Bayesian Inference with Optimal Maps

Tarek A. El Moselhy; Youssef M. Marzouk

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Bayesian Inference with Optimal Maps

Tarek A. El Moselhy, Youssef M. Marzouk

arXiv:1109.1516v3Sep 7, 2011stat.COmath.NAmath.ST

TL;DR

Bayesian inference for noisy, limited observations often relies on computationally intensive MCMC with correlated samples and heuristic convergence diagnostics. The paper instead constructs a deterministic, monotone map from prior to posterior through optimization, yielding independent sampling, algebraic posterior moments, convergence information, and automatic evidence evaluation. Across linear, nonlinear ODE, and high-dimensional PDE inverse problems, the approach is reported as more reliable and efficient than MCMC, while bounded-support posteriors and triangular-map monotonicity remain limitations.

Problem
Bayesian inverse problems require uncertainty quantification, but MCMC can be computationally intensive, produce correlated samples, and lack clear convergence criteria.
Method
The paper constructs a deterministic monotone map that pushes the prior measure to the posterior and computes it through an optimization problem.
Results
The method is demonstrated on linear, nonlinear ODE, and high-dimensional nonlinear PDE inverse problems, with greater reliability and efficiency than MCMC reported particularly for high-dimensional problems.
Takeaways & Limitations
The map representation enables independent posterior samples, algebraic posterior moments, clear convergence monitoring, and automatic computation of the marginal likelihood.
Takeaways & Limitations
Bounded-support posteriors near boundaries require reparameterization or future methods for directly bounding the map range, and triangular-map monotonicity remains theoretically unresolved.

Abstract

from arXiv · show

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.

1. Introduction

The paper motivates replacing MCMC for Bayesian inverse problems because posterior computation can be costly, correlated, difficult to diagnose, and inconvenient for uncertainty propagation. It introduces map-based inference, which constructs a prior-to-posterior transformation and provides independent sampling, posterior moments, convergence information, and evidence evaluation.

Motivation: Bayesian inverse problems require uncertainty quantification because observations can be indirect, noisy, and limited, while parameter uncertainty propagates to predictions.
Limitations of MCMC: MCMC is widely used for posterior simulation but produces correlated samples, reducing effective sample sizes and increasing posterior-estimation errors.
Limitations of MCMC: MCMC can require thousands or millions of posterior evaluations and lacks a clear convergence criterion or principled burn-in rule.
Limitations of MCMC: Sample-based posterior representations can make uncertainty propagation impractical for computationally intensive models, including sequential Monte Carlo methods that retain weighted samples.
Map-based inference: The proposed method constructs a polynomial map that pushes the prior measure to the posterior, avoids Markov chain simulation, and exploits likelihood or forward-model gradients when available.
Advantages: Map optimization automatically computes the marginal likelihood, supplies a convergence criterion, enables arbitrarily many independent posterior samples, and permits algebraic evaluation of posterior moments.
Related approaches: The paper contrasts direct approximation of random variables through maps with variational Bayesian methods that approximate distributions within selected parameterized families.
Evaluation: Experiments cover a linear problem, nonlinear ODE systems with non-Gaussian posteriors, and high-dimensional nonlinear PDE inverse problems.

2. Formulation

The formulation seeks a monotone map that transforms the prior measure into the posterior and computes it by minimizing a map-dependent discrepancy. Optimal-transport constraints establish uniqueness, while alternative parameterizations and regularization support finite-dimensional optimization.

2.1. Bayesian framework: Bayesian inference models parameters with a prior, observations with a likelihood, and the posterior as the parameter distribution conditioned on observed data.
2.1. Bayesian framework: The forward model maps parameters to idealized observations, while additive Gaussian noise is a common likelihood construction but is not required by the map-based method.
2.2. Inference with a map: The core formulation seeks a map f whose pushforward of the prior measure equals the posterior measure, with the map depending on the data and likelihood.
2.2. Inference with a map: The transformed posterior density can be compared with the prior through its Jacobian, and monotonicity provides a condition for constructing the inferential map.
2.2. Inference with a map: The map is parameterized componentwise, and Hellinger or KL discrepancies are rewritten into optimization objectives involving likelihoods, prior densities, and the map Jacobian.
2.2. Inference with a map: The cornerstone quantity T(x; f) should be constant over the prior support, motivating minimization of its variance; the evidence is recovered afterward without being required during optimization.
2.3. The optimal transport formulation: Optimal transport makes the map unique almost everywhere under weak conditions by selecting the gradient of a convex function, which is monotone.
2.3. The optimal transport formulation: The approach may regularize the map with an optimal-transport-inspired penalty, but it targets accurate measure transformation rather than true L2-optimal transport.

3. Solution Algorithm

The solution algorithm approximates a prior-to-posterior map with prior-orthogonal multivariate polynomials and computes its coefficients through stochastic optimization. It uses derivative-based solvers, staged polynomial enrichment, and composite maps for transformations that require high polynomial degree.

Optimization formulation: The optimization problems are stochastic and potentially infinite-dimensional because they involve prior expectations and maps represented in function spaces.Likelihood evaluations may require computationally intensive forward-problem solutions.
Polynomial approximation: The map is approximated with multivariate polynomials orthogonal with respect to the prior measure, using a finite multi-index set to truncate the expansion.The construction assumes independent prior variables for the displayed tensor-product polynomial form, while dependent-variable expansions are also possible.
Polynomial approximation: The polynomial map supports analytical posterior moments and uncertainty propagation without additional sampling or forward-model solves.Posterior covariance is computed from weighted outer products of expansion coefficients, while higher moments use expectations of products of orthogonal polynomials.
Optimization algorithms: Newton’s method and nonlinear least squares solve for map coefficients, with Newton’s method requiring first and second derivatives of the objective and derivatives of the forward map.Adjoint techniques can compute gradients and Hessian-vector products at a cost of O(1) forward solves independent of parameter dimension.
Composite map: Composite maps use intermediate distributions, such as progressively reduced noise covariance, to approximate high-degree prior-to-posterior transformations more efficiently.Intermediate-map errors can be tolerated only when the construction preserves the required measure transformations; otherwise, errors propagate into the final map.

4. Results

The numerical results show that optimal-map inference accurately recovers posterior structure across linear, nonlinear, and distributed-parameter inverse problems. Polynomial-map construction converges reliably, agrees with MCMC where compared, and supports high-dimensional posterior analysis.

4.1. Linear-Gaussian model: 15 iterations yielded maps matching the analytical linear-Gaussian solution in both optimization formulations.The evidence also converged to its analytical value within machine precision.
4.1. Linear-Gaussian model: The triangular map matched the posterior Cholesky factor with relative Frobenius error below 10^-6.A symmetric optimal-transport map also matched the posterior covariance square root with relative error below 10^-6 when symmetry was enforced and the penalty was reduced to zero.
4.2. Nonlinear ODE systems: In the two-parameter ODE example, posterior samples concentrated near k2 = 2k1 and the stable-trajectory region, while only 0.08% had negative Jacobian determinants.The negative determinants were confined to tails of the prior, indicating that monotonicity was lost only in those regions.
4.4. Distributed-parameter inverse problems: A four-stage cascade of third-order maps was required to accurately capture the high-order posterior in the distributed-parameter problem.The resulting marginal posterior densities showed good agreement with a long MCMC run, and only 861 of 10^6 prior samples had negative Jacobian determinants.
4.4. Distributed-parameter inverse problems: For permeability-field inference, optimal-map and MCMC posterior means, standard deviations, realizations, and covariances were broadly similar.The map also enabled analytical computation of posterior covariance, while MCMC failed to converge in almost half of attempted Case III runs and the map reliably reached tolerance δ.

5. Conclusions

The paper presents deterministic map-based Bayesian inference with convergence monitoring, analytical posterior quantities, and inexpensive independent sampling. Numerical examples support its reliability and efficiency, while the conclusions identify boundaries involving hierarchical, bounded-support, ill-posed, and high-dimensional problems.

Core approach: The method constructs a deterministic measure-preserving map from prior to posterior by solving an optimization problem, rather than generating correlated MCMC samples.Existence and uniqueness of a monotone map are established under weak conditions.
Core approach: Var[T(X)] → 0 provides a convergence criterion for terminating optimization or adaptively enriching the map’s function space.The criterion supports a specified target accuracy.
Computational outputs: The optimization computes the posterior evidence, while orthogonal-polynomial coefficients yield analytical posterior moments and minimal-cost independent posterior samples.Arbitrary numbers of independent samples can be generated by applying the map to prior samples.
Computational outputs: Gradient-based Newton or quasi-Newton methods with adjoints can solve the stochastic optimization objective, and compositions of low-order maps can represent complex changes of measure economically.The objective involves prior expectations and is therefore stochastic.
Numerical demonstrations: The approach converges rapidly in a linear problem and is demonstrated on nonlinear ODE and high-dimensional, ill-posed nonlinear inverse problems involving an elliptic PDE.The examples include real experimental data and complicated, non-Gaussian posteriors.
Overall assessment: Map inference is reported as more reliable and efficient than MCMC, especially for high-dimensional inverse problems; typical map costs equate to roughly 200 independent MCMC samples.The comparison is described as incomplete because maps provide additional information and diagnostics.
Extensions: The map’s polynomial-chaos representation supports posterior uncertainty propagation and extensions to filtering, prediction, and alternative base measures.The paper notes that a third base measure could map to both prior and posterior when prior sampling is inconvenient or the prior is improper.
Scope and limitations: The conclusions identify unfinished development for hierarchical models, bounded-support posteriors, ill-posed problems, and high-dimensional nonlinear problems without a natural variable ordering.Bounded domains are currently handled by reparameterization, while high-dimensional cases may require high-order maps.

Appendix A. Relation between variance and Kullback-Leibler divergence near convergence

This appendix relates the convergence diagnostic Var[T(X)] to posterior approximation error measured by Kullback-Leibler divergence. Near convergence, small variance corresponds asymptotically to a linear relationship with the KL divergence.

Variance–divergence relation: When Var[T(X)] is small, the KL divergence from the target posterior p to its approximation ˜p is also small.The relationship is established for the near-convergence regime.
Variance–divergence relation: For small perturbations of T around its expectation T0 = E[T], the variance–KL relationship is asymptotically linear.The appendix begins the derivation from the KL divergence.

Appendix B. Linear-Gaussian model

The linear-Gaussian case admits a closed-form Gaussian posterior and an affine prior-to-posterior map. The map’s linear component is not uniquely determined by the covariance constraint.

Closed-form solution: A linear forward model with additive Gaussian noise and a Gaussian prior has a closed-form Gaussian posterior.The prior mean is taken to be zero without loss of generality.
Closed-form solution: The posterior mean, covariance, and evidence are represented by μ, Σ, and β, respectively.These quantities arise by equating the posterior-density expressions.
Map construction: The prior-to-posterior map is specified by an affine transformation whose offset is the posterior mean.The map expression is introduced after deriving the Gaussian relations.
Map construction: The linear map component Z1 must satisfy Σ = Z1ΣPZ1^T, but this condition does not uniquely determine Z1.With identity prior covariance, right multiplication by any orthonormal matrix produces another valid map.

Appendix C. Nonlinear model with Gaussian prior and Gaussian additive noise

For nonlinear forward models with Gaussian priors and additive Gaussian noise, the paper derives optimization derivatives for polynomial map representations. The implementation must handle expensive second derivatives and their tensor storage carefully.

Assumptions and setup: The appendix derives derivatives with respect to optimization parameters for nonlinear forward models with Gaussian priors and additive Gaussian observational noise.The prior is specialized to zero mean and identity covariance, and the map uses multivariate polynomials.
Assumptions and setup: The problem setup uses prior density p, posterior density q, and data d, with the map Jacobian entering the transformed-density formulation.The quantity T is defined from the Jacobian determinant and the posterior formulation.
Optimization formulation: The finite-dimensional map is represented with multivariate polynomials orthogonal under the prior measure, forming a polynomial chaos expansion.The basis approximation makes the otherwise infinite-dimensional optimization computationally feasible.
Derivative computation: Newton’s method requires second derivatives of the logarithm of the Jacobian determinant and of the nonlinear forward-model terms.The forward-model derivative with respect to optimization variables is the principal expensive quantity identified in the implementation discussion.
Derivative computation: Second derivatives of the vector-valued forward model are best stored as an m × ℓ × ℓ third-order tensor, although a reduced matrix representation is possible for a specific norm term.Here m is the number of observations and ℓ is the number of optimization variables.