Factorial Mixture of Gaussians and the Marginal Independence Model Ricardo Silva Joint work-in-progress with Zoubin Ghahramani

Goal To model sparse distributions subject to marginal independence constraints

Why? X1 Y1 Y2Y3Y4Y5 X2

Why? X1 Y1 X2 Y2 X3 Y3 H 12 H 23

Why?

How? X1 Y1 X2 Y2 X3 Y3

How?

Context Y i = f i (X, Y) + E i, where E i is an error term E is not a vector of independent variables Assumed: sparse structure of marginally dependent/independent variables Goal: estimating E-like distributions

Why not latent variable models? Requires further decisions How many latents? Which children? Silly when marginal structure is sparse In the Bayesian case: – Drag down MCMC methods with (sometimes much) extra autocorrelation – Requires priors over parameters that you didn’t even care in the first place (Note: this talk is not about Bayesian inference)

Example

Bi-directed models: The story so far Gaussian models – Maximum likelihood (Drton and Richardson, 2003) – Bayesian inference (Silva and Ghahramani, 2006, 2008) Binary models – Maximum likelihood (Drton and Richardson, 2008)

New model: mixture of Gaussians Latent variables: mixture indicators – Assumed #levels is decided somewhere else No “real” latent variables

Caveat emptor I think you should buy this, but be warned that speed of computation is not the primary concern of this talk

Simple? Y1Y2Y3 C Y1, Y2, Y3 jointly Gaussian with sparse covariance matrix  c indexed by C

Not really Y1Y2Y3 C

Required: a factorial mixture of Gaussians c1 c2 c3

A parameterization under latent variables Assume Z variables are zero-mean Gaussians, C variables are binary

A parameterization under latent variables

Implied indexing  ij should be indexed only by those cliques containing both Yi and Yj

Implied indexing  i should be indexed only by those cliques containing Yi

Factorial mixture of Gaussians and the marginal independence model The general case for all latent structures Constraints: in the indexing whenever c and c’ agree on clique indicators in the intersection of the cliques with Yi and Yj

Factorial mixture of Gaussians and the marginal independence model The parameter pool (besides mixture probs.): – {  c[i] }, the mean vector – {  ii c[i] }, the variance vector – {  ij c[ij] }, the covariance vector For Yi and Yj linked in the bi-directed graph (since covariance is zero otherwise): marginal independence constraints Given c, we assemble the corresponding mean and covariance

Size of the parameter space Let L[i  j] be the size of the largest clique intersection, L[i] largest clique Let k be the maximum number of values any mixture indicator can take Let p be the number of variables, e the number of edges in bi-directed graph Total number of parameters: O(e k L[i  j] + pk L[i] )

Size of the parameter space Notice this is not a simple function of sparseness: – Dependence on the number of clique intersections – Non-sparse models can have few cliques In decomposable models, number of clique intersections is given by the branch factor of the junction tree

Maximum likelihood estimation An EM framework

Maximum likelihood estimation An EM framework

Algorithms First, solving the exact problem (exponential in the number of cliques) Constraints – Positive definite constraints – Marginal independence constraints Gradient-based methods – Moving in all dimensions quite unstable Violates constraints – Move over a subset while keeping part fixed

Iterative conditional fitting: Gaussian case See Drton and Richardson (2003) Choose some Yi Fix the covariance of Y \i Fit the covariance of Yi with Y \i, and its variance Marginal independence constraints introduced directly

 12 =  3,12 Iterative conditional fitting: Gaussian case Y1Y1 Y2Y2 Y3Y3  11  12  12  22 Y1Y1 Y2Y2 Y3Y3 11 2 22 33 3 Y1Y1 Y2Y2 Y3Y =  31  32 = 0   11 +  12 = = f(,  12 ) 23   b b b b b3b3 b b bb b b

Iterative conditional fitting: Gaussian case Y1Y1 Y2Y2 Y3Y3 Y1Y1 Y2Y2 Y3Y3 33 Y1Y1 Y2Y2 Y3Y3 23 R 2.1 Y3 = R  3, where R 2.1 is the residual of the regression of Y2 on Y1 23 b b

How does it change in the mixture of Gaussians case? Y1Y1 Y2Y2 Y3Y3 C 12 C 23 Y1Y1 Y2Y2 Y3Y3 Y1Y1 Y2Y2 Y3Y3 Y1Y1 Y2Y2 Y3Y3 C 12 C 23

Parameter expansion Yi = b 1 c R 1 + b 2 c R  c c does vary over all mixture indicators That is, we create an exponential number of parameters – Exponential in the number of cliques Where do the constraints go?

Parameter constraints Equality constraints are back Similar constraints for the variances b 1 c  R1j c + b 2 c  R2j c b k c  Rkj c = b 1 c’  R1j c’ + b 2 c’  R2j c’ b k c’  Rkj c’

Parameter constraints Variances of  c,  c, have to be positive Positive definiteness for all c is then guaranteed (Schur’s complement)

Constrained EM Maximize expected conditional of Y i given everybody else subjected to – An exponential number of constraints – An exponential number of parameters – Box constraints on gamma What does this buy us?

Removing parameters Covariance equality constraints are linear Even a naive approach can work: – Choose a basis for b (e.g., one b ij c corresponding to each non-zero  ij c[ij] ) – Basis is of tractable size (under sparseness) – Rewrite EM function as a function of basis only b 1 c  R1j c + b 2 c  R2j c b k c  Rkj c = b 1 c’  R1j c’ + b 2 c’  R2j c’ b k c’  Rkj c’

Quadratic constraints Equality of variances introduce quadratic constraints tying  s and bs Proposed solution: – fix all  ii c[i] first – fit b ij s with such fixed parameters Inequality constraints, non-convex optimization – Then fit  s given b – Number of parameters back to tractable Always an exponential number of constraints Note: reparameterization also takes an exponential number of steps

Relaxed optimization Optimization still expensive. What to do? Relaxed approach: fit bs ignoring variance equalities – Fix , fit b – Quadratic program, linear equality constraints I just solve it in closed formula –  s end up inconsistent Project them back to solution space without changing b – always possible May decrease expected log-likelihood – Fit  given bs Nonlinear programming, trivial constraints

Recap Iterative conditional fitting: maximize expected conditional log-likelihood Transform to other parameter space – Exact algorithm: quadratic inequality constraints “instead” of SD ones – Relaxed algorithm: no constraints No constraints?

Approximations Taking expectations is expensive what to do? Standard approximations use a ``nice’’  ’(c) – E.g., mean-field methods Not enough!

A simple approach The Budgeted Variational Approximation As simple as it gets: maximize a variational bound forcing most combinations of c to give a zero value to  ’(c) – Up to a pre-fixed budget How to choose which values? This guarantees positive-definitess only of those  (c) with non-zero conditionals  ’(c) – For predictions, project matrices first into PD cone

Under construction Implementation and evaluation of algorithms – (Lesson #1: never, ever, ever, use MATLAB quadratic programming methods) Control for overfitting – Regularization terms The first non-Gaussian directed graphical model fully closed under marginalization?

Under construction: Bayesian methods Prior: product of experts for covariance and variance entries (times a BIG indicator function) MCMC method: a M-H proposal based on the relaxed fitting algorithm – Is that going to work well? Problem is “doubly-intractable” – Not because of a partition function, but because of constraints – Are there any analogues to methods such as Murray/Ghahramani/McKay’s?

Thank You