Efficient Bayesian Inference in Strictly Semi-parametric Linear Inverse Problems

Abstract: We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal $f$ corrupted by statistical noise, with the transformation $K_θ$ being linear but unknown up to a scalar $θ$. We adopt a Bayesian approach and put a prior on the pair $(θ,f)$ and prove a Bernstein-von Mises theorem for the marginal posterior of $θ$ under regularity conditions on the operators $K_θ$ and on the prior. We apply our results to the recovery of location parameters in semi-blind deconvolution problems and to the recovery of attenuation constants in X-ray tomography.

• The paper establishes a Bernstein–von Mises theorem, showing the marginal posterior of the parameter converges to a Gaussian distribution with efficient information bounds.
• It derives the semi-parametric efficient Fisher information and identifies the least favourable direction, ensuring optimal uncertainty quantification in complex inverse problems.
• The study validates the framework through applications like semi-blind deconvolution and X-ray tomography, demonstrating practical implications for ill-posed settings.

Efficient Bayesian Inference in Strictly Semi-parametric Linear Inverse Problems

Problem Formulation and Theoretical Setting

The paper rigorously investigates Bayesian inference for finite-dimensional parameters in inverse problems where observations take the form

$X^{(n)} = K_\theta f + \frac{1}{\sqrt{n}}\dot{W},$

with a family of linear operators $K_\theta: H_1 \rightarrow H_2$ indexed by an unknown scalar $\theta$, and an unknown function $f$ in an infinite-dimensional Hilbert space. The inverse problem is strictly semi-parametric: uncertainty in both the finite-dimensional ($\theta$) and infinite-dimensional ($f$) components is acknowledged, but their interaction is restricted to the action of $K_\theta$ on $f$.

The study's cornerstone is the Bernstein-von Mises (BvM) theorem for the marginal posterior of $\theta$. Under regularity and identifiability conditions, the limiting distribution is Gaussian with covariance matching the inverse efficient information, yielding asymptotic frequentist validity for Bayesian credible sets for $\theta$.

Semi-parametric Fisher Information and Least Favourable Directions

A major technical accomplishment is the detailed characterization of semi-parametric efficient Fisher information for $\theta$. The authors derive the least favourable direction (LFD) in the nonparametric component $f$ that minimizes Fisher information for $\theta$. For any parametric submodel direction $g\in H$,

$$\text{Fisher Information} = \|\dot{K}_\theta f + K_\theta g\|_{H_2}^2.$$

Projecting $-\dot{K}_\theta f$ onto the closure of $K_\theta H$ yields the efficiency bound and identifies the LFD, which is vital for both the BvM theorem and practical implementation.

Strict positive efficient information $\tilde{i}_{\theta_0, f_0} > 0$ is shown necessary for asymptotic normality, with explicit verification in canonical inverse problems such as semi-blind deconvolution and X-ray tomography.

General Bernstein-von Mises Theorem and Regularity Structure

The methodology employs modern empirical process theory and nonparametric Bayes techniques. The BvM theorem is established under:

• Posterior contraction of the coupled parameter $(\theta, f)$ and relevant functionals at appropriately fast rates.
• Regularity and stability conditions on $K_\theta$ (uniform bounds, differentiability, Taylor expansions, and finite metric entropy for image classes).
• Priors that possess insensitivity to Cameron–Martin translations in the LFD direction.

A pivotal aspect is the distinction between contraction of the full posterior at $(\theta_0, f_0)$ versus contraction in smoother coordinates, such as at $K_{\theta_0} f_0$, which is attainable even in severely ill-posed problems due to the regularizing properties of the forward operator.

Gaussian Process Priors and Contraction Analysis

The authors meticulously treat the scenario where $\pi_f$ is a (rescaled) Gaussian process prior on $f$. They show that, provided appropriate Sobolev-type regularity and the operator's smoothing properties, one obtains polynomial contraction rates for the posterior, even when $f_0$ is only moderately smooth. The scaling of the prior is matched to the ill-posedness and the desired rate via classical small-ball and covering number analyses in the relevant Hilbert or Sobolev spaces.

Key is the demonstration that with proper re-scaling, the prior supports the necessary regularity and ensures that posterior mass concentrates around "small" balls in strong enough topologies to guarantee the uniform LAN (local asymptotic normality) expansions needed for the BvM theorem.

Applications

Semi-blind Deconvolution

The paper investigates the recovery of shift parameters in semi-blind deconvolution, where the convolution operator $K_\theta f = g * f(\cdot - \theta)$, and both $g$ and $f$ are unknown up to prescribed structural constraints (e.g., symmetry, zero location). For symmetric $f$, the Fisher information loss due to $f$ vanishes, achieving the parametric (ideal) information lower bound; for zero-location $f$, a characterization of the LFD and identification of attainable information is provided. The theory is instantiated with explicit Gaussian series priors for $f$ adapted to the Sobolev scale dictated by $g$'s regularity.

Attenuated X-ray Tomography

For inference on the attenuation parameter in generalizations of the geodesic X-ray transform, the paper gives a comprehensive functional analytic framework using Zernike-Sobolev scales and manages the complexity of non-commutative geometric flows on manifolds with boundary. The efficient Fisher information and LFDs are characterized, and the necessary regularity assumptions are verified using precise operator-theoretic estimates and mapping properties. The approach is robust to mild ill-posedness and leverages advanced functional analysis and microlocal analysis as needed.

Implications and Further Developments

The results provide a systematic recipe for semi-parametric inference in non-linear and nonparametric inverse problems of moderate ill-posedness, with rigorous uncertainty quantification via Bayesian credible sets for the finite-dimensional parameter. The strong specification of contraction rates and LFD approximability in Gaussian process frameworks is of substantial practical use for uncertainty quantification in inverse problems, notably signal processing and (medical) tomography. The explicit BvM theorems hold at nearly parametric rates in certain ill-posed settings, articulating precisely when regularization helps rather than hinders a Bayesian approach.

From a theoretical standpoint, future developments might extend these efficiency results to larger classes of nonlinearity in $(\theta, f)\mapsto K_\theta f$, higher-dimensional parameter spaces, or priors with more involved dependence structures. The analytic machinery developed may see application to statistical inverse problems governed by more complex PDEs or underlying physical processes.

Conclusion

This work establishes a comprehensive theoretical framework for efficient Bayesian inference on finite-dimensional parameters within semi-parametric linear inverse problems, rigorously deriving conditions for the nonparametric Bernstein-von Mises theorem in the presence of operator and signal uncertainty. The implications span both concrete inverse problems and foundational statistical theory, with detailed analyses for Gaussian process priors, regularity/stability structures, and application to semi-blind deconvolution and X-ray tomography. The results clarify the interplay between operator regularity, prior geometry, and information bounds, and provide practitioners with conditions to achieve asymptotic normality and efficient uncertainty quantification for parameters of interest in high-dimensional and ill-posed settings (2602.00901).