An Introduction to Variational Methods for Graphical Models

1. An Introduction to Variational Methods for Graphical Models

2. Introduction (1) • Problem of Probabilistic Inference • H : set of hidden nodes • E : set of evidence nodes • P(E) : likelihood • Provide satisfactory solution to inference and learning • cases where time or space complexity is unacceptable

3. Introduction (2) • Variational Method • provide approach to the design of approximate inference algorithm • deterministic approximation procedures • Intuition • “Complex graphs can be probabilistically simple.”

4. Exact Inference (1) • Overview of Exact inference for graphical model • Junction Tree Algorithm • Directed Graphical Model (Bayesian network) • joint probability distribution of all of the N nodes • P(S) = P(S1, S2, …, SN)

5. Exact Inference (2) • Undirected Graphical Model • potential  • function on the set of configuration of a clique that associates a positive real number with each configuration • joint probability distribution

6. Exact Inference (3) • Junction Tree Algorithm • Moralization Step • Compiles directed graphical model into undirected graphical models • Triangulation Step • input : moral graph • output : undirected graph in which additional edges have been added • allow recursive calculation of probabilities to take place • triangulated graph • data structure known as junction tree

7. Junction Tree • running intersection property • “If a node appears in any two cliques in the tree, it appears in all cliques that lie on the path between the two cliques.” • local consistency  global consistency • time complexity of probabilistic calculation • depends on the size of the cliques • for discrete data, the number of values to represent the potential is exponential in the number of nodes in the clique.

8. The QMR-DT database (1) • large-scale probabilistic database • intended to be used as a diagnostic aid • bipartite graphical model • upper layer : diseases • lower layer : symptoms • 600 disease nodes and 400 symptom nodes

9. The QMR-DT database (2) • finding • observed symptoms • f : the vector of findings • d : vector of diseases • all nodes are binary

10. The QMR-DT database (3) • The joint probability over diseases and findings: • Prior probabilities of the diseases are obtained from archival data. • Conditional Probabilities were obtained from expert assessments under a “noisy-OR” model • qij are parameters obtained form the expert assessment.

11. The QMR-DT databse (4) • Rewrite the nosy-OR model • Joint probability distribution • Negative findings • benign with respect to the inference problem • Positive findings • cross products terms  couple the diseases • coupling terms  exponential growth in inferential complexity • Diagnostic calculation under QMR-DT model is generally infeasible.

12. Neural networks as graphical models • Neural networks • layered graphs endowed with a nonlinear “activation” function at each node • Activation function • bounded zero and one • f(z) = 1 / (1+e-z) • Treat neural network as a graphical model • associating a binary variable Si with each node • interpreting the activation of the node as the probability that the associated binary variable takes one of its two values

13. Neural networks (2) • Example (sigmoid belief network) • ij : parameter associated with the edges between parent nodes j and node i • i0 : bias

14. Neural networks (3) • Exact inference is infeasible in general layered neural network models • a node has as parents all of the nodes in the preceding layer. • Thus the moralized neural network graph has links between all of the nodes in this layer • hidden units in the penultimate layer become probabilistic dependent, as do their ancestors in the preceding hidden layers

15. Factorial hidden Markov models (1) • FHMM • composed of a set of M chains • : state node for the mth chain at time i • A(m) : transition matrix for the mth chain

16. FHMM (2) • Overall transition probability • effective state space for the FHMM • the Cartesian product of the state space associated with the individual chains • Represent a large effective state space with a much smaller number of parameters

17. FHMM (3) • Emission probabilities of the FHMM • Ghahramani and Jordan • B(m) and  : matrices of parameters • the states become stochastically coupled when the outputs are observed.

18. FHMM (4) • Time complexity • N : the number states in each chain • cliques for the hidden state : size N3 • time complexity of exact inference O(N3T) • triangulation creates cliques of size N4 • complexity of exact inference : O(N4T) • in general : M is number of chain : O(NM+1T)

19. Hidden Markov decision trees • HMDT (Hidden Markov Decision Tree) • Make decisions in decision tree conditional not only the current data point, but also on the decisions at the previous moment in time • Dependency is assumed to be level-specific. • The probabilities of a decision depends only on the previous decision at the same level of the decision tree • Problem • Given a sequence of input vector Ui and sequence of output vector Yi, compute the conditional probability distribution over the hidden states. • intractable (including FHMM)

20. Basics of variational methodology • Variational Methods • Converts a complex problem into a simpler problem • The simpler problem is generally characterized by a decoupling of the degrees of freedom in the original problem • Decoupling is achieved via an expansion of the problem to include additional parameters(variational parameter) that must be fit to the problem.

21. Examples (1) • Express the logarithm function variationally: •  : variational parameter

22. Examples (2) • For any given x, for all  • Variational transformation provides a family of upper bounds on the logarithm. • The minimum over bounds is the exact value of the logarithm • Pragmatic justification • nonlinear function  linear function • cost : a free parameter  for each x • if we set  well we can obtain a good bound

23. Example (3) • For binary-valued nodes, it is common to represent the probability that the node takes one of its values via a monotonic nonlinearity • Example: logistic regression model • x : weighted sum of the values of the parents of a node • neither convex nor concave  simple linear bound will not work • log concave

24. Example (4) • Bound the log logistic function with linear functions • bound the logistic function by the exponential • H() : binary entropy function • Good choice of  provide better bound.

25. Example (5) • Significance of the advantage of the transformation • For conditional probabilities represented with logistic regression, we obtain product of functions of the form f(x) = 1 / (1+e-x). • Augment network representation by including variational parameters, a bound on the joint probability is obtained by taking products of exponentials.

26. Convex duality (1) • General fact of convex analysis • concave function f(x) can be represented via a conjugate or dual function • conjugate function f*()

27. Convex duality (2) • f(x) and linear function x for a particular  • shift x vertically by an amount which is the minimum of the value x - f(x) • obtain upper bounding line with slope  that touches f(x) at a single point

28. Convex duality (3) • Framework of convex duality applies equally well to lower bound. • Convex duality is not restricted to linear bounds.

29. Approximation for join probabilities and conditional probabilities • Directed Graphs • Suppose we have a lower bound and upper bound for each of the local conditional probabilities . • We have forms and . • let E and H be a disjoint partition of S

30. Approximation (2) • Given that upper bound hold for any settings of values the variational parameter , it holds in particular for optimizing settings of the parameters. • The right hand side of the equations • a function to be minimized with respect to . • Distinction between joint prob. and marginal prob. • joint probabilities • If we allow the variational parameters to be set optimally for each value of the argument S, then it is possible to find optimizing settings of the variational parameters that recover the exact value of the joint probability • marginal probabilities • generally not able to recover exact value