PaperlayerView exact revision on arXiv

Source-linked AI summary

On Prior-to-Posterior Stability in the Wasserstein Metric for Bayesian Inverse Problems

Lianghao Cao

arXiv:2609.07197v2math.NAcs.NA

TL;DR

Prior approximations in Bayesian inverse problems raise the question of how prior errors affect posterior distributions and predictions. The paper establishes uniform, Hölder, and Lipschitz stability in Wp under conditions on likelihoods and prior classes, with different rates arising for different prior families.

  • Problem

    Approximations to priors in Bayesian inverse problems make it important to understand how prior errors affect posterior distributions and downstream quantities of interest.

    Source: §1 · p. 1

  • Method

    The paper derives Wp stability estimates under conditions on likelihood regularity and prior classes, including Poincaré and essential-oscillation bounds for Lipschitz stability.

    Source: §3.2 · p. 7, §5.3 · pp. 14, 17

  • Results

    The paper establishes uniform, sharp-exponent Hölder, and Lipschitz prior-to-posterior stability; Darcy results show linear sensitivity for Gaussian priors and square-root sensitivity for two-point priors.

    Source: §7 · p. 20

  • Takeaways & Limitations

    For p ≥ 2, posterior Wp stability controls errors in the means and standard deviations of Lipschitz quantities of interest.

    Source: §1 · p. 2, §7 · p. 20

    For p > 1, likelihood regularity and moment bounds alone need not ensure Lipschitz stability, even for priors supported in a common bounded set.

    Source: §4 · p. 11

Abstract

from arXiv · show

Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior map where both prior and posterior perturbations are measured in the same Wasserstein metric $W_p$, $p\geq1$. We identify verifiable conditions on likelihood regularity and admissible prior classes that ensure uniform, H\"older, and Lipschitz stability. For bounded likelihoods that are uniformly continuous on bounded sets, uniform stability holds over prior classes with uniformly integrable $p$-th moments and a common positive evidence lower bound. With global H\"older regularity of the likelihood and uniform bounds on higher prior moments, a coupling argument leads to a H\"older estimate with a sharp exponent. For $p>1$, a Lipschitz likelihood need not give Lipschitz stability, even for priors with bounded support. We establish Lipschitz stability through an interpolation argument under uniform Poincar\'e bounds and a globally Lipschitz potential with uniformly bounded essential oscillation under the priors. For Gaussian priors with additive Gaussian noise and bounded Lipschitz forward models, these estimates give posterior $W_2$ bounds even for mutually singular prior perturbations. Numerical experiments for a Darcy inverse problem illustrate the predicted H\"older and Lipschitz rates and the resulting control of errors in the posterior mean and standard deviation of a Lipschitz quantity of interest.

1. Introduction.

Bayesian inverse problems update uncertainty from noisy, indirect observations, and approximated priors can affect posteriors and downstream quantities of interest. This work studies prior-to-posterior stability in the same Wasserstein metric and develops uniform, Hölder, and Lipschitz results under verifiable assumptions.

1. Introduction.

  • Prior approximations from discretization, hyperparameter estimation, or learned samples can affect posterior distributions and downstream quantities of interest.For p ≥ 2, Wp stability controls errors in both means and standard deviations of Lipschitz quantities of interest.Source: §1 · pp. 1–2

1.1. Contributions and Paper Organization.

  • The paper establishes uniform, Hölder, and Lipschitz prior-to-posterior stability in Wp under distinct likelihood and prior-class conditions.The Hölder exponent is sharp; the Lipschitz result for p > 1 uses Poincaré bounds and bounded essential oscillation of the potential.Source: §1 · p. 2, §1.1 · pp. 2–3

1.2. Related Work.

  • Related work addresses prior sensitivity, Bayesian brittleness, and posterior stability, with prior-to-posterior bounds often using different input and output metrics.Existing results include qualitative Wp stability and bounds that propagate W2 prior error to W1 posterior error.Source: §1.2 · pp. 3–4

2. Preliminary.

This section sets the notation for continuity and Wasserstein stability, then reviews qualitative Wp stability and W1-Lipschitz stability under a bounded ground metric.

2. Preliminary.

  • The preliminary section introduces the setting and reviews existing qualitative Wp and bounded-ground-metric W1 stability results.The W1 result is attributed to Sprungk.Source: §2 · p. 4

2.1. Setting and Notation.

  • Local continuity means uniform control on bounded balls, while global continuity uses constants on the whole space.Figure 1 distinguishes unconditional implications from implications requiring sufficient conditions; for β = 1, Lipschitz and Hölder notions coincide.Source: §2.1 · p. 5, Figure 1 · p. 5

2.2. Existing Results on Wp Stability.

  • For continuous likelihoods, prior-to-posterior maps are qualitatively Wp-continuous when evidence is positive; a W1-Lipschitz result assumes a bounded ground metric.The bounded-metric estimate does not directly extend to an unbounded ground metric because transport beyond distance D has bounded cost in that metric.Source: §2.2 · pp. 5–6

3. Qualitative Uniform Stability.

The section establishes uniform Wp stability under bounded likelihood, bounded-set uniform continuity, uniformly integrable prior p-th moments, and a common positive evidence lower bound. A coupling estimate supports the result, while examples show why its assumptions matter.

3.1. Posterior Coupling Estimate.

  • Reweighting a prior-optimal coupling may not give both posterior marginals, so the estimate retains compatible mass and controls residual probability mass.The lemma bounds posterior Wp distance using priors with evidence at least ζ > 0.Source: §3.1 · p. 6

3.2. Main Result.

  • A bounded likelihood uniformly continuous on bounded sets gives uniform Wp stability when prior p-th moments are uniformly integrable and evidence has a common positive lower bound.These are the stated sufficient conditions for uniform continuity of the prior-to-posterior map.Source: §3 · p. 6, §3.2 · p. 7

3.3. Dissecting the Assumptions.

  • In infinite-dimensional spaces, continuity alone is insufficient: a continuous likelihood can fail to be uniformly continuous on bounded sets.The section gives points in the unit ball whose distance vanishes while their likelihood values remain separated.Source: §3.3 · p. 8
  • An unbounded likelihood can turn vanishing prior tail mass into an order-one posterior transport cost, so boundedness cannot be omitted.The example uses a globally Lipschitz likelihood L(x) = 1 + |x|.Source: §3.3 · pp. 8–9
  • Uniform p-th moment bounds do not replace uniform integrability, even when prior distances vanish and the likelihood is bounded and globally Lipschitz.The example's Bayesian update increases tail mass and changes the posterior transport cost.Source: §3.3 · p. 9
  • Without a uniform evidence lower bound, prior distances can vanish while posterior distance remains 1/2, even for priors supported on [0, 1].In the example, evidence tends to zero and the update removes the shared atom at 0.Source: §3.3 · p. 9

4. H¨older Stability.

The section derives a power-law Wp stability estimate from global Hölder regularity of the likelihood and a uniform higher-moment bound on priors. Its exponent has two distinct sharpness limits, and local Lipschitz regularity alone gives no uniform power modulus.

4. H¨older Stability.

  • The Hölder estimate has exponent α = min{β/p, 1 − p/q} for a globally β-Hölder likelihood and priors with a uniform q-moment bound, q > p.The theorem also assumes a positive uniform evidence lower bound.Source: §4 · p. 9

4.1. Likelihood Regularity and Exponent Sharpness.

  • The 1 − p/q branch reflects small prior mass far away; its exponent is sharp when 1 − p/q ≤ β/p.The example yields posterior distance asymptotic to 2^-1/p times the prior distance to the power 1 − p/q.Source: §4 · p. 10, §4.1 · p. 12

4. H¨older Stability.

  • For globally Lipschitz likelihoods, moment-based exponents approach 1 as q grows when p = 1, but are at most 1/p for p > 1, even with bounded support.The section therefore turns to Poincaré bounds and bounded potential oscillation for Lipschitz stability.Source: §4 · p. 11

4.1. Likelihood Regularity and Exponent Sharpness.

  • Local Lipschitz continuity does not ensure any uniform power modulus: the example's posterior distance decays polynomially while prior distance decays exponentially.The section contrasts this with the globally Hölder likelihood assumption used for the power-law estimate.Source: §4.1 · p. 11
  • The β/p branch reflects likelihood reweighting that creates an order h^β mass imbalance transported over distance one, and it remains sharp for bounded-support priors.The example rules out any larger uniform exponent when β/p ≤ 1 − p/q.Source: §4.1 · p. 12

5. Lipschitz Stability.

For p > 1, Wp-Lipschitz stability follows under a globally Lipschitz potential and uniform bounds on prior Poincaré constants and potential essential oscillations. The result includes Gaussian priors with additive Gaussian noise, even when prior perturbations produce mutually singular measures.

5.1. Poincar´e Constant.

  • Uniform Lr-Poincaré bounds with r = p/(p − 1) control function variation and are translation invariant, unlike moment bounds.Uniform distributions on [n, n + 1] have the same finite Poincaré constant despite diverging moments, whereas a two-point support has an infinite Poincaré constant despite bounded moments.Source: §5.1 · p. 13

5.2. µ-Essential Oscillation.

  • A uniform bound on potential essential oscillation controls likelihood ratios on each prior's support, without requiring the potential to be globally bounded.For additive Laplace noise and uniform priors on intervals of width D, the oscillation is at most D/b uniformly over the intervals, although the potential is unbounded on R.Source: §5.2 · pp. 13–14

5.3. Main Result.

  • Theorem 5.2 gives Wp-Lipschitz stability for p > 1 under a globally Lipschitz potential and uniform prior Poincaré and potential essential-oscillation bounds.Its proof reweights a Wp-optimal coupling through a geometric interpolation of the likelihood and controls distances along the interpolating curve of marginals.Source: §5.3 · p. 14
  • For pushforward priors, checking conditions on a shared reference measure suffices when the generating maps have uniformly bounded Lipschitz constants and potential oscillations.Theorem 5.2 bounds the posterior error by the Lp norm of the pointwise distance between the generating maps; γp < 2 and γp = 1 when p = 2.Source: §5.3 · pp. 14, 17

5.4. Gaussian Priors and Additive Gaussian Noise.

  • With additive Gaussian noise and a bounded globally Lipschitz forward map, Gaussian priors with uniformly bounded covariance operators satisfy a W2-Lipschitz posterior bound.The bound remains valid when mean or covariance perturbations make the Gaussian priors mutually singular.Source: §5.4 · pp. 17–18

6. A Numerical Example: Two-Layer Darcy Flow.

The Darcy-flow experiment contrasts linear posterior sensitivity for Gaussian priors with square-root sensitivity for two-point priors. It also illustrates bounds on posterior quantity-of-interest statistics for Gaussian perturbations.

6.1. Observation Model.

  • The experiment uses a two-layer Darcy-flow inverse problem with a two-dimensional parameter and additive Gaussian observation noise.The forward map is the pressure at the interface between the layers.Source: §6 · p. 18, §6.1 · p. 18

6.2. Gaussian Prior Perturbations.

  • For Gaussian priors, the posterior W2 error has an approximately linear rate for small perturbations, while the bound uses Cstab ≈ 7.0867.For h ≤ 10^-2, the fitted log–log slope is approximately 1 and W2(ρ0, ρh)/h ≈ 0.9153; the bound captures the rate but its constant is conservative.Source: §6.2 · p. 19
  • The errors in the posterior mean and standard deviation of Q(x) = κ(x1) both lie below their shared bound, with standard-deviation error larger at every tested h.Figure 2 reports the shared bound as Cstabh/2.Source: §6.2 · p. 19, Figure 2 · p. 19

6.3. Comparison with Two-Point Priors.

  • Posterior error scales as h for Gaussian priors and h1/2 for two-point priors as h ↓ 0 under the same likelihood.The two-point family attains the Hӧlder exponent 1/2, contrasting with the linear rate for the Gaussian family.Source: Figure 2 · p. 19, §6.3 · p. 19

7. Conclusion.

The paper establishes uniform, Hölder, and Lipschitz prior-to-posterior stability in the same Wasserstein metric under explicit likelihood and prior-class assumptions. It identifies limits of the Lipschitz conditions and acknowledges AI assistance in the work.

7. Conclusion.

  • The stability results control posterior approximation errors and, for p ≥ 2, errors in means and standard deviations of Lipschitz quantities of interest.The Hölder exponent is sharp under the stated assumptions.Source: §7 · p. 20
  • Likelihood regularity and moment bounds alone do not ensure Lipschitz stability for p > 1, even for priors supported in a common bounded set.The sufficient Lipschitz conditions also require uniform prior Poincaré bounds and uniform bounds on potential essential oscillation under the priors.Source: §7 · p. 20
  • The Lipschitz result does not cover continuous unbounded potentials when the prior has full support.The paper identifies relaxing these assumptions as future research and asks whether learned-prior approximations can maintain uniform bounds on generating-map Lipschitz constants and potential oscillation.Source: §7 · p. 20

Declaration.

  • The author reports AI assistance with numerical experiments, example suggestions, and development of the Lipschitz-stability conditions and proof strategy.The author takes responsibility for the manuscript materials.Source: Declaration · p. 20

Appendix A. Proof of Lemma 3.1.

The proof constructs a coupling of the posterior measures by retaining the compatible mass of a prior coupling and adding a residual coupling when needed.

Appendix A. Proof of Lemma 3.1.

  • The proof normalizes likelihoods under the two priors and uses them to define two unit-mass measures on a prior coupling.Their shared component γ0 has marginals dominated by the two posteriors.Source: Appendix A · p. 22
  • When the residual mass r is positive, adding the scaled product of residual measures completes the coupling of the posteriors.When r = 0, the shared component alone is used.Source: Appendix A · p. 22
  • The proof bounds the residual cost and then substitutes into the preceding estimates to obtain the claimed bound.The resulting bound also holds when the residual term is absent.Source: Appendix A · p. 23

Appendix B. Supplementary Lemmas for Theorem 5.2.

The supplementary lemmas support Theorem 5.2 by controlling changes in tilted measures and translating derivative bounds into Wasserstein bounds. Their proof combines a centering inequality with a dual criterion for Lipschitz curves in Wp.

Appendix B. Supplementary Lemmas for Theorem 5.2.

  • The centering inequality controls the covariance from differentiating tilted measures, with γ2 = 1 and γp < 2 for p > 1.The cited bound gives γp ≤ 2|1−2/p| < 2.Source: Appendix B · pp. 23–24
  • The dual criterion adapts Hopf–Lax duality to arbitrary p > 1, using power-cost estimates.The proof strategy uses the Hopf–Lax inequality to offset the changing-measure contribution under assumption (ii).Source: Appendix B · p. 24
  • The dual criterion yields Wp(ρs, ρt) ≤ Λ(t−s) when the curve is uniformly dominated and its bounded Lipschitz test-function expectations satisfy the stated derivative bound.This is the conclusion of Lemma B.2.Source: Appendix B · p. 24
  • The proof bounds the dual expression along an interpolating curve by separating changes in the test function and measure, then applying the Hamilton–Jacobi inequality and Young’s inequality.Uniform domination justifies the time-dependent test functions; the resulting derivative bound is integrated to obtain the Wasserstein estimate.Source: Appendix B · pp. 24–26

C.1. Stability Constant.

The section evaluates the stability bound using bounds on the potential and its Lipschitz constant. At σ = 0.05, the bound is far above the computed distance ratio.

C.1. Stability Constant.

  • The section applies Theorem 5.2 with a bounded, Lipschitz potential and a Poincaré constant bounded by 5/4.The potential bounds are 0 ≤ Φ ≤ M and Lip(Φ) ≤ K.Source: §C.1 · p. 26
  • At σ = 0.05, Cstab ≈ 2.05×10^5 while W2(ρ0, ρh)/h at h = 10−4 is approximately 0.676, so the bound can be conservative.The model has 0 ≤ Φ ≤ M, Lip(Φ) ≤ K, and CP,2(µh) ≤ 5/4.Source: §C.1 · p. 26

C.2. Posterior Wasserstein Distance.

The posterior Wasserstein distance is evaluated by one-dimensional quadrature, and the analysis establishes a linear rate in h. The reported computations include grid-refinement checks and reference posterior moments.

C.2. Posterior Wasserstein Distance.

  • The posterior distance is computed by one-dimensional quadrature, using the latent posterior CDF and monotone coupling for the stated cost.The CDF is computed by a composite trapezoidal rule on 80,001 points, then inverted by linear interpolation; the cost integral uses composite Simpson quadrature.Source: §C.2 · p. 26
  • Monotone coupling therefore minimizes the cost, while a coefficient tends to a positive limit as h ↓ 0; together with (6.5), this gives W2(ρ0, ρh) ≍ h.Source: §C.2 · p. 26
  • At h = 0, the reference posterior mean is approximately 1.4565 and the standard deviation is approximately 0.2411; grid refinements changed posterior distances by less than 10−7 relatively.The refinements halved both grid spacings and enlarged the integration intervals.Source: §C.2 · p. 26

C.3. Two-Point Priors.

For y > 1/2, the likelihood decreases along the specified ray, and the resulting weight ah is below a0 for h > 0. The priors in this example have bounded support and evidence bounded below, but infinite Poincaré constants.

C.3. Two-Point Priors.

  • For y > 1/2, the decreasing likelihood gives ah < a0 for h > 0, and its derivative at zero is negative.The priors are supported in the unit ball and have evidence at least e−M, while their Poincaré constants are infinite.Source: §C.3 · p. 27
Loading 2609.07197v2…