Posterior Sampling in Bayesian Inference

• Posterior sampling is a Bayesian method that generates random draws from the posterior distribution, allowing for full uncertainty quantification and robust statistical inference.
• Techniques such as MCMC, surrogate modeling, and Langevin dynamics enhance sampling efficiency in high-dimensional or non-convex settings while ensuring theoretical convergence.
• Applications in inverse problems, reinforcement learning, and differential privacy benefit from posterior sampling’s practical insights, optimized computation, and rigorous error analysis.

Posterior sampling is a family of Bayesian computational methodologies that draw random samples from the posterior distribution over parameters or latent variables, conditional on observed data and prior information. It underpins uncertainty quantification in inverse problems, decision-making under uncertainty, and exploration in sequential optimization, and has seen significant technical advancement in areas such as high-dimensional inference, generative modeling, reinforcement learning, and differentially private data analysis.

1. Bayesian Formulation and Motivation

In the Bayesian paradigm, given prior density $\pi(\theta)$ and likelihood $\pi(d\mid\theta)$, the posterior distribution over parameters $\theta$ given observed data $d$ is

$$\pi(\theta\mid d) \propto \pi(d\mid\theta)\,\pi(\theta).$$

Direct computation of posterior quantities (mean, mode, credible intervals) is often intractable in high dimensions or non-convex models. Posterior sampling via Monte Carlo methods provides a general mechanism for uncertainty quantification and facilitates full Bayesian inference over model parameters, latent states, or predictions, enabling posterior model averaging, credible intervals, and hypothesis testing far beyond what optimization-based point estimates (e.g., MAP) can offer (Villani et al., 26 Nov 2024).

2. Core Methodologies in Posterior Sampling

2.1 Markov Chain Monte Carlo (MCMC)

Standard posterior sampling often uses variants of MCMC such as Metropolis–Hastings, Langevin dynamics, Hamiltonian Monte Carlo, and Gibbs sampling, each constructing a Markov chain whose stationary law is the target posterior (Piccioli et al., 2023, Yrjänäinen et al., 4 Aug 2025). Recent work achieves rigorous accuracy guarantees even in challenging non-log-concave or multimodal regimes by pairing MCMC with problem-specific strategies—e.g., measure decomposition for high-dimensional sparse regression (Montanari et al., 27 Jun 2024), Polya–Gamma augmentation for logistic models (Yrjänäinen et al., 4 Aug 2025), and blocked Gibbs for neural networks with intermediate-noise models (Piccioli et al., 2023).

2.2 Surrogate Models and Active Design

When each evaluation of the forward model (e.g., PDE solve) is costly, surrogate modeling with Gaussian processes (GPs) replaces expensive simulations. Posterior samples are then drawn from the GP-induced posterior. A fully adaptive greedy strategy incrementally builds the surrogate by optimally allocating computation across input locations and evaluation tolerances, with design decisions tailored to maximizing the fidelity of posterior estimation for a fixed budget (Villani et al., 26 Nov 2024).

2.3 Diffusion Processes and Langevin Dynamics

Increasingly, diffusion models and Langevin-based posterior sampling have found traction in generative modeling and Bayesian inversion. These include latent-space Langevin dynamics driven by pre-trained generative priors (Purohit et al., 2 Oct 2024), plug-and-play conditional diffusion, and annealed Langevin MCMC for high-dimensional inverse problems. Provably efficient polynomial-time posterior samplers are available for log-concave priors under well-defined error metrics (Xun et al., 30 Oct 2025, Chang et al., 8 Dec 2025); conversely, cryptographic hardness results establish worst-case intractability for general posteriors even when unconditional generation remains tractable (Gupta et al., 20 Feb 2024).

2.4 Discrete-State and Structured Models

For discrete data domains, split Gibbs sampling schemes alternate likelihood-guided and prior-guided updates under discrete diffusion (Chu et al., 3 Mar 2025), yielding theoretically convergent plug-and-play posterior samplers for tasks such as DNA design and music infilling.

2.5 Plug-and-Play and Data-Driven Priors

In imaging and high-dimensional inference, posterior sampling frameworks increasingly incorporate plug-and-play denoisers or deep CNN priors, within unadjusted Langevin algorithms (PnP-ULA) (Renaud et al., 2023) or Stein variational gradient descent (PnP-SVGD) (Izzatullah et al., 2022), enabling image-based regularization and efficient sample diversity.

3. Theoretical Guarantees, Error Analysis, and Hardness

Posterior sampling workflows are supported by strong theoretical guarantees under various regimes:

• Polynomial-time convergence for globally or locally log-concave posteriors using annealed Langevin MCMC, with explicit bounds in total variation, Wasserstein distance, or KL divergence controlled via score-estimation accuracy and mixing time analysis (Xun et al., 30 Oct 2025, Chang et al., 8 Dec 2025).
• Intractability results for general posteriors under cryptographically motivated constructions, establishing that no robust black-box posterior sampler (diffusion-based or otherwise) can succeed in polynomial time in the worst case, whereas unconditional sampling remains easy (Gupta et al., 20 Feb 2024).
• Instance-optimal compressed sensing via posterior sampling achieves minimal measurement complexity, robust recovery under arbitrary priors, and provable robustness to prior mismatch quantified in Wasserstein distance (Jalal et al., 2021).

Error models such as posterior-L2 pseudometrics and bounds on sampling drift quantitatively characterize sensitivity to measurement and prior mismatches, enabling precise control of sampling error and fidelity (Renaud et al., 2023).

4. Adaptive Design, Surrogates, and Computational Efficiency

Posterior sampling is often limited by computational cost, especially when forward model evaluations are expensive (e.g., PDE-based inversion, seismic tomography).

• Adaptive greedy design strategies maximize posterior accuracy per unit cost by sequentially selecting evaluation points and numerical tolerance levels for surrogate modeling; fully adaptive AGP methods achieve a 30–60% reduction in expensive evaluations compared to static or position-adaptive approaches (Villani et al., 26 Nov 2024).
• Surrogate training with GP regression scales as $O(s^3)$ with the number of points $s$, but in Bayesian inversion, typically $s=O(10^2)$ suffices.
• Efficient latent-space Langevin samplers exploit the structure of generative models for amortized sampling with constant cost per sample (Purohit et al., 2 Oct 2024).

5. Posterior Sampling in Sequential Decision and Reinforcement Learning

Posterior sampling is foundational in sequential decision-making and exploration-exploitation tradeoff algorithms such as Thompson Sampling and Bayesian RL.

• In multi-armed bandits, posterior sampling matches or surpasses regret bounds for confidence-based approaches (UCB), with general Bayesian regret scaling as $O(\sqrt{KT\log T})$ (Russo et al., 2013, Kalvit et al., 20 Feb 2024).
• Tabular and deep RL extensions include posterior sampling for Q-learning (PSQL, achieving $\tilde{O}(H^2\sqrt{SAT})$ regret) (Agrawal et al., 1 Jun 2025), scalable deep RL (PSDRL) with model-based uncertainty quantification (Sasso et al., 2023), and constrained RL via posterior-sampled MDP optimization (Provodin et al., 2022).

These approaches leverage Gaussian or Dirichlet posteriors to guide exploration, policy selection, and occupancy measure optimization in both unconstrained and constrained MDPs.

6. Applications, Practical Considerations, and Extended Domains

Bayesian Inverse Problems

Posterior sampling is integral to geophysical and cosmological inverse problems, enabling credible interval quantification, non-Gaussian uncertainty analysis, and image recovery under spherical geometry via proximal MCMC and wavelet priors (Marignier et al., 2021).

Probabilistic Embedding Models

Blocked Gibbs sampling with Polya–Gamma augmentation achieves correct uncertainty quantification in large-scale probabilistic word embeddings, outperforming mean-field variational inference and MAP estimates on hold-out likelihoods (Yrjänäinen et al., 4 Aug 2025).

Differential Privacy

Posterior sampling itself constitutes a differentially private mechanism under global or stochastic Lipschitz assumptions on the likelihood and/or prior, with quantitative (ε,δ)-privacy and explicit utility/distinguishability bounds (Dimitrakakis et al., 2013).

Compressible and Sparse Models

Measure decomposition techniques yield efficient posterior samplers for sparse regression in the regime $n/d$ above a constant threshold; posterior draws admit rigorous coverage guarantees and practical diagnostic criteria (Montanari et al., 27 Jun 2024).

7. Limitations, Extensions, and Open Questions

• Posterior sampling is fundamentally intractable for arbitrary priors and measurement models, barring structure (e.g., log-concavity, strong regularity) (Gupta et al., 20 Feb 2024).
• Fidelity is limited by the accuracy of generative or surrogate priors, requiring careful model selection and validation.
• Practical extensions include multi-output kernels, gradient-enhanced GPs, multi-fidelity surrogates, adaptive SDE integrators, and alternate acquisition strategies.
• Open questions focus on tightening sampling error bounds under regularity assumptions, average-case hardness for realistic data priors, scaling to infinite-dimensional spaces, fully data-driven surrogate modeling, and robust handling of non-Gaussian or structured noise.

Posterior sampling thus continues to be a central methodological pillar across Bayesian inference, uncertainty quantification, and statistical decision theory, but its practical achievability, computational efficiency, and robustness require careful attention to prior structure, model approximation, and the computational landscape.