditional independence in factor graph The Markov blanket for our factor graphs is very similar to MRFs The Markov blanket of a variable node in a factor graph is given by the variables second-neighbour nodes Consider the nearest-neighbours of variable d: Set exclusion notation: denotes the neighbours of x with variable d excluded a b Consider the second-neighbours of variable d: f 1 f2 f 3 c d e
14 Message notations On each edge of the factor graph, there are two messages traveling in opposite directions We use function to denote messages The messages are scalar functions of the variable nodes The subscript denotes the origin and the destination of these messages, eg: factor-to-variable variable-to-factor
15 The sum product algorithm on factor graph Two rules in the sum-product algorithm: Variable-to-factor messages: Let be a subset of the factor nodes in a factor graph connected to variable xi, Incoming messages There are N factor nodes in total connected to xi:
16 The sum product algorithm on factor graph Two rules in the sum-product algorithm: Factor-to-variable messages: Let be a subset of the variable nodes in a factor graph connected to factor fs Incoming messages Note that: There are M variable nodes in total connected to fs:
17 The sum product algorithm on factor graph Two rules in the sum-product algorithm: Factor-to-variable messages: Variable-to-factor messages:
18 The sum product algorithm on factor graph How to start the sum-product algorithm: Choose a node in the factor graph as the root node Compute all the leaf-to-root messages Compute all the root-to-leaf messages
19 The sum product algorithm on factor graph How to start the sum-product algorithm: Choose a node in the factor graph as the root node Compute all the leaf-to-root messages Compute all the root-to-leaf messages Initial conditions (ie the nearest-neighbours exclude the outgoing node is empty set or ): Starting from a factor leaf/root node, the initial factor-to-variable message is the factor itself Starting from a variable leaf/root node, the initial variable-to-factor message is a vector of ones
20 Compute probabilities with messages How to convert messages to actual probabilities: Define unnormalized probabilities as g That is, g is positive everywhere and may not sum to 1 We can normalize g to obtain the actual joint and marginal probabilities: where Z is the normalization constant to get proper probability distributions
21 Compute probabilities with messages How to convert messages to actual probabilities: Compute unnormalized probabilities using messages (the sum-product algorithm): unnormalized marginal probabilities: the product of all the incoming messages from the adjacent factors normalization constant: marginal probabilities:
22 Compute probabilities with messages How to convert messages to actual probabilities: Let be a subset of the variables we wish to compute joint distribution over Compute unnormalized probabilities using messages (the sum-product algorithm): unnormalized joint probabilities: normalization constant: marginal probability of a subset of variables:
23 Compute probabilities with messages How to convert messages to actual probabilities: Let be a subset of the variables we wish to compute joint distribution over Compute unnormalized probabilities using messages (the sum-product algorithm): unnormalized joint probabilities: eg When all the variables in are connected to the same factor We make the computation efficient normalization constant: marginal probability of a subset of variables:
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B-Trees Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Motivation for B-Trees So far we have assumed that we can store an entire data structure in main memory.