Infinite-Dimensional Bayesian Framework

• Infinite-dimensional Bayesian framework is a rigorous formulation for inference with function-valued parameters defined on Banach or quasi-Banach spaces.
• It extends classical methods by incorporating heavy-tailed, α-stable priors that enable robust modeling of sparse, discontinuous structures.
• Mesh-invariant sampling algorithms like pCN and MALA ensure that posterior stability and convergence persist under discretization refinement.

The infinite-dimensional Bayesian framework describes the rigorous formulation and analysis of Bayesian inference and uncertainty quantification where the parameter of interest is a function, field, or any other object governed by infinite-dimensional mathematical structures such as (quasi-)Banach or Hilbert spaces. Unlike finite-dimensional Bayesian analysis, this framework is designed to be invariant under discretization, providing well-posedness and stability properties that persist in the function-space limit. Central challenges include prior modeling, posterior consistency, computational tractability, and the extension to non-Gaussian and heavy-tailed priors.

1. Mathematical Structure: Function-Space Parametrization and Posterior Definition

The framework is formulated on real separable Banach or quasi-Banach spaces $X$ (for parameters) and $Y$ (for data), equipped with Borel σ-algebras. The unknown parameter $u \in X$ is related to observations $y \in Y$ through a forward map $G: X \to Y$ and an observation model, frequently with additively separable noise $y = G(u) + \eta$, e.g., $\eta \sim N(0, \Sigma)$ for Gaussian noise.

The prior $\mu_0$ is a probability measure on $X$. The likelihood is encoded via the negative log-likelihood (potential) $\Phi(u;y)$, such as $$\Phi(u;y) = \frac{1}{2}\| \Sigma^{-1/2}(G(u) - y) \|_Y^2$$ for Gaussian noise.

The Bayesian posterior is then defined by the Radon–Nikodym derivative with respect to the prior:

$$\frac{d\mu^y}{d\mu_0}(u) = \frac{1}{Z(y)}\exp(-\Phi(u;y))$$

where $Z(y) = \int_X \exp(-\Phi(u;y)) \mu_0(du)$ (Sullivan, 2016).

Well-posedness in infinite dimensions requires verification of regularity, integrability, and lower-boundedness properties for $\Phi$ and $\mu_0$. For Gaussian priors, Fernique's theorem gives the necessary exponential integrability; for stable priors, only power-type or logarithmic integrability can be obtained, requiring corresponding modifications in the analysis pipelines (Sullivan, 2017, Sullivan, 2016).

2. Heavy-Tailed Stable Priors and Quasi-Banach Space Extensions

The infinite-dimensional Bayesian framework has been extended to accommodate heavy-tailed priors, such as α-stable laws, on quasi-Banach spaces (Sullivan, 2017, Sullivan, 2016). In this context, the parameter space $X$ may be a quasi-Banach space (e.g., $\ell^p$ or $L^p$ for $0 < p < 1$), and the prior $\mu_0$ is constructed through an infinite α-stable series expansion:

$$u = \sum_{n=1}^\infty u_n \psi_n, \quad u_n \sim \text{Stable}(\alpha, \beta_n, \gamma_n, \delta_n)$$

Convergence and measure support properties are established via frame inequalities and moment estimates, ensuring that such priors are properly defined in infinite dimensions even without finite second moments. For these priors, weaker conditions on the potential $\Phi$ (e.g., only logarithmic decay in the tails is required) suffice for the existence and uniqueness of the posterior measure.

Moreover, the framework guarantees that the resulting posterior depends locally Lipschitz-continuously (in the Hellinger and total variation metrics) on the data and on perturbations of the forward model, under suitable integrability assumptions reflecting the moments of the prior (Sullivan, 2017, Sullivan, 2016).

3. Well-Posedness, Stability, and Metrics

Existence and stability of the posterior are governed by a set of core assumptions. In both Hilbert/Banach and quasi-Banach spaces, it is required that:

• The negative log-likelihood $\Phi$ is measurable, locally bounded, and lower-bounded by a function $M_{1,R}(\|u\|_X)$ for all $y$ in bounded balls,
• For all $R > 0$, $\exp(-M_{1,R}(\|u\|_X)) \in L^1(X, \mu_0)$,
• $\Phi(u;y)$ is locally Lipschitz-continuous in $y$, possibly with a Lipschitz constant growing logarithmically or polynomially with $\|u\|_X$.

Under these, the posterior normalization constant is finite and positive, and the posterior measure is well-defined and Radon (Sullivan, 2016).

Lipschitz dependence on the data is quantified by Hellinger and total variation distances. If suitable integrability holds (allowing for logarithmic, not just polynomial, tails due to the lack of higher moments), one has

$d_H(\mu^y, \mu^{y'}) \leq C \|y - y'\|_Y$

for all $y, y'$ in bounded balls (Sullivan, 2017, Sullivan, 2016). Equivalent bounds hold for the total variation distance. This provides a rigorous continuity and stability theory for Bayesian inference in infinite-dimensional settings, even under heavy-tailed priors.

4. Posterior Consistency: Discretization and Limiting Behavior

The infinite-dimensional framework ensures that if all analysis is performed in function space and not merely after finite-dimensional discretization, mesh refinement does not alter the definition or stability of the Bayesian inference problem (Sullivan, 2017, Sullivan, 2016). For discretization schemes via spectral or finite element expansions, convergence of the finite-dimensional approximate posteriors to the true infinite-dimensional posterior in strong metrics (such as Hellinger) is established under mild assumptions.

This property, sometimes termed "discretization-invariance," guarantees that computations and theory performed at the discrete level reflect true function-space inference and do not introduce artificial regularization, concentration, or loss of uncertainty due to finite truncation effects.

5. Sampling, Algorithms, and Computational Considerations

Infinite-dimensional Bayesian models impose distinct challenges and features for parameter exploration algorithms:

• MCMC Algorithms: Standard random-walk Metropolis proposals suffer from vanishing acceptance rates with increasing discretization dimension. Instead, mesh-invariant methods such as preconditioned Crank-Nicolson (pCN), pCN-GM, MALA, and dimension-independent independence samplers are developed, in which proposal distributions and acceptance probabilities remain stable as the function-space limit is approached (Hu et al., 2015, Sullivan, 2017). For heavy-tailed priors, these algorithms require careful adjustment but remain applicable due to the function-space definition of the prior and likelihood.
• Posterior Computation: In the presence of non-Gaussian priors, such as α-stable laws, posterior sampling requires only mild regularity and integrability, since proposal moves and acceptance ratios are constructed directly in the function-space setting and make use of series expansions in an appropriate (quasi-)Banach basis.
• Significance for Sparsity and Robustness: Heavy-tailed stable priors enable modeling of compressible, sparse, or discontinuous fields in infinite dimensions, as the framework permits priors with only weak (log-moment) regularity. This is particularly important in applications such as imaging, where sparse structures and jumps are prevalent.

6. Implications, Limitations, and Extensions

The extension of the infinite-dimensional Bayesian framework to non-Hilbertian parameter spaces and heavy-tailed stable priors preserves the core well-posedness, stability, and discretization-invariance properties established in the Gaussian case but substantially broadens the class of admissible models and phenomena.

Notably, the requirement that misfit lower bounds need only grow as −C log ‖u‖ for large ‖u‖ (as opposed to quadratic or higher polynomial growth in Gaussian settings) means that the inference protocol is robust to priors with infinite variance or even absence of moments above order α for α-stable laws (Sullivan, 2017, Sullivan, 2016).

A plausible implication is that this framework may serve as a solid mathematical and computational foundation for Bayesian inference in problems where phenomena such as outliers, jumps, or sparse representation are of primary importance, and where classical Gaussian assumptions are inappropriate or insufficient.

7. Summary Table: Key Mathematical Ingredients

Feature	Gaussian Case	Stable/Heavy-tailed Case
Parameter space	Hilbert space (e.g., L²)	Quasi-Banach (e.g., L^p, p<1)
Prior support	Exponential moment	Only fractional or log moments
Well-posedness condition on Φ	Quadratic or faster growth	Logarithmic growth sufficient
Posterior stability (Hellinger, TV)	Requires exp(Lipschitz bound)	Requires log-Lipschitz bound
Discretization-invariant sampling	Yes (pCN, etc.)	Yes (under modified integrability)

The infinite-dimensional Bayesian framework thus rigorously establishes the key properties—existence, stability, and computational tractability—of Bayesian inference for function-valued unknowns, and robustly extends to non-Gaussian, heavy-tailed prior constructions and quasi-Banach spaces, while maintaining practical relevance for modern applications such as sparse recovery, imaging, and spatial-temporal modeling.