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Factorial Mixture of Gaussians and the Marginal Independence Model Ricardo Silva silva@statslab.cam.ac.uk Joint work-in-progress with Zoubin Ghahramani

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Goal To model sparse distributions subject to marginal independence constraints

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Why? X1 Y1 Y2Y3Y4Y5 X2

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

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Why?

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How? X1 Y1 X2 Y2 X3 Y3

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How?

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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

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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)

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Example

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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)

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

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

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

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Not really Y1Y2Y3 C

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Required: a factorial mixture of Gaussians c1 c2 c3

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

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A parameterization under latent variables

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

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

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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

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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

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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] )

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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

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Maximum likelihood estimation An EM framework

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Maximum likelihood estimation An EM framework

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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

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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

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12 = 3,12 Iterative conditional fitting: Gaussian case Y1Y1 Y2Y2 Y3Y3 11 12 12 22 Y1Y1 Y2Y2 Y3Y3 11 2 22 33 3 Y1Y1 Y2Y2 Y3Y3 23 13 23 = 31 32 = 0 32 13 11 + 12 = 0 23 13 = f(, 12 ) 23 b b b b b3b3 b b bb b b

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Iterative conditional fitting: Gaussian case Y1Y1 Y2Y2 Y3Y3 Y1Y1 Y2Y2 Y3Y3 33 Y1Y1 Y2Y2 Y3Y3 23 R 2.1 Y3 = R 2.1 + 3, where R 2.1 is the residual of the regression of Y2 on Y1 23 b b

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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

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Parameter expansion Yi = b 1 c R 1 + b 2 c R 2 +... + 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?

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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’

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

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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?

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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’

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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

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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

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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?

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

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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

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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?

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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?

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Thank You

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