Bayesian Inference

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Foundations and workflow

• Bayesian Data Analysis - Third-edition reference on hierarchical models, computation, model checking and predictive evaluation; the linked electronic edition incorporates later corrections. Gelman et al. 2013.
• Bayesian Methods for Hackers - Notebook-based introduction to Bayesian modeling and probabilistic programming through worked examples. Davidson-Pilon 2015.
• Bayesian workflow - Connects iterative model construction, inference, checking, comparison and computational troubleshooting; posterior computation is one part of the analysis. Gelman et al. 2020.
• Visualization in Bayesian workflow - Uses graphics throughout model building, inference and checking, extending diagnostic visualization beyond MCMC trace plots. Gabry et al. 2019.
• General Bayesian updating - Weights the prior by exp(−w × cumulative loss). The scale w sets prior-data weighting and changes the target; it needs justification. Negative log likelihood with w = 1 recovers ordinary Bayes. Bissiri et al. 2016.

Markov chain Monte Carlo

• Introduction to MCMC - Geyer's handbook chapter covers Markov-chain sampling, Monte Carlo error and practical analysis of dependent draws. Geyer 2011.
• Metropolis sampling - Samples molecular configurations with Boltzmann probabilities using accepted or rejected random moves, avoiding direct simulation of the physical dynamics. Metropolis et al. 1953.
• Hastings' generalization - Generalizes Metropolis sampling and examines implementation and Monte Carlo error estimation for Markov-chain methods. Hastings 1970.
• Understanding Metropolis-Hastings - Derives the algorithm and its implementation; shows how block updates and Gibbs sampling fit within the Metropolis-Hastings framework. Chib and Greenberg 1995.
• Hamiltonian Monte Carlo - Neal's review uses auxiliary momenta and Hamiltonian trajectories to propose distant states; explains volume preservation, discretization and acceptance corrections. Neal 2011.
• Geometric foundations of HMC - Develops HMC through measures on smooth manifolds, connecting the theory to implementations and generalizations. Betancourt et al. 2014.
• Conceptual introduction to HMC - Explains the geometry behind HMC, practical implementations and failure modes; useful when a sampler runs but explores the posterior poorly. Betancourt 2017.
• No-U-Turn Sampler - Adapts HMC trajectory length by stopping expansion when paths turn back; also introduces dual-averaging step-size adaptation. Hoffman and Gelman 2014.
• Parallel tempering - Runs chains at different temperatures and proposes state swaps, helping explore separated modes. Samples from the target-temperature chain retain the intended distribution; temperature spacing and swap mixing affect efficiency. Earl and Deem 2005.

Stochastic-gradient samplers

• SGLD - Adds noise to minibatch gradient updates and anneals the step size toward posterior sampling. The limiting argument does not make finite-step implementations exact. Welling and Teh 2011.
• SGHMC - Uses momentum and friction to account for stochastic-gradient noise. Simply replacing HMC's full gradients with minibatch estimates can give the wrong target distribution. Chen et al. 2014.
• A recipe for SG-MCMC - Constructs continuous-time samplers through diffusion and curl matrices, unifying existing dynamics and motivating stochastic-gradient Riemannian HMC. Ma et al. 2015.

Sequential Monte Carlo

• SMC Methods in Practice - Edited reference on particle methods, convergence and applications to tracking, filtering, time series and model selection. Doucet et al. 2001 🔒.
• SMC samplers - Propagates weighted particles through distributions on a common space; supports sequential estimation and ratios of normalizing constants beyond state-space filtering. Del Moral et al. 2006.
• Introduction to SMC - Unifies particle filtering and smoothing with Feynman-Kac models, importance sampling and resampling; covers particle MCMC and implementation in Python. Chopin and Papaspiliopoulos 2020 🔒.

Particle MCMC and noisy likelihoods

• Pseudo-marginal MCMC - Nonnegative, unbiased likelihood estimates can preserve the posterior as an invariant marginal on an extended space. The estimate is retained on rejection; finite chains still have sampling error and can mix poorly. Andrieu and Roberts 2009.
• Particle MCMC - Uses SMC likelihood estimates in particle marginal Metropolis-Hastings and conditional SMC in particle Gibbs. With valid support and resampling, finite particle counts can preserve the invariant posterior. Andrieu et al. 2010.

Approximate Bayesian computation

• ABC in population genetics - Uses simulated summary statistics and local-linear regression adjustment to approximate parameter posteriors without evaluating the likelihood. Beaumont et al. 2002.
• MCMC without likelihoods - Builds likelihood-free MCMC using simulated data; summary-statistic matching within a nonzero tolerance targets an approximation to the full-data posterior. Marjoram et al. 2003.
• SMC without likelihoods - Introduces ABC-PRC with sequential particles. Read the appended 2009 correction: the original importance weights can bias posterior samples, particularly in the tails. Sisson et al. 2007.
• Nonlinear regression ABC - Fits nonlinear heteroscedastic regressions of parameters on summary statistics, then refines estimation with importance sampling. Blum and Francois 2009.
• Likelihood-free MCMC review - Reviews ABC-MCMC construction and practical choices, including summary statistics, discrepancy kernels and tolerance; these choices define the approximation. Sisson and Fan 2010.
• ABC in practice - Reviews ABC algorithms and applications in evolution and ecology; emphasizes model formulation, fitting and improvement within a Bayesian analysis. Csilléry et al. 2010.
• Hamiltonian ABC - Approximates ABC gradients using forward simulations and incorporates simulator random seeds into the chain state to support Hamiltonian-style exploration. Meeds et al. 2015.
• Random forests for ABC model choice - Treats model choice as classification from simulated summaries, then uses a second random-forest stage to estimate the selected model's posterior probability. Pudlo et al. 2014.

Variational inference

• Variational parameter estimation - Uses variational transformations for logistic regression with Gaussian priors and binary graphical models, yielding tractable posterior approximations. Jaakkola and Jordan 2000.
• Gaussian variational approximation - Relates Gaussian VI to Laplace approximation and simplifies its parameterization for Gaussian priors with factorized likelihoods; applies it to GP regression with non-Gaussian observations. Opper and Archambeau 2009.
• Stochastic VI - Scales variational posterior approximation through stochastic optimization, demonstrated with large-corpus topic models. Hoffman et al. 2012.
• Black-box VI - Estimates variational-objective gradients from Monte Carlo draws and reduces their variance, avoiding a new analytic derivation for every model. Ranganath et al. 2013.
• Doubly stochastic variational Bayes - Uses stochastic optimization and joint-density gradients for non-conjugate inference in continuous parameter spaces. Titsias and Lázaro-Gredilla 2014.
• VI for statisticians - Reviews KL-based density approximation, mean-field methods, exponential-family models and stochastic optimization; discusses limits and open statistical questions. Blei et al. 2017.
• ADVI - Fits mean-field or full-rank Gaussians after unconstraining continuous latent variables. Requires differentiable log densities and suitable support transforms; discrete latent variables need marginalization or another treatment. Kucukelbir et al. 2017.
• Advances in VI - Organizes methods by scalability, applicability beyond conjugacy, richer approximating families and amortized inference networks. Zhang et al. 2017.
• Stein variational gradient descent - Transports interacting particles toward a target distribution using functional gradient descent on KL divergence and a kernelized Stein construction. Liu and Wang 2016.
• Pathfinder - Builds Gaussian approximations along quasi-Newton optimization paths and selects by estimated ELBO; multiple paths are combined with importance resampling. Produces approximate draws or initial states for MCMC. Zhang et al. 2022.

Rich posterior approximations

• VI with normalizing flows - Transforms a simple density through invertible maps to construct flexible variational posteriors, extending beyond factorized approximation families. Rezende and Mohamed 2015.
• Normalizing-flow review - Surveys invertible distribution transformations, expressiveness and computational tradeoffs for density estimation and approximate inference. Papamakarios et al. 2021.
• Model-Informed Flows - Connects Variationally Inferred Parameters plus a full-rank Gaussian to augmented autoregressive flows; builds prior information, translation and hierarchical ordering into the flow. Ko and Domke 2025.
• Expectation propagation - Iteratively matches selected moments of approximate belief states, linking assumed-density filtering with loopy belief propagation; supports hybrid discrete/continuous networks. Minka 2001.
• EP for partitioned data - Updates local likelihood approximations while sharing information through the global approximation and prior; a framework for distributed inference, not one fixed local solver. Vehtari et al. 2014.
• INLA - Approximates posterior marginals in latent Gaussian models using nested Laplace approximations and integration over hyperparameters. The original method exploits sparse precision matrices and few hyperparameters; likelihoods can be non-Gaussian. Rue et al. 2009.
• Bayesian backpropagation framework - Uses local Gaussian posterior approximations and Bayesian evidence for neural-network regularization, model comparison and uncertainty estimates. MacKay 1992.
• Laplace Redux - Reviews scalable variants of neural-network Laplace approximation and introduces a PyTorch library; results depend on the approximation and predictive configuration. Daxberger et al. 2021.

Neural simulation-based inference

• SBI overview - Reviews inference for scientific simulators with intractable likelihoods and the role of learned surrogates in solving inverse problems. Cranmer et al. 2020.
• Neural posterior estimation - Learns conditional posterior densities from simulations and uses preliminary inference to guide later simulations, replacing ABC's explicit tolerance test with density estimation. Papamakarios and Murray 2016.
• Automatic Posterior Transformation - Extends sequential neural posterior estimation to dynamically updated proposals and expressive flow-based density estimators. Greenberg et al. 2019.
• Sequential Neural Likelihood - Fits an autoregressive-flow likelihood surrogate using simulations concentrated in high-posterior regions; includes calibration and goodness-of-fit diagnostics. Papamakarios et al. 2019.
• Classifier-based likelihood ratios - Uses calibrated discriminative classifiers to approximate likelihood ratios from simulator samples, supporting hypothesis tests without explicit likelihood evaluation. Cranmer et al. 2015.
• Benchmarking SBI - On 10 tasks with 1k-100k simulations, sequential neural estimators generally improve simulation efficiency. Rejection ABC has lower overhead and can remain competitive on low-dimensional problems with cheap simulators. Lueckmann et al. 2021.
• SBI evaluation - On Two Moons, median-heuristic MMD obscures posterior differences. Predictive median distance can reward point estimates without recovering uncertainty; prior-initialized slice-sampling chains missed modes in tested neural pipelines. Lueckmann et al. 2021.

Diffusion priors and inverse problems

• Score-based SDEs - Relates noise injection and reverse-time generation through learned scores; includes predictor-corrector sampling and applications to inverse problems. Song et al. 2021.
• Diffusion posterior sampling - Combines diffusion sampling with measurement-informed gradients for noisy linear and nonlinear inverse problems; the posterior sampling procedure is approximate. Chung et al. 2023.

Posterior accuracy

• Diffusion-sampling benchmark - Tests three samplers on 64D Lévy signals and four linear inverse problems, with Gibbs references. Learned-denoiser settings are tuned for reconstruction MSE, not posterior coverage. Zach et al. 2026.
• Diffusion coverage - At nominal 90%, C-DPS, DiffPIR and DPnP with learned denoisers cover 0-58% in Gaussian/Laplace cases; all reach 100% with Bernoulli-Laplace increments. These averages over signal-measurement pairs do not ensure coverage for each observation. Zach et al. 2026.

Applications

• Diffusion for optical tomography - Blends learned and model-based scores. Its geometric density-mixture approximation holds locally at small diffusion times under finite-dimensional smoothness and positivity assumptions; it does not establish globally exact sampling. Schneider et al. 2026.

Neural-network and Gaussian-process uncertainty

• Bayes by Backprop - Learns a distribution over network weights by optimizing a variational free-energy objective, allowing weight uncertainty to enter predictions and exploration. Blundell et al. 2015.
• Dropout as Bayesian approximation - Interprets dropout as approximate inference in a deep Gaussian process. Predictive uncertainty uses stochastic test-time passes, which add computation. Gal and Ghahramani 2016.
• Deep ensembles - Combines independently trained neural predictors for uncertainty estimation; a practical comparison method alongside explicit posterior approximations. Lakshminarayanan et al. 2017.
• Aleatoric and epistemic uncertainty - Combines input-dependent observation noise with model uncertainty for depth regression and semantic segmentation, distinguishing two sources of predictive uncertainty. Kendall and Gal 2017.
• Bayesian view of generalization - Studies marginalization across and within neural-network solution basins, function-space priors and tempering; connects ensemble predictions with approximate Bayesian averaging. Wilson and Izmailov 2020.
• Bayesian computation in deep learning - Reviews SG-MCMC and VI for neural-network weights and latent variables in deep generative models, focusing on their computational challenges. Chen et al. 2025.
• Gaussian Processes for Machine Learning - Reference on GP regression, classification, covariance functions, model selection and approximate inference. Rasmussen and Williams 2006.
• Sparse variational GPs - Introduces globally shared inducing variables and stochastic variational inference for GP models, including non-Gaussian likelihoods. Hensman et al. 2013.

Evidence and predictive model comparison

• Dynamic nested sampling - Adapts the number of live points during nested sampling to allocate computation between posterior estimation and evidence accuracy, rather than using a fixed live-point count. Higson et al. 2018.
• Bridge sampling - Estimates a marginal likelihood using posterior samples and samples from a proposal distribution. Unlike predictive LOO scores, this targets model evidence; overlap, normalized priors and posterior sampling quality affect reliability. Gronau et al. 2017.
• LOO and WAIC - Estimate predictive accuracy for a specified held-out unit. In hierarchical data, withholding an observation or an entire group answers different questions. Unstable PSIS weights can require explicit refits or K-fold validation. Vehtari et al. 2016.

Diagnostics and calibration

Computational diagnostics and reference tests

• Rank-normalized R-hat - Improves convergence diagnostics for heavy tails and unequal chain variances; adds local efficiency measures and Monte Carlo error estimates for quantiles. Vehtari et al. 2021.
• Simulation-based calibration - Checks inference implementations with repeated simulations from the model and posterior-rank diagnostics. It assesses computation under that model, not whether the model fits reality. Talts et al. 2018.
• SBC sensitivity - Returning the prior can pass parameter-only rank checks. Data-dependent quantities, including the joint likelihood, can detect failures those checks miss. Passing a finite set of checks does not certify the posterior. Modrák et al. 2025.
• Pareto-smoothed importance sampling - Fits a generalized Pareto tail to stabilize importance weights; provides effective-sample-size, Monte Carlo error and tail-shape diagnostics. Vehtari et al. 2024.
• posteriorDB - Pairs models and data with reference draws where available. Compares posterior accuracy against computation; reference samples are numerical approximations, and some difficult targets lack them. Magnusson et al. 2025.
• VI defaults - On a 1D Gaussian test, PyMC 5.25.1 default ADVI overestimates posterior SD by about 5%; more gradient samples or Adam mitigate it. The study tests specific versions and simple targets, not general framework superiority. Andersen et al. 2026.

Predictive calibration

• The well-calibrated Bayesian - Relates coherent Bayesian forecasting to long-run calibration; the theorem concerns a forecaster's probabilistic expectations, not automatic empirical calibration of every fitted model. Dawid 1982.
• Multiclass probability calibration - Zadrozny and Elkan's study of transforming classifier scores into multiclass probability estimates; a historical reference for post-processing predictions. Zadrozny and Elkan 2002.
• Predicting good probabilities - Compares distortions in classifier probabilities and correction by Platt scaling or isotonic regression; calibration method and calibration-set size affect results. Niculescu-Mizil and Caruana 2005.
• Nearly-isotonic regression - Relaxes monotonicity through a convex penalty and computes a path of fits; intermediate solutions help examine whether strict monotonicity is appropriate. Tibshirani et al. 2011.
• Ensembles of calibration models - Studies ensembles of near-isotonic regression for binary probability calibration and compares them with linear-trend calibration; relaxes strict isotonicity. Pakdaman Naeini and Cooper 2017.
• Beta calibration - Introduces a parametric calibration family for binary classifiers that includes the identity map and accommodates skewed score distributions. Kull et al. 2017.
• Temperature scaling - Evaluates modern neural-network calibration and a single-parameter post-processing baseline; effectiveness on the evaluated datasets does not guarantee calibration after distribution shift. Guo et al. 2017.
• Verified uncertainty calibration - Combines parametric scaling with binning and studies finite-sample calibration-error estimation; distinguishes measurable calibration from apparently smooth reliability plots. Kumar et al. 2019.
• CRUDE - Calibrates regression uncertainty using empirical errors; assumes a common residual-distribution shape after shifting by predicted means and scaling by predicted standard deviations. Zelikman et al. 2020.
• Uncertainty under dataset shift - Benchmarks predictive accuracy and calibration as input distributions change; shows that successful in-distribution post-hoc calibration need not survive shift. Ovadia et al. 2019.

Software

Modeling languages and inference frameworks

• Stan - Probabilistic modeling language with R, Python, Julia and command-line interfaces; supports hierarchical and time-series models and an ecosystem for validation and visualization. docs.
• PyMC - Python probabilistic modeling with MCMC and VI, including NUTS and ADVI; uses PyTensor for computation. code.
• Pyro - Python probabilistic programming built on PyTorch, integrating neural networks with probabilistic models and customizable inference. docs.
• NumPyro - Probabilistic programming with Pyro-style primitives and JAX differentiation/compilation; includes HMC and NUTS. code.
• TensorFlow Probability - Distributions, invertible transformations and probabilistic inference components for TensorFlow, with a JAX substrate. code.
• Turing.jl - Julia probabilistic programming with MCMC, VI, maximum-likelihood and MAP estimation, plus interfaces for custom inference. code.
• DynamicPPL - Turing's backend and modeling language for defining, conditioning, running and manipulating probabilistic models. code.
• NIMBLE - Extends BUGS model syntax and compiles models plus user-written algorithms to C++. Supports customizable MCMC, particle methods and deterministic approximations; model specification and inference algorithms can be changed independently. docs.
• JAGS - Cross-platform MCMC engine for Bayesian hierarchical models in the BUGS language, with extensible functions, distributions and samplers. docs.
• greta - Defines statistical models directly in R and fits them by MCMC or optimization, using TensorFlow and TensorFlow Probability internally. code.
• Gen - Probabilistic modeling and inference components for custom algorithms combining neural proposals, VI, SMC and MCMC. docs.
• Soss.jl - Julia probabilistic programming with model transformations and inference integrations; models can be rewritten for predictive use. code.
• Brancher - PyTorch-based symbolic probabilistic modeling and stochastic variational inference for differentiable Bayesian models. code.

Regression interfaces

• brms - R formulas generate Stan models for multilevel and distributional regression, including nonlinear and multivariate responses. Unlike rstanarm's precompiled models, brms generates model-specific Stan code. docs.
• rstanarm - Fits precompiled Bayesian regression models through R formulas and data frames, with additional prior specifications. code.
• Bambi - High-level interface built on PyMC for Bayesian regression and mixed-effects models. code.
• BRMP - Generates Bayesian regression models and design matrices from formula syntax, targeting Pyro or NumPyro. code.

Sampling and posterior approximations

• BlackJAX - JAX sampling components for user-supplied log densities; integrates with probabilistic programs without providing a modeling language itself. code.
• nutpie - Rust-backed NUTS sampler for PyMC and Stan models, with compiled log densities, sampler diagnostics and experimental flow adaptation. It supplies an inference backend rather than a separate modeling language. docs.
• LaplacesDemon - R environment for Bayesian inference with multiple sampling algorithms. code.
• laplace-torch - Laplace approximations for full networks, subnetworks or final layers, with marginal-likelihood and predictive computations. code.
• BAT.jl - Julia toolkit for posterior sampling, mode estimation and integration, with plotting and input/output support. code.

Simulation-based inference and density estimation

• sbi - PyTorch toolkit for amortized or sequential posterior, likelihood and ratio estimation from simulations. Includes calibration and coverage diagnostics. Boelts et al. 2024; code.
• BayesFlow 2 - Amortizes posterior, likelihood and ratio estimation across simulated datasets. Uses Keras 3 with JAX, PyTorch or TensorFlow backends and includes calibration diagnostics. Kühmichel et al. 2026; code.
• FlowJAX - Distributions, bijections and normalizing flows built with JAX and Equinox; supports conditional density estimation. code.
• Zuko - PyTorch normalizing flows with trainable conditional distributions and transformations. code.

Diagnostics and reference targets

• ArviZ - Convergence diagnostics, predictive checks, model comparison and posterior visualization across inference libraries. Separates statistical analysis and visualization from the sampler that produced the draws. docs.
• posteriorDB tools - Versioned models, data, reference draws and interfaces for inference testing. Check which targets have reference samples before treating the collection as a ground-truth benchmark. code.

Browser and graphical interfaces

• WebPPL - Probabilistic programming for the web, also runnable through Node.js with external packages. code.
• StatSim - Browser-based visual stochastic modeling and Bayesian inference, with simulation, MCMC and variational methods. docs.
• JASP - Graphical statistics software offering classical and Bayesian analyses, with tables and plots for reporting results. docs.

Historical systems

• BUGS - Historical review of Bayesian inference using Gibbs sampling in BUGS, WinBUGS and OpenBUGS; discusses both modeling flexibility and limitations. Lunn et al. 2009.
• Edward - TensorFlow-based library for probabilistic modeling, inference and model criticism; the linked documentation describes the original Edward framework. docs.
• ZhuSuan - TensorFlow probabilistic programming for Bayesian deep learning, including VI, HMC and stochastic-gradient samplers. Its installation guide specifies TensorFlow 1.13 or later; check this older API before reusing examples. code.

Related topics

• Bayesian Networks
• Missing Data
• Optimization
• Neural Networks
• Generative Models