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Dimension-independent likelihood-informed MCMC

Tiangang Cui, Kody J. H. Law, Youssef M. Marzouk

arXiv:1411.3688v2stat.COmath.NAstat.ME

TL;DR

High-dimensional Bayesian inference over discretized functions can make standard MCMC increasingly slow and can violate function-space validity conditions. The paper introduces operator-weighted, likelihood-informed proposals built from an inhomogeneous Langevin discretization, and reports substantial efficiency gains for nonlinear inverse problems. Its approach assumes a Gaussian reference structure and focuses on global operators, leaving combinations with local operators as future work.

  • Problem

    Standard MCMC can slow under function discretization refinement, while many conventional proposals are not well defined in the infinite-dimensional limit.

  • Method

    The paper constructs function-space operator-weighted proposals that exploit a global likelihood-informed subspace and use different Langevin discretizations on that subspace and its complement.

  • Results

    The DILI samplers offer significant gains in sampling efficiency over current state-of-the-art algorithms across the demonstrated nonlinear inverse problems.

  • Takeaways & Limitations

    The resulting samplers provide dimension-independent MCMC performance while capturing finite-direction departures from the Gaussian prior, including non-Gaussianity.

  • Takeaways & Limitations

    The work focuses on global operators independent of the local parameter value, with combining global and local operators left as future work.

Abstract

from arXiv · show

Many Bayesian inference problems require exploring the posterior distribution of high-dimensional parameters that represent the discretization of an underlying function. This work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt to the particular structure of a posterior distribution over functions. Two distinct lines of research intersect in the methods developed here. First, we introduce a general class of operator-weighted proposal distributions that are well defined on function space, such that the performance of the resulting MCMC samplers is independent of the discretization of the function. Second, by exploiting local Hessian information and any associated low-dimensional structure in the change from prior to posterior distributions, we develop an inhomogeneous discretization scheme for the Langevin stochastic differential equation that yields operator-weighted proposals adapted to the non-Gaussian structure of the posterior. The resulting dimension-independent, likelihood-informed (DILI) MCMC samplers may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure. Two nonlinear inverse problems are used to demonstrate the efficiency of these DILI samplers: an elliptic PDE coefficient inverse problem and path reconstruction in a conditioned diffusion.

1. Introduction

The paper targets Bayesian function-space inference problems where standard MCMC slows as discretizations become finer. It combines dimension-independent operator-weighted proposals with likelihood-informed structure to improve posterior exploration.

  • Standard MCMC convergence can degrade as high-dimensional function representations are refined, including in PDE inverse problems and stochastic differential equation path reconstruction.
  • The framework uses operator-weighted proposal distributions that are well defined on function space and support discretization-independent performance.
  • Likelihood-informed subspaces identify directions where posterior structure differs most strongly from the prior, particularly when local Hessians have aligned low-rank dominant eigenspaces.
  • The paper extends likelihood-informed subspaces to infinite dimensions while seeking exact posterior sampling and using them as convergence-acceleration mechanisms.
  • DILI proposals arise from an inhomogeneous Langevin SDE discretization that treats the likelihood-informed subspace and its complement differently.

2. Theoretical background

The theoretical background formulates Bayesian inverse problems over function spaces and explains why valid, dimension-independent MCMC proposals require careful measure-theoretic construction. It motivates adapting proposals to posterior geometry and prior-to-posterior differences.

  • 2.1. Bayesian inference framework: The unknown is modeled as a function in a separable Hilbert space, with a Gaussian prior whose covariance operator is self-adjoint, positive definite, and trace-class.
  • 2.1. Bayesian inference framework: The observation model uses a forward operator with additive Gaussian noise, while regularity assumptions support bounded data misfit and posterior domination by the prior.
  • 2.2. MCMC on function space: Metropolis-Hastings requires mutual absolute continuity of forward and reverse measure pairs to define a valid acceptance probability and transition kernel.
  • 2.2. MCMC on function space: Many conventional proposals violate this condition as dimension tends to infinity, causing convergence rates to deteriorate under mesh refinement.
  • 2.3. Preconditioned semi-implicit proposal: pCN achieves dimension-independent autocorrelation but uses a uniform prior-covariance scale, making moves overly conservative in prior-dominated directions.
  • 2.3. Preconditioned semi-implicit proposal: Operator-weighted proposals introduce direction-dependent step sizes and can reduce autocorrelation, but a single-point Hessian cannot represent all non-Gaussian posterior structure.
  • 2.4. Explicit proposals using local geometry: A globalized likelihood-informed partition can combine effective finite-dimensional proposals with function-space sampling on the infinite-dimensional complement.

3. Likelihood-informed MCMC

The paper combines operator-weighted proposals with likelihood-informed subspaces to build adaptive, dimension-independent MCMC methods for function-space posteriors.

  • The framework combines dimension-independent operator-weighted proposals, likelihood-informed subspaces, low-rank posterior covariance approximations, and adaptive proposal construction.These components are integrated through an inhomogeneous discretization of a preconditioned Langevin SDE.
  • The resulting proposals adapt MCMC updates to posterior structure while preserving function-space validity.
  • The section presents several operator-weighted proposals and an adaptive posterior sampling framework combining LIS construction, covariance approximation, and automatic proposal adaptation.

3.1. Operator-weighted proposals

Operator-weighted proposals provide a flexible class of function-space MCMC updates whose operators can be chosen to exploit posterior structure while maintaining dimension-independent sampling.

  • The proposal class uses bounded self-adjoint operators A, B, and G defined through a common orthonormal eigenbasis.Their spectral representations support coordinate-wise operator weighting in transformed parameter space.
  • Theorem 3.1 gives eigenvalue conditions ensuring that the operator-weighted proposal defines a valid Metropolis-Hastings algorithm on function space.These conditions include bounded eigenvalues, nondegenerate noise coefficients, and compatibility when components are not updated.
  • Transformations to whitened coordinates simplify proposal derivations and can improve finite-dimensional implementation efficiency.The transformed formulation requires applying the prior covariance square root once to transform back for forward-model evaluation and once to compute the transformed gradient.
  • The framework permits different step sizes in different parameter directions by selecting operators using Hessian and prior-covariance information.

3.2. Likelihood-informed subspace

The likelihood-informed subspace identifies directions where the likelihood most strongly changes the prior, extending local Hessian-based constructions to global, infinite-dimensional inverse problems.

  • The LIS captures posterior departures from the prior that are concentrated in a finite-dimensional parameter subspace.This structure reflects smoothing by the forward operator, limited observations, and prior smoothing.
  • For nonlinear problems, the global LIS is formed from posterior expectations of local Hessian information rather than a single parameter value.This addresses variation of the Hessian across parameter space and extends the construction to an infinite-dimensional complementary subspace.
  • The prior-preconditioned Gauss-Newton Hessian ranks directions by the relative strength of likelihood information versus prior information.Large eigenvalues correspond to likelihood-dominated directions, while small eigenvalues correspond to prior-dominated directions.
  • The low-rank posterior covariance approximation requires sample counts determined by LIS rank r rather than discretized dimension N.Its eigendecomposition costs O(r^3), and the approximation’s storage and factorization are dominated by the LIS representation.
  • The Förstner distance and weighted subspace distance provide consistent criteria for monitoring convergence of the global LIS.

3.3. DILI proposals

DILI proposals use the global LIS to apply posterior-adapted Langevin updates in informed directions and prior-preserving updates in the complementary subspace.

  • The global LIS and low-rank posterior covariance approximation yield operator-weighted proposals with discretization-invariant mixing.
  • The parameter space is partitioned into the finite-dimensional LIS and an infinite-dimensional complementary subspace containing prior-dominated directions.
  • An inhomogeneous Langevin discretization assigns separate update behavior to the LIS and complementary subspace.The tuning parameters switch between Langevin and random-walk proposals in the LIS and complementary subspace.
  • The proposals depend on the global LIS basis, the diagonalized LIS covariance, and separate time steps for the LIS and complementary subspace.
  • LI-Prior uses a Crank-Nicolson scheme in the LIS, while LI-Langevin uses a Langevin proposal there; both use pCN updates in the complementary subspace.

The acceptance probability of this proposal has the form

The paper combines likelihood-informed subspace updates with operator-weighted proposals to construct function-space MCMC schemes. These schemes include Metropolis-within-Gibbs variants, adaptive LIS construction, and benchmark proposals using prior or Hessian information.

  • 3.3.2. Metropolis-within-Gibbs scheme: Metropolis-within-Gibbs updates separately propose changes in the likelihood-informed subspace and its complement, then accept or reject each candidate.The LIS update uses a Langevin or prior-invariant proposal, while the complementary-space update uses a separate operator-weighted proposal.
  • 3.3.2. Metropolis-within-Gibbs scheme: MGLI-Prior uses a likelihood-informed proposal invariant with respect to a shifted prior distribution, with both updates sharing the acceptance probability in (38).
  • 3.3.2. Metropolis-within-Gibbs scheme: MGLI-Langevin applies a Langevin proposal in the LIS and uses acceptance probabilities in (40) and (38) for the LIS and complementary-space updates.
  • 3.4. Adaptive posterior sampling framework: The adaptive framework constructs and updates a global LIS basis and empirical posterior covariance from posterior samples, then integrates these with operator-weighted proposals.
  • 3.4. Adaptive posterior sampling framework: The adaptive operator-weighted framework incurs linear-in-discretization-dimension proposal application costs, while acceptance evaluations require likelihood calculations and Langevin gradients.
  • 3.4. Adaptive posterior sampling framework: LIS adaptation is monitored through a convergence diagnostic, limited by a maximum number of eigendecompositions, and followed by fixed-LIS sampling with covariance updates restricted to the LIS.
  • 3.4. Adaptive posterior sampling framework: Each adaptive iteration proposes an LIS candidate, accepts or rejects it, proposes a complementary-space candidate, and accepts or rejects that second update.
  • 3.5. Benchmark algorithms: Benchmark samplers include pCN-RW and H-Langevin, with the latter using a constant low-rank MAP Hessian preconditioner rather than recomputing local Hessians.

4. Example 1: Elliptic PDE

The elliptic PDE experiment infers a transmissivity field from noisy potential observations on a discretized domain. Across both noise levels, operator-weighted proposals improve mixing, particularly in prior-dominated directions, while global LIS construction further improves sampling.

  • 4. Example 1: Elliptic PDE: The experiment infers an elliptic PDE transmissivity field and tests operator-weighted proposals against pCN-RW and H-Langevin.It also evaluates posterior-sample-based LIS construction and LIS behavior under grid refinement.
  • 4.1. Problem setup: The forward model solves a Poisson equation with transmissivity κ(s), potential p(s), and forcing f(s) on Ω = [0, 1]2.The PDE uses Neumann boundary conditions and is discretized with bilinear finite elements on a uniform 40 × 40 grid.
  • 4.1. Problem setup: The data contain 25 potential measurements under SNR10 and SNR50, with the two noise levels used to examine their effects on LIS structure.The transmissivity has a log-normal prior with exponential-kernel covariance, while the truth is not directly drawn from that prior.
  • 4.2. Sampling efficiency: MGLI-Langevin generally mixes better than competing proposals, especially for high-index prior eigenfunctions that are less constrained by the data and more aligned with the complementary subspace.On low-index modes, operator-weighted and H-Langevin proposals have similar mixing; pCN-RW decorrelates slowly.
  • 4.2. Sampling efficiency: The OMF autocorrelation results show shorter autocorrelation times for operator-weighted proposals than for H-Langevin and pCN-RW.
  • 4.2. Sampling efficiency: The same pattern holds for SNR50: MGLI-Langevin resembles H-Langevin on smooth directions but significantly outperforms it on high-index directions and in OMF mixing.
  • 4.2. Sampling efficiency: Across all components, MGLI-Langevin autocorrelations decrease for high-index modes and are lower on average than H-Langevin autocorrelations.
  • 4.3. Global versus local LIS: The global LIS improves mixing over a single local LIS in both noise cases, with dimensions r = 66 for SNR10 and r = 193 for SNR50.The larger SNR50 LIS is attributed to greater information carried by the data.

5. Example 2: Conditioned diffusion

The conditioned diffusion example reconstructs a particle path and driving force from noisy observations, then evaluates DILI samplers against other MCMC proposals. DILI methods show substantially stronger mixing, with global LIS construction outperforming a local MAP-based LIS.

  • Problem: Noisy observations of a particle diffusing in a double-well potential are used to infer the driving force and reconstructed path.The setup uses 20 equispaced observations over [0, 10], with σ = 0.1 and an Euler-Maruyama discretization of dimension N = 1000.
  • Posterior setup: Figure 11 compares the true trajectory, noisy observations, posterior mean, pointwise quantiles, and marginalized posterior densities.The time interval is long enough to capture two transitions between the potential wells.
  • Sampling performance: DILI samplers outperform pCN and show dramatically improved mixing over H-Langevin in OMF traces and autocorrelations.H-Langevin performs almost as poorly as pCN in this conditioned diffusion example.
  • Sampling performance: MGLI-Langevin produces much faster lag-1 autocorrelation decay than H-Langevin across the first 1000 prior eigenfunctions.Figure 13 reports componentwise lag-1 autocorrelations for the v parameter.
  • Global versus local LIS: A global LIS constructed adaptively yields faster OMF autocorrelation decay and better lag-1 autocorrelations than a local MAP-based LIS.Both comparisons use the MGLI-Langevin algorithm.

6. Conclusions

The paper develops function-space operator-weighted proposals and uses global likelihood-informed subspaces to construct dimension-independent samplers adapted to posterior structure. Experiments on two nonlinear inverse problems suggest significant sampling-efficiency gains, while several extensions remain open.

  • Contributions: Operator-weighted proposals are well defined on function space and remain dimension-independent under refinement of the function discretization.The proposal class includes earlier dimension-independent algorithms as special cases.
  • Contributions: A global LIS captures finite-dimensional directions containing departures from the prior, including posterior non-Gaussianity.The LIS is identified by approximating the posterior expectation of the Hessian of a preconditioned data-misfit functional.
  • Contributions: The four studied DILI proposals combine geometric information with local gradient information and Metropolis-within-Gibbs updates.An adaptive strategy also yields computationally affordable approximations of the posterior covariance.
  • Results: Numerical experiments on two nonlinear inverse problems suggest significant sampling-efficiency gains over current state-of-the-art algorithms.The examples are an elliptic PDE coefficient inverse problem and conditioned-diffusion path reconstruction.
  • Future directions: The paper leaves alternative LIS constructions, gradient-free approaches, and combinations of global and local operators for future work.These directions include active subspaces, adaptive covariance estimates, and locally preconditioned proposals.

Appendix A. Feldman-Hajek Theorem

The Feldman-Hajek theorem characterizes when two Gaussian measures are equivalent. The characterization requires matching Cameron-Martin spaces, an admissible mean difference, and a covariance condition.

  • Equivalence conditions: Two Gaussian measures are equivalent only if they have the same Cameron-Martin space.This is the first stated Feldman-Hajek condition.
  • Equivalence conditions: Their mean-function difference must belong to the Cameron-Martin space.This is the second stated condition for equivalence.

Appendix B. Proof of Theorem 3.1

The appendix proves that the operator-weighted proposal defines a valid Metropolis-Hastings algorithm for the infinite-dimensional posterior. The proof establishes equivalence of the forward and reverse joint Gaussian measures using Feldman-Hajek conditions.

  • Proposal representation: The proposal is represented using Gaussian reference variables and a shifted reference mean before simplifying its transition distribution.The simplified proposal is equivalent to the original proposal.
  • Measure equivalence: The proof compares forward and reverse joint Gaussian measures associated with the proposal and posterior.Equivalence is required for a well-defined infinite-dimensional Metropolis-Hastings algorithm.
  • Proof strategy: Feldman-Hajek conditions are verified by decomposing the space into LIS and complementary components and analyzing the corresponding covariance operators.The block structure makes some covariance conditions automatic under the theorem assumptions.
  • Conclusion: The operator difference is Hilbert-Schmidt under the theorem assumptions, completing the equivalence argument.Therefore the proposal yields a well-defined Metropolis-Hastings algorithm for the infinite-dimensional posterior equivalent to the prior.

Appendix C. Proof of Corollary 3.5

The appendix derives the acceptance probability for operator-weighted proposals by transforming variables and comparing associated Gaussian measures. It also verifies almost-sure finiteness and handles fixed components in the conditional update.

  • Conditional update: Components with bi = 0 are fixed by the conditional update and do not affect the acceptance probability.The proof therefore assumes |bi| ≥c > 0 for all i ∈N.
  • Proposal measures: The proof defines forward and reverse proposal measures and expresses the acceptance probability through their associated measure pairs.The measures are constructed using the posterior and Gaussian reference measures.
  • Transformation: The transformation (15) rewrites the proposals in transformed parameters as w′ = DAw + DBξ.This transformation is used to simplify the derivation of the acceptance ratio.
  • Gaussian measure comparison: The transformed proposal measures are shown to follow Gaussian laws, enabling derivation of the required Radon-Nikodym derivatives.The proof invokes the unitary nature of transformation (15) and Gaussian-measure representations.
  • Finiteness: Equivalence of the Gaussian laws for q(w, dw′) and ˜q(w, dw′) guarantees that the logarithm of their density ratio is µ0-a.s. finite.This finiteness follows under Theorem 3.1 and Assumption 2.1(3), including almost-sure finiteness of the relevant white-noise inner product.
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