Bayesian Multiple linear regression

Arnold Zellner (econometric formulation); broader development by Harold Jeffreys and Gelman et al.

Bayesian Multiple Linear Regression models a continuous outcome as a linear combination of several predictors, but instead of producing a single point estimate it yields a full posterior distribution over all regression coefficients and the error variance. This makes uncertainty quantification explicit and allows seamlessly incorporating prior knowledge from theory or previous studies.

continuous outcome, continuous or categorical predictors

Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. Wiley.

ols-regression bayesian-simple-linear-regression ridge-regression bayesian-generalized-linear-model lasso-regression bayesian-hierarchical-linear-model


Bayesian Multiplex Network Analysis

Bayesian Multiplex Network Analysis (Probabilistic Inference on Multi-Layer Networks)

probabilistic multiplex network inference

Bayesian multiplex network analysis applies probabilistic generative modelling to networks that carry more than one type of relational tie simultaneously — such as friendship, collaboration, and communication links among the same set of actors. By placing priors over community memberships, edge probabilities, and layer interdependencies, the framework yields posterior distributions rather than point estimates, supporting principled uncertainty quantification across all inferred network properties.

Probabilistic generative model for multiplex networks

Multi-layer adjacency matrices / edge lists across network layers

De Bacco, C., Power, E. A., Larremore, D. B., & Moore, C. (2017). Community detection, link prediction, and layer interdependence in multilayer networks. Physical Review E, 95(4), 042317.

Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., & Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2(3), 203-271.

multiplex-network-analysis bayesian-stochastic-block-model bayesian-community-detection multilayer-network-analysis exponential-random-graph-model stochastic-block-model


Bayesian spatially varying coefficient model

Fotheringham, Yang & Kang (MGWR); Bayesian extension by Li and co-authors

Bayesian Multiscale Geographically Weighted Regression (Bayesian MGWR) extends the MGWR framework by placing Bayesian priors on each spatially varying coefficient. Each predictor is allowed its own bandwidth — its own geographic scale of influence — while Bayesian inference replaces classical back-fitting with posterior sampling, yielding full uncertainty quantification for every local coefficient surface.

Spatially varying coefficient regression

Georeferenced cross-sectional or panel data with continuous outcome

Fotheringham, A. S., Yang, W., & Kang, W. (2017). Multiscale Geographically Weighted Regression (MGWR). Annals of the American Association of Geographers, 107(6), 1247-1265.

Li, Z., Fotheringham, A. S., Li, W., & Oshan, T. (2020). Fast Geographically Weighted Regression (FastGWR): a scalable algorithm to investigate spatial process heterogeneity in millions of observations. International Journal of Geographical Information Science, 33(1), 155-175.


Bayesian Naive Bayes

Naive Bayes with Bayesian parameter estimation

1960s (base); Bayesian parameter treatment formalized 2000s

Naive Bayes: Maron & Kuhns (1960); full Bayesian treatment formalized by Murphy (2012) and Bishop (2006)

Bayesian Naive Bayes applies a fully Bayesian treatment to the parameters of the classic Naive Bayes classifier: instead of estimating class-conditional distributions by maximum likelihood, it places conjugate priors (typically Dirichlet for categorical data or Gaussian-Gamma for continuous data) over the parameters and integrates them out, producing predictive posterior distributions that naturally quantify uncertainty and avoid overfitting on small datasets.

Categorical, count, or continuous features with discrete or binary labels

Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective (Ch. 3, 4). MIT Press.

Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Ch. 8). Springer.

gaussian-process k-nearest-neighbors semi-supervised-naive-bayes logistic-regression-ml support-vector-machine bayesian-logistic-regression


Bayesian NARDL

Bayesian Nonlinear Autoregressive Distributed Lag Model

Bayesian NARDL
Bayesian nonlinear ARDL
Bayesian asymmetric ARDL
B-NARDL

2014 (NARDL); Bayesian extension c. 2015–2020

Shin, Yu & Greenwood-Nimmo (NARDL base); Bayesian extension developed in subsequent applied literature

Bayesian NARDL combines the Nonlinear Autoregressive Distributed Lag framework of Shin, Yu, and Greenwood-Nimmo (2014) with Bayesian posterior inference. It models asymmetric long-run cointegration — allowing positive and negative shocks to a regressor to have different equilibrium effects — while incorporating prior knowledge and producing full posterior distributions over all parameters, including the asymmetry gap.

Nonlinear cointegrating model with Bayesian inference

Time series; continuous variables with potential asymmetric long-run adjustment

Shin, Y., Yu, B., & Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In W. C. Horrace & R. C. Sickles (Eds.), Festschrift in Honor of Peter Schmidt: Econometric Methods and Applications (pp. 281–314). Springer.

Koop, G. (2003). Bayesian Econometrics. Wiley.

nonlinear-ardl bayesian-ardl-bounds-test bayesian-vecm arellano-bond-gmm-estimator vector-error-correction-model panel-nardl


Bayesian Nash Equilibrium

Bayesian Nash Equilibrium with Incomplete Information

BNE
Perfect Bayesian Equilibrium
Type-Contingent Equilibrium

Bayesian Nash Equilibrium (BNE) extends Nash Equilibrium to games with incomplete information, where players lack full knowledge of others' payoff functions. Introduced by John Harsanyi in 1967, BNE models strategic interaction under uncertainty by representing unknown payoffs as players' private types drawn from a probability distribution. Equilibrium is found by solving for type-contingent strategies that are best responses to all possible type realizations.

Harsanyi, J. C. (1967). Games with incomplete information played by Bayesian players, Parts I, II, and III. Management Science, 14(3), 159-182.

Harsanyi, J. C. (1968). Games with incomplete information played by Bayesian players. Management Science, 14(7), 486-502.

nash-equilibrium vcg-mechanism principal-agent-model first-price-auction


Bayesian Negative Binomial Regression

Bayesian NB regression
Bayesian negbin model
Bayesian overdispersed count regression
Bayesian NB-2 model

Gelman, Carlin, Stern, Dunson, Vehtari & Rubin; Cameron & Trivedi

Bayesian Negative Binomial Regression models non-negative integer count outcomes that exhibit overdispersion — where the variance exceeds the mean — by placing a negative binomial likelihood on the data and specifying prior distributions over the regression coefficients and the dispersion parameter. Posterior inference is typically performed via Markov chain Monte Carlo (MCMC) or variational methods, yielding full posterior distributions rather than point estimates.

Non-negative integer (count) outcome with overdispersion

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

Cameron, A. C., & Trivedi, P. K. (2013). Regression Analysis of Count Data (2nd ed.). Cambridge University Press.

bayesian-poisson-regression negative-binomial-regression zero-inflated-model bayesian-zero-inflated-model poisson-regression bayesian-generalized-linear-model


Bayesian nested case-control

Bayesian sampled case-control within cohort

1977 (nested case-control); Bayesian adaptation developed through 1990s–2010s

Nested case-control: D. C. Thomas (1977); Bayesian extension: various authors in biostatistics

A Bayesian nested case-control study embeds a case-control sampling scheme within a defined prospective cohort and then estimates exposure-outcome associations using Bayesian inference. Cases are individuals in the cohort who develop the outcome of interest; controls are sampled from the risk set at the time each case is identified. The Bayesian framework allows incorporation of prior knowledge — from earlier studies, expert opinion, or biological plausibility — and produces full posterior distributions for effect estimates rather than single-point estimates with confidence intervals.

Observational analytical study design with Bayesian inference

Time-to-event data, exposure measurements, biomarker data from an established cohort

Thomas, D. C. (1977). Addendum to: Methods of cohort analysis: Appraisal by application to asbestos mining. Journal of the Royal Statistical Society, Series A, 140(4), 469–491.

Wakefield, J. (2007). Disease mapping and spatial regression with count data. Biostatistics, 8(2), 158–183.

nested-case-control case-control-study bayesian-case-control-study cohort-study bayesian-cohort-study competing-risks-analysis


Bayesian Network Diffusion Analysis

Bayesian Network Diffusion Analysis (Probabilistic Inference on Contagion and Spreading Processes)

Bayesian diffusion model
probabilistic network diffusion
Bayesian spreading process inference
BNDA

Gomez Rodriguez, M.; Leskovec, J.; and related network science community

Bayesian Network Diffusion Analysis applies Bayesian probabilistic inference to the study of how information, diseases, behaviors, or innovations propagate through a network. By placing priors over diffusion parameters and updating them with observed cascade data, it quantifies transmission rates, identifies influential spreaders, reconstructs latent propagation pathways, and provides full uncertainty estimates — all within a principled statistical framework.

Probabilistic inference on network spreading processes

Observed contagion cascades, edge/node timestamps, network topology

Gomez Rodriguez, M., Leskovec, J., & Scholkopf, B. (2012). Structure and Dynamics of Information Pathways in Online Media. Proceedings of the 6th ACM International Conference on Web Search and Data Mining (WSDM), 23–32.

Kitsak, M., Gallos, L. K., Havlin, S., Liljeros, F., Muchnik, L., Stanley, H. E., & Makse, H. A. (2010). Identification of influential spreaders in complex networks. Nature Physics, 6(11), 888–893.

network-diffusion-analysis bayesian-exponential-random-graph-model temporal-network-diffusion-analysis exponential-random-graph-model bayesian-stochastic-block-model social-network-analysis


Bayesian Network with Measurement Error

Bayesian Network with Measurement Error (Errors-in-Variables Graphical Model)

Bayesian graphical model with measurement error

1988 (Bayesian networks); measurement-error extension: 1990s

Judea Pearl (Bayesian networks); measurement-error extension developed in epidemiology and psychometrics through the 1990s–2000s

A Bayesian network with measurement error is a probabilistic directed acyclic graphical model in which one or more node variables are observed with error rather than exactly. Latent true-value nodes are introduced for mismeasured variables, and the model jointly infers the network's conditional probability parameters and the unobserved true values from the noisy observations.

Probabilistic graphical model with latent variables

Continuous, binary, or categorical observed variables with known or estimated error structure

Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann.

Richardson, S. & Gilks, W. R. (1993). A Bayesian approach to measurement error problems in epidemiology using conditional independence models. American Journal of Epidemiology, 138(6), 430–442.


Bayesian Network

Bayes network
belief network
probabilistic graphical model
directed graphical model
Bayesian Ağı

A Bayesian network is a probabilistic graphical model, introduced by Judea Pearl in 1988, that encodes a set of variables and their conditional dependencies as a directed acyclic graph (DAG). Each node represents a variable; each directed edge encodes a direct probabilistic influence. By combining Bayes' rule with the graph's conditional independence structure, the model supports reasoning under uncertainty — computing the probability of any variable given observed evidence about others.

relationship / prediction / classification

exact (variable elimination, belief propagation) / approximate (MCMC, variational)

conditional probability tables / posterior marginals

bayesian-regression mcmc hierarchical-bayes dag-identification structural-equation-modeling


Bayesian Nonparametric Methods

Bayesian Nonparametric Methods (Dirichlet Process / Gaussian Process)

Bayesian Nonparametrik (Dirichlet Process / Gaussian Process)

Ferguson (Dirichlet Process, 1973); Rasmussen & Williams (GP, 2006)

Bayesian nonparametric methods are a family of flexible Bayesian models in which model complexity is not fixed in advance but grows automatically with the data. The two most widely used members are the Dirichlet Process Mixture (DPM), which clusters observations without pre-specifying the number of clusters, and Gaussian Process (GP) regression, which places a prior directly over functions and performs regression or classification without committing to a parametric form. Both frameworks were formalised in the Bayesian nonparametric literature, with the canonical GP treatment given by Rasmussen and Williams (2006).

posterior distributions / credible intervals / predictive distributions

Rasmussen, C.E. & Williams, C.K.I. (2006). Gaussian Processes for Machine Learning. MIT Press.

Müller, P. & Quintana, F.A. (2004). Nonparametric Bayesian Data Analysis. Statistical Science, 19(1), 95–110.

bayesian-regression mcmc hierarchical-bayes gaussian-process mixture-models kernel-methods


Bayesian NSGA-II

Bayesian Surrogate-Assisted Non-dominated Sorting Genetic Algorithm II

B-NSGA-II
Surrogate-Assisted NSGA-II
Gaussian Process NSGA-II
Bayesian Multi-Objective EA

Emmerich, M. T. M. et al. (surrogate-assisted MO-EA); Deb et al. (NSGA-II base)

Bayesian NSGA-II integrates Gaussian process surrogate models (Bayesian metamodels) into the NSGA-II evolutionary loop to solve expensive multi-objective optimization problems. By replacing costly true function evaluations with fast probabilistic predictions, it discovers high-quality Pareto-front approximations with far fewer real evaluations than standard NSGA-II.

Surrogate-assisted multi-objective evolutionary algorithm

Continuous or mixed decision variables; expensive black-box objective functions

Deb, K., Pratap, A., Agarwal, S., Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197.

Emmerich, M. T. M., Giannakoglou, K. C., Naujoks, B. (2006). Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Transactions on Evolutionary Computation, 10(4), 421–439.

nsga-ii bayesian-optimization multi-objective-optimization gaussian-process-regression surrogate-assisted-optimization multi-objective-genetic-algorithm


Bayesian Observational Quantitative Research Design

Bayesian non-experimental quantitative design

1990s–2000s (systematic application to observational research)

Thomas Bayes (foundational theorem, 1763); modern applied form developed by Sander Greenland, Andrew Gelman, and colleagues (1990s–2000s)

Bayesian observational quantitative research applies Bayesian statistical inference to data collected without experimental manipulation — surveys, administrative records, registries, or secondary datasets. Instead of relying solely on p-values and confidence intervals, the analyst encodes prior knowledge about parameters as probability distributions, updates them with observed data via Bayes' theorem, and reports conclusions as posterior probability statements. The approach is especially valued in epidemiology, social science, and health services research where randomisation is impossible or unethical.

Quantitative non-experimental research design with Bayesian inference

Quantitative observational data (surveys, administrative records, registries, secondary datasets)

Greenland, S. (2006). Bayesian perspectives for epidemiological research: I. Foundations and basic methods. International Journal of Epidemiology, 35(3), 765–775.

bayesian-inference observational-study propensity-score-matching multilevel-modeling structural-equation-modeling cross-sectional-survey


Bayesian OLS

Bayesian Ordinary Least Squares Regression

Bayesian linear regression
Bayesian normal regression
BLR
Bayesian least squares

Bayesian OLS combines the classical linear regression likelihood with prior distributions over the coefficients and error variance. Rather than reporting point estimates, it produces full posterior distributions that quantify both estimated effects and their uncertainty. The approach is especially valuable when prior knowledge is available or when samples are small.

Cross-sectional or time-series continuous outcome with continuous or categorical predictors

Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. Wiley.

Koop, G. (2003). Bayesian Econometrics. Wiley-Interscience.

ols-regression bayesian-var-model bayesian-gls ridge-regression bayesian-random-effects-model bayesian-fixed-effects-model


Bayesian one-class SVM

Bayesian One-Class Support Vector Machine

Bayesian OCSVM
Bayesian one-class classifier
probabilistic one-class SVM
Bayes-OCSVM

Scholkopf et al. (base OCSVM); Bayesian extension via Tipping and others

Bayesian one-class SVM combines the classical one-class support vector machine — which learns a tight boundary around normal training examples — with Bayesian inference to produce calibrated probability estimates of anomaly, rather than only a binary flag. This allows uncertainty quantification over the novelty decision, making the approach more suitable when downstream actions depend on how confident the model is that a new observation is anomalous.

Continuous or mixed features; unlabeled or single-class data

Scholkopf, B., Platt, J. C., Shawe-Taylor, J., Smola, A. J., & Williamson, R. C. (2001). Estimating the support of a high-dimensional distribution. Neural Computation, 13(7), 1443–1471.

Tipping, M. E. (2001). Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1, 211–244.

one-class-svm gaussian-process autoencoder-anomaly-detection isolation-forest robust-one-class-svm bayesian-gaussian-process


Bayesian one-sample t-test

Bayesian single-sample t-test
Bayes factor one-sample t-test
JZS one-sample Bayes factor
Bayesian location test

The Bayesian one-sample t-test compares a single group's mean against a fixed reference value using a Bayes factor rather than a p-value. It quantifies the evidence the data provide for the null hypothesis (mean equals the reference) versus the alternative, and yields a full posterior distribution over the effect size — enabling statements about practical magnitude, not just a binary reject-or-retain decision.

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16(2), 225–237.

Kruschke, J. K. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd ed.). Academic Press.

one-sample-t-test bayesian-independent-samples-t-test bayesian-paired-samples-t-test bayes-factor wilcoxon-signed-rank-test bayesian-pearson-correlation


Bayesian one-way ANOVA

Bayesian ANOVA
BF ANOVA
Bayes factor one-way ANOVA
Bayesian F-test

1961 (foundations); 2012 (ANOVA Bayes factors)

Harold Jeffreys (foundations); Jeffrey Rouder et al. (default priors for ANOVA)

Bayesian one-way ANOVA tests whether the means of three or more independent groups differ by computing a Bayes factor — a ratio that quantifies how much more likely the data are under a model that allows group differences than under the null model that assumes equal means. Unlike the classical F-test, it provides direct evidence for or against the null hypothesis rather than merely rejecting or retaining it.

Continuous dependent variable, one categorical independent variable (3+ groups)

Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012). Default Bayes factors for ANOVA designs. Journal of Mathematical Psychology, 56(5), 356–374.

Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford University Press.

one-way-anova bayesian-two-way-anova bayesian-independent-samples-t-test bayesian-repeated-measures-anova kruskal-wallis-test welch-corrected-one-way-anova


Bayesian Online Learning

Bayesian Online Learning (Sequential Posterior Update)

online Bayesian inference
sequential Bayesian learning
recursive Bayesian estimation
BOL

Opper, M.; Sato, M. (among key contributors)

Bayesian online learning applies Bayesian inference sequentially: each time a new observation arrives, the current posterior over model parameters becomes the prior for the next update. The result is a principled probabilistic framework that maintains calibrated uncertainty estimates throughout, making it well-suited for streaming and non-stationary data settings.

Sequential / streaming data (continuous, binary, count)

Opper, M. (1998). A Bayesian approach to on-line learning. In D. Saad (Ed.), On-Line Learning in Neural Networks (pp. 363–378). Cambridge University Press.

Sato, M. (2001). Online model selection based on the variational Bayes. Neural Computation, 13(7), 1649–1681.

online-learning bayesian-gaussian-process gaussian-process semi-supervised-learning bayesian-logistic-regression variational-inference


Bayesian Optimization

Bayesian Optimization (Hyperparameter Tuning)

Bayesçi Optimizasyon (Hyperparameter Tuning)

Mockus (1975); popularised for ML by Snoek, Larochelle & Adams (2012)

Bayesian Optimization is a sequential, model-based strategy for finding the optimum of expensive black-box functions with as few evaluations as possible. Rooted in the work of Mockus (1975) and brought to mainstream machine-learning practice by Snoek, Larochelle, and Adams (2012), it fits a probabilistic surrogate model — typically a Gaussian Process — to past observations and uses an acquisition function to decide where to probe next, balancing exploration of unknown regions with exploitation of promising ones.

Sequential model-based black-box optimization

Expected Improvement (EI), Upper Confidence Bound (UCB), Probability of Improvement (PI)

Continuous and ordinal hyperparameter spaces with predefined bounds

Snoek, J., Larochelle, H., & Adams, R.P. (2012). Practical Bayesian Optimization of Machine Learning Algorithms. Advances in Neural Information Processing Systems (NeurIPS), 25.

Frazier, P.I. (2018). A Tutorial on Bayesian Optimization. arXiv:1807.02811.

random-search grid-search gaussian-process-regression hyperparameter-tuning neural-architecture-search stochastic-optimization


Bayesian Ordinal Logistic Regression

Bayesian Ordinal Logistic Regression (Proportional Odds Model)

Bayesian proportional odds model
Bayesian cumulative logit model
Bayesian ordered logit
Bayesian cumulative link model

Johnson & Albert (1999); Bayesian proportional odds framework

Bayesian ordinal logistic regression extends the classical proportional odds model by placing prior distributions on the regression coefficients and threshold parameters and updating them with observed data via Bayes' theorem. The result is a full posterior distribution over all parameters, enabling uncertainty quantification without relying on large-sample approximations.

Ordinal outcome (Likert scale, graded categories), continuous or categorical predictors

Johnson, V. E., & Albert, J. H. (1999). Ordinal Data Modeling. Springer.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.


Bayesian Ordinary Kriging

Bayesian kriging
BOK
geostatistical Bayesian interpolation
Bayesian spatial prediction

Handcock & Stein (1993); Diggle & Ribeiro (2007)

Bayesian Ordinary Kriging is a geostatistical interpolation method that combines classical ordinary kriging with a Bayesian framework to jointly estimate the spatial covariance parameters and produce predictions at unsampled locations. Unlike plug-in kriging, it propagates uncertainty about variogram parameters through to the predictive distribution, yielding more honest uncertainty quantification.

Continuous georeferenced point observations

Diggle, P. J., & Ribeiro, P. J. (2007). Model-Based Geostatistics. Springer.

Handcock, M. S., & Stein, M. L. (1993). A Bayesian analysis of kriging. Technometrics, 35(4), 403-410.

ordinary-kriging bayesian-kriging bayesian-universal-kriging bayesian-co-kriging kriging spatial-autocorrelation


Bayesian PageRank

Bayesian PageRank (Probabilistic Ranking on Networks)

Bayesian PR
probabilistic PageRank
uncertainty-aware PageRank
stochastic PageRank

1999 (PageRank); 2000s (Bayesian extension)

Page, L. & Brin, S. (PageRank); Bayesian extension by multiple authors

Bayesian PageRank extends the classic PageRank algorithm by embedding it within a Bayesian probabilistic framework. Instead of returning a single deterministic rank score for each node, it quantifies uncertainty over rank estimates — particularly valuable when the network is incomplete, noisy, or observed with error. It is used in web analysis, citation networks, and social network research where rank uncertainty matters.

Directed (weighted or unweighted) networks

Page, L., Brin, S., Motwani, R., & Winograd, T. (1999). The PageRank citation ranking: Bringing order to the web. Stanford InfoLab Technical Report.

directed-pagerank temporal-pagerank multilayer-pagerank eigenvector-centrality bayesian-network-diffusion-analysis bayesian-community-detection


Bayesian Panel Data Analysis

Bayesian panel model
Bayesian longitudinal model
hierarchical panel model
Bayesian multilevel panel

Zellner (1971); Hsiao, Pesaran, and Tahmiscioglu (1999)

Bayesian panel data analysis applies Bayesian inference to models with repeated observations on multiple units. By placing prior distributions on coefficients and variance components, it merges prior knowledge with the observed panel likelihood to produce full posterior distributions for fixed or random effects, slope heterogeneity, and variance parameters — rather than point estimates and asymptotic standard errors.

Balanced or unbalanced panel (cross-sectional units observed over time)

Hsiao, C. (2003). Analysis of Panel Data (2nd ed.). Cambridge University Press.

Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. Wiley.

fixed-effects-model random-effects-model panel-data-analysis bayesian-fixed-effects-model bayesian-random-effects-model panel-hausman-test


Bayesian Panel Event Study

Bayesian event-study estimator
Bayesian dynamic DiD
Bayesian panel ES
Bayes event study

Developed from panel event-study literature (Sun & Abraham 2021; Freyaldenhoven et al. 2021) combined with Bayesian estimation frameworks

Bayesian Panel Event Study is a causal inference design that estimates dynamic treatment effects around a datable event using panel data, replacing classical frequentist estimation with Bayesian posterior inference. It produces period-by-period effect estimates with full probability distributions, enabling principled uncertainty quantification, regularization of noisy pre-trend coefficients, and probabilistic tests of parallel trends.

Panel / longitudinal data with a datable treatment event

Freyaldenhoven, S., Hansen, C., Shapiro, J. M., & Teso, E. (2021). Visualization, Identification, and Estimation in the Linear Panel Event-Study Design. NBER Working Paper No. 29170. National Bureau of Economic Research.

Jakiela, P. (2021). Simple Diagnostics for Two-Way Fixed Effects. Working Paper. Center for Global Development.

panel-event-study difference-in-differences bayesian-difference-in-differences dynamic-difference-in-differences event-study-design panel-data-event-study-design


Bayesian Panel Research

Bayesian longitudinal panel study
Bayesian panel data analysis
BPD research
Bayesian repeated-measures panel design

Building on Bayes (1763) and panel data econometrics; systematised by Hsiao, Lancaster, and others in the 1990s–2000s

Bayesian panel research combines the longitudinal structure of panel data — where the same units (individuals, firms, countries) are observed at multiple time points — with Bayesian statistical inference. Rather than relying solely on the observed data and point estimates, it incorporates prior knowledge via probability distributions, updates those priors with repeated-measures data, and produces full posterior distributions over model parameters. This yields richer uncertainty quantification and principled handling of individual heterogeneity across waves.

Quantitative longitudinal research design with Bayesian inference

Repeated measures (numeric) from the same individuals, firms, or units observed across multiple time points

Lancaster, T. (2004). An Introduction to Modern Bayesian Econometrics. Blackwell Publishing.

panel-research longitudinal-research bayesian-longitudinal-research multilevel-modeling bayesian-confirmatory-research bayesian-correlational-research


Bayesian Particle Swarm Optimization

Bayesian Particle Swarm Optimization — Probabilistic prior-guided swarm search

Bayesian PSO
BPSO
Probabilistic Swarm Optimization
Prior-guided PSO

Higashi, N., Iba, H. (extending Kennedy and Eberhart's PSO)

Bayesian Particle Swarm Optimization (Bayesian PSO) integrates Bayesian probabilistic reasoning into the standard particle swarm framework. Particles update their velocities and positions guided not only by personal and global best positions but also by a Bayesian posterior that encodes prior knowledge about the solution space, enabling more directed and statistically principled exploration of complex optimization landscapes.

Hybrid metaheuristic — Bayesian probabilistic swarm search

Continuous or mixed-variable parameter spaces with prior knowledge

Higashi, N., Iba, H. (2003). Particle swarm optimization with Gaussian mutation. Proceedings of the 2003 IEEE Swarm Intelligence Symposium, Indianapolis, IN, USA, pp. 72-79.

Kennedy, J., Eberhart, R. (1995). Particle swarm optimization. Proceedings of ICNN'95 — International Conference on Neural Networks, Perth, WA, Australia, vol. 4, pp. 1942-1948.


Bayesian Pathway Enrichment Analysis

Bayesian gene-set testing
Bayesian GSEA
Bayesian functional enrichment
BGSEA

Pierre Baldi, Anthony Long; Michael Newton et al. (foundational Bayesian gene-set frameworks)

Bayesian pathway enrichment analysis tests whether a predefined set of genes — a biological pathway — is systematically overrepresented among genes that show evidence of differential activity in an experiment. Unlike classical over-representation tests, it encodes prior biological knowledge as a prior distribution and updates it with the observed expression data, yielding posterior probabilities of enrichment rather than p-values. This probabilistic framing naturally handles small samples, multiple pathways, and uncertainty propagation in a coherent statistical framework.

RNA-seq count data, microarray expression values, pre-ranked gene lists, curated pathway databases (KEGG, Reactome, GO)

Baldi, P., & Long, A. D. (2001). A Bayesian framework for the analysis of microarray expression data: regularized t-test and statistical inferences of gene changes. Bioinformatics, 17(6), 509–519.

Newton, M. A., Quintana, F. A., Den Boon, J. A., Bhattacharya, S., & Ahlquist, P. (2004). Random-set methods identify distinct aspects of the enrichment signal in gene-set analysis. The Annals of Applied Statistics, 1(1), 85–106.


Bayesian Phase I clinical trial

Bayesian Phase I Clinical Trial (Dose-Finding Design)

Bayesian dose-finding trial
CRM trial
continual reassessment method trial
Bayesian dose-escalation study

O'Quigley, Pepe & Fisher (Continual Reassessment Method)

A Bayesian Phase I clinical trial uses prior probability models and sequential Bayes updating to find the maximum tolerated dose (MTD) of a new agent. Unlike the traditional 3+3 rule-based escalation, the Bayesian approach revises a dose-toxicity curve continuously as each patient's outcome is observed, allowing faster convergence to the true MTD while minimising exposure of patients to unsafe or subtherapeutic doses.

Binary toxicity outcomes (dose-limiting toxicity per patient cohort)

O'Quigley, J., Pepe, M., & Fisher, L. (1990). Continual reassessment method: a practical design for phase 1 clinical trials in cancer. Biometrics, 46(1), 33–48.

Chevret, S. (Ed.). (2006). Statistical Methods for Dose-Finding Experiments. Wiley.

phase-i-clinical-trial adaptive-phase-i-clinical-trial bayesian-randomized-clinical-trial dose-response-analysis competing-risks-analysis adaptive-randomized-clinical-trial


Bayesian Phase II Clinical Trial

Bayesian phase 2 trial
Bayesian single-arm phase II study
Bayesian early-phase efficacy trial
Bayes phase II

Peter Thall, Richard Simon, Donald Berry (key contributors)

A Bayesian Phase II clinical trial applies Bayesian statistical inference to the standard Phase II objective of evaluating whether an experimental treatment shows sufficient early-phase efficacy to justify progression to a Phase III trial. By combining prior information with accumulating trial data, it enables principled interim monitoring, flexible stopping rules, and updated probability statements about treatment effect — all without the multiple-testing penalties that burden frequentist sequential designs.

Binary or continuous efficacy outcomes; sequential patient-level data

Thall, P. F., & Simon, R. (1994). Practical Bayesian guidelines for phase IIB clinical trials. Biometrics, 50(2), 337–349.

Berry, D. A. (2006). Bayesian clinical trials. Nature Reviews Drug Discovery, 5(1), 27–36.

phase-ii-clinical-trial adaptive-phase-ii-clinical-trial bayesian-randomized-clinical-trial bayesian-phase-i-clinical-trial adaptive-randomized-clinical-trial dose-response-analysis


Bayesian Phase III Clinical Trial

Bayesian Phase III Confirmatory Clinical Trial

Bayesian confirmatory trial
Bayesian RCT Phase III
Bayesian pivotal trial
BayesCT

Donald A. Berry; David J. Spiegelhalter (formalization in clinical context)

A Bayesian Phase III clinical trial is a large-scale, confirmatory randomized controlled trial that uses Bayesian statistical inference rather than conventional frequentist hypothesis testing to evaluate whether an experimental treatment meets pre-defined efficacy and safety thresholds. By combining prior evidence with accumulating trial data, it quantifies the probability that the treatment effect exceeds a clinically meaningful threshold, enabling more transparent decision-making under uncertainty.

Confirmatory randomized controlled trial with Bayesian inference

Randomized patient outcome data (continuous, binary, time-to-event)

Spiegelhalter, D. J., Abrams, K. R., & Myles, J. P. (2004). Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Wiley.

randomized-clinical-trial phase-iii-clinical-trial adaptive-phase-iii-clinical-trial bayesian-randomized-clinical-trial bayesian-survival-analysis sequential-analysis


Bayesian Phase IV study

Bayesian post-marketing surveillance study

1980s–1990s (formalized application to post-marketing settings)

Donald A. Berry and colleagues (applied Bayesian framework to clinical trials)

A Bayesian Phase IV study is a post-marketing research design that applies Bayesian statistical inference to accumulate evidence about a drug or device already approved for clinical use. By formally combining prior evidence from earlier development phases with emerging real-world data, it enables continuous, probabilistic updating of safety and effectiveness estimates — moving beyond the binary hypothesis tests of conventional frequentist surveillance.

Observational or interventional post-marketing study with Bayesian inference

Registry data, electronic health records, spontaneous adverse event reports, or controlled follow-up data

bayesian-adaptive-trial pharmacovigilance post-marketing-surveillance signal-detection comparative-effectiveness-research registry-based-study


Bayesian Phylogenetic Analysis

Bayesian Phylogenetic Analysis using Markov Chain Monte Carlo

Bayesian phylogenetics
Bayesian inference of phylogeny
MCMC phylogenetics
Bayesian molecular phylogenetics

Rannala & Yang (1996); operationalized by Huelsenbeck et al. (MrBayes, 2001)

Bayesian phylogenetic analysis uses Bayes' theorem and Markov chain Monte Carlo (MCMC) sampling to estimate the posterior probability distribution over phylogenetic trees and model parameters given observed sequence data. Unlike bootstrapped maximum-likelihood methods that return a single best tree, Bayesian inference yields a credible set of trees with associated posterior probabilities, providing a principled measure of phylogenetic uncertainty. It is the dominant framework for estimating divergence times and ancestral relationships in molecular evolution.

Aligned nucleotide or amino acid sequence data (DNA, RNA, protein)

Ronquist, F., & Huelsenbeck, J. P. (2003). MrBayes 3: Bayesian phylogenetic inference under mixed models. Bioinformatics, 19(12), 1572–1574.

Drummond, A. J., & Rambaut, A. (2007). BEAST: Bayesian evolutionary analysis by sampling trees. BMC Evolutionary Biology, 7(1), 214.

phylogenetic-analysis sequence-alignment maximum-likelihood-phylogenetics molecular-clock-analysis ancestral-sequence-reconstruction bayesian-gwas


Bayesian Placebo Test

Bayesian Placebo Test for Causal Inference

Bayesian falsification test
Bayesian permutation placebo
Bayesian robustness check
Bayesian in-time placebo

Brodersen, Gallusser, Koehler, Remy & Scott (Bayesian causal impact context); Abadie, Diamond & Hainmueller (placebo permutation tradition)

The Bayesian Placebo Test is a falsification strategy for causal inference that applies Bayesian inference to placebo scenarios — either fake treatments in the pre-intervention period, on unaffected units, or at fictitious cut-offs — to verify that observed treatment effects cannot plausibly arise by chance or from a misspecified model. It integrates prior information and yields posterior distributions of placebo effects for direct probabilistic comparison.

Time series, panel, or cross-sectional observational data

Brodersen, K. H., Gallusser, F., Koehler, J., Remy, N., & Scott, S. L. (2015). Inferring causal impact using Bayesian structural time-series models. Annals of Applied Statistics, 9(1), 247-274.

Abadie, A., Diamond, A., & Hainmueller, J. (2010). Synthetic control methods for comparative case studies: Estimating the effect of California's tobacco control program. Journal of the American Statistical Association, 105(490), 493-505.

placebo-test bayesian-causal-impact-analysis bayesian-synthetic-control-method sensitivity-analysis-for-causality bayesian-difference-in-differences causal-impact-analysis


Bayesian Poisson Regression

Bayesian log-linear count model
Bayesian GLM Poisson
Poisson regression with priors
Bayesian count regression

1989 (GLM foundation); Bayesian treatment formalized in 1990s–2000s

Gelman et al. (BDA); classical Poisson GLM from McCullagh & Nelder (1989)

Bayesian Poisson regression models non-negative integer count outcomes using a Poisson likelihood with a log link, placing prior distributions on the regression coefficients. Posterior inference — combining prior beliefs with the data likelihood — produces full probability distributions over the coefficients rather than single-point estimates, enabling coherent uncertainty quantification and incorporation of domain knowledge.

Bayesian generalized linear model for count data

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall.

poisson-regression bayesian-negative-binomial-regression bayesian-generalized-linear-model negative-binomial-regression zero-inflated-model bayesian-multiple-linear-regression


Bayesian Power Analysis

Bayesian Power Analysis (Assurance / Bayesian Sample Size Determination)

Bayesian Güç Analizi (Assurance / Bayesian Sample Size)

Spiegelhalter & Freedman (1986); O'Hagan, Stevens & Campbell (2005)

Bayesian power analysis — also called assurance — is a sample size determination method that replaces the frequentist notion of power with a probability-weighted average over a prior distribution on the effect size. First formalised by Spiegelhalter and Freedman (1986) and further developed by O'Hagan, Stevens and Campbell (2005), it answers the question: given our current uncertainty about the true effect, what sample size gives us a high overall probability of obtaining a statistically significant result?

O'Hagan, A., Stevens, J.W. & Campbell, M.J. (2005). Assurance in Clinical Trial Design. Pharmaceutical Statistics, 4(3), 187–201.

Spiegelhalter, D.J. & Freedman, L.S. (1986). A Predictive Approach to Selecting the Size of a Clinical Trial, Based on Subjective Clinical Opinion. Statistics in Medicine, 5(1), 1–13.

simulation-based-power sequential-analysis frequentist-power-analysis bayesian-t-test prior-elicitation


Bayesian PP unit root test

Bayesian PP test
Bayesian Phillips-Perron test
Bayesian nonparametric unit root test
Bayes PP unit root

Phillips & Perron (classical test, 1988); Bayesian framework: Sims & Uhlig (1991)

The Bayesian Phillips-Perron unit root test combines the nonparametric long-run variance correction of the classical Phillips-Perron test with a Bayesian inferential framework. Instead of a p-value, it yields a posterior probability or Bayes factor quantifying evidence for or against a unit root, allowing researchers to incorporate prior economic knowledge and obtain probability statements directly about the persistence of a time series.

Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75(2), 335-346.

Sims, C. A., & Uhlig, H. (1991). Understanding unit rooters: A helicopter tour. Econometrica, 59(6), 1591-1599.

phillips-perron-unit-root-test augmented-dickey-fuller-unit-root-test bayesian-adf-unit-root-test bayesian-kpss-test zivot-andrews-structural-break-test bayesian-var-model


Bayesian Principal Component Analysis

probabilistic PCA with Bayesian inference

Bayesian principal component analysis embeds probabilistic PCA within a Bayesian framework, placing priors over the loading matrix so that irrelevant components are automatically pruned. It handles missing data naturally and provides principled uncertainty estimates for both the latent scores and the dimensionality of the representation.

Bayesian latent variable / dimension reduction

Continuous multivariate data (complete or incomplete)

Bishop, C. M. (1999). Bayesian PCA. In M. S. Kearns, S. A. Solla & D. A. Cohn (Eds.), Advances in Neural Information Processing Systems 11 (pp. 382–388). MIT Press.

Tipping, M. E. & Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society: Series B, 61(3), 611–622.


Bayesian Probit model

Bayesian probit regression
probit model with data augmentation
Gibbs sampling probit
Albert-Chib probit

Albert & Chib (data augmentation formulation)

The Bayesian Probit model is a binary regression method that models the probability of a binary outcome using the normal CDF (probit link) within a Bayesian framework. It assigns prior distributions to regression coefficients and updates them with observed data, yielding a full posterior distribution rather than a single point estimate. The Albert-Chib data-augmentation algorithm makes posterior sampling computationally efficient via Gibbs sampling.

Binary outcome, continuous or categorical predictors

Albert, J. H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88(422), 669-679.


Bayesian Process Capability Analysis

Bayesian PCA
Bayesian capability indices
Bayesian Cp/Cpk estimation
Bayesian process performance analysis

Classical PCA: 1986; Bayesian extensions: 1990s–2000s

Bayesian extensions developed by multiple authors including Bernardo, Smith, and Vannman; classical PCA by Juran and Kane (1986)

Bayesian Process Capability Analysis integrates Bayesian inference with classical capability indices (Cp, Cpk, Cpm) to estimate how well a production process meets specification limits. Rather than relying solely on observed sample data, it incorporates prior knowledge about process parameters — yielding more stable and credible estimates of process capability, especially under small sample sizes common in manufacturing and quality engineering.

Continuous process measurement data (numeric)

Kotz, S., & Johnson, N. L. (2002). Process Capability Indices — A Review, 1992–2000. Journal of Quality Technology, 34(1), 2–19.

Vannman, K., & Kulahci, M. (2006). Bayesian Estimation of Process Capability Indices. Quality and Reliability Engineering International, 22(4), 393–412.

process-capability-analysis statistical-process-control control-chart bayesian-statistical-process-control six-sigma-dmaic bayesian-reliability-analysis


Bayesian Propensity Score Matching

Bayesian Propensity Score Matching Estimator

Bayesian PSM
BPSM
Bayesian matching estimator
Bayesian propensity weighting

Kaplan & Chen (2012); foundational PSM by Rosenbaum & Rubin (1983)

Bayesian Propensity Score Matching (Bayesian PSM) extends classical propensity score matching by placing a prior distribution over the propensity model parameters and propagating posterior uncertainty through the matching and outcome stages. Introduced formally by Kaplan and Chen (2012), it offers a principled account of estimation uncertainty that frequentist matching commonly ignores, and allows incorporation of substantive prior knowledge about treatment selection.

Observational cross-sectional or panel data with binary treatment

Kaplan, D., & Chen, J. (2012). A Two-Step Bayesian Approach for Propensity Score Analysis: Simulations and Case Study. Psychometrika, 77(3), 581-609.

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41-55.

propensity-score-matching bayesian-difference-in-differences inverse-probability-weighting coarsened-exact-matching entropy-balancing doubly-robust-estimation


Bayesian Propensity Score Weighting

Bayesian Propensity Score Weighting for Causal Inference

Bayesian PSW
Bayesian IPW
Bayesian inverse probability weighting
Bayesian propensity weighting

Bayesian Propensity Score Weighting estimates causal treatment effects in observational data by combining a Bayesian model for the propensity score with inverse probability weighting. By placing a prior over propensity-score parameters and propagating posterior uncertainty through the weighting step, this approach yields fully probabilistic uncertainty intervals for the average treatment effect, accounting for the uncertainty in both the score model and the outcome.

Observational data with binary or multi-valued treatment, continuous or binary outcome

McCandless, L. C., Gustafson, P., & Austin, P. C. (2009). Bayesian propensity score analysis for observational data. Statistics in Medicine, 28(1), 94–112.

Saarela, O., Stephens, D. A., Moodie, E. E. M., & Klein, M. B. (2015). On Bayesian estimation of marginal structural models. Biometrics, 71(2), 279–288.

propensity-score-weighting bayesian-difference-in-differences inverse-probability-weighting propensity-score-matching marginal-structural-model doubly-robust-estimation


Bayesian Proteomics Analysis

Bayesian Statistical Analysis of Proteomics Data

Bayesian protein quantification
Bayesian peptide inference
probabilistic proteomics
Bayesian mass spectrometry analysis

Multiple contributors; foundational statistical frameworks by Nesvizhskii, Kall, Choi, and colleagues

Bayesian proteomics analysis applies probabilistic models to mass spectrometry data to identify peptides, infer protein presence, and quantify differential protein abundance across conditions. By encoding prior knowledge and propagating uncertainty through each step of the pipeline, Bayesian approaches produce calibrated posterior probabilities of identification and quantification rather than simple point estimates, enabling more principled control of false discovery rates and more honest reporting of uncertainty than purely frequentist alternatives.

Mass spectrometry data (peptide spectrum matches, protein abundance measurements)

Kall, L., Canterbury, J. D., Weston, J., Noble, W. S., & MacCoss, M. J. (2008). Semi-supervised learning for peptide identification from shotgun proteomics datasets. Nature Methods, 5(11), 923–925.

Choi, H., & Nesvizhskii, A. I. (2008). Semisupervised model-based validation of peptide identifications in mass spectrometry-based proteomics. Journal of Proteome Research, 7(1), 254–265.


Bayesian Quality Function Deployment

Bayesian QFD
Probabilistic QFD
Bayesian House of Quality
Bayesian Voice of the Customer Analysis

QFD: 1966–1972; Bayesian QFD extensions: 2000s–present

Yoji Akao (QFD); Bayesian extension developed by multiple researchers including Fung, Tang, and colleagues

Bayesian Quality Function Deployment (Bayesian QFD) integrates Bayesian probabilistic inference into the classical House of Quality framework to handle uncertainty in customer preference data and relationship matrices. By expressing relationship weights and importance ratings as probability distributions rather than point estimates, it propagates uncertainty through the planning process and yields more defensible engineering prioritization decisions under incomplete or conflicting customer information.

Probabilistic customer-driven design planning method

Customer preference ratings, engineering parameter data, expert judgments (continuous and ordinal)

Tang, J., Fung, R. Y. K., Xu, B., & Wang, D. (2002). A new approach to quality function deployment planning with financial consideration. Computers & Operations Research, 29(11), 1447–1463.


Bayesian QQR
Bayesian QQ regression
Bayes quantile-on-quantile
BQQ regression

Bayesian QQ framework combines Sim & Zhou (2015) QQ regression with Bayesian quantile regression (Yu & Moyeed, 2001)

Bayesian Quantile-on-Quantile (BQQ) Regression extends the Sim-Zhou quantile-on-quantile framework by replacing frequentist local linear estimation with Bayesian posterior inference. For each pair of quantiles (theta of the outcome, tau of the predictor), the method yields a full posterior distribution over the slope, enabling uncertainty quantification across the entire bivariate quantile surface — a key advantage when sample sizes are moderate and tail quantiles are sparse.

Nonparametric quantile regression with Bayesian estimation

Continuous time-series or cross-sectional data; bivariate or multivariate

Sim, N., & Zhou, H. (2015). Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking and Finance, 55, 1–8.

Yu, K., & Moyeed, R. A. (2001). Bayesian quantile regression. Statistics and Probability Letters, 54(4), 437–447.

quantile-on-quantile-regression quantile-regression bayesian-var-model bayesian-ardl-bounds-test nonlinear-ardl bayesian-vecm


Bayesian Quantile Regression

BQR
Bayesian quantile regression model
asymmetric Laplace Bayesian regression
posterior quantile regression

Kozumi & Kobayashi; building on Yu & Moyeed (2001)

Bayesian Quantile Regression estimates the full posterior distribution of regression coefficients at any chosen quantile of the outcome. By combining the asymmetric Laplace likelihood with prior distributions over the coefficients, it delivers uncertainty-quantified estimates of conditional quantiles — such as the median, the 10th, or the 90th percentile — without assuming Gaussian errors.

Continuous, count, or skewed outcome; any scale predictors

Kozumi, H., & Kobayashi, G. (2011). Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81(11), 1565–1578.

Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics – Theory and Methods, 34(9–10), 1867–1879.

quantile-regression bayesian-multiple-linear-regression bayesian-robust-regression robust-quantile-regression bayesian-generalized-linear-model bayesian-tobit-model


Bayesian Quantitative Content Analysis

Bayesian content analysis
Bayesian text analysis
probabilistic content analysis
BQCA

1990s–2000s (convergence of content analysis and Bayesian statistics)

Integration of Krippendorff's content analysis framework with Bayesian statistical inference (Gelman et al.)

Bayesian quantitative content analysis systematically codes and counts features in textual or media content, then quantifies patterns and tests hypotheses using Bayesian statistical inference. Unlike classical frequency-based content analysis, it incorporates prior knowledge or domain expectations into the estimation process, producing posterior probability distributions over content parameters rather than single point estimates with p-values. The approach is particularly valuable when prior research, expert knowledge, or pilot data exist and when uncertainty quantification around content proportions and category frequencies is important.

Coded text, media content, documents, categorical/frequency data

Krippendorff, K. (2018). Content Analysis: An Introduction to Its Methodology (4th ed.). Sage.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.


Bayesian Queueing Simulation

Bayesian Queueing Simulation — Bayesian parameter inference integrated with stochastic queueing simulation

BQS
Bayesian Queue Simulation
Bayesian Stochastic Queueing
Bayesian Queuing Analysis

Bayesian Queueing Simulation combines Bayesian statistical inference with stochastic queueing simulation to model waiting-line systems under parameter uncertainty. Instead of treating arrival and service rates as fixed known values, it places prior distributions over them, updates these with observed data to obtain posteriors, and propagates the resulting parameter uncertainty through repeated simulation runs to produce probabilistic predictions of system performance metrics such as queue length, waiting time, and server utilisation.

Bayesian inference + stochastic simulation

Observed arrival/service counts, prior distributions on rates

Armero, C., & Bayarri, M. J. (1994). Bayesian prediction in M/M/1 queues. Queueing Systems, 15(1–4), 401–417.

Insua, D. R., Wiper, M., & Ruggeri, F. (1998). Bayesian analysis of M/Er/1 and M/H_k/1 queues. Queueing Systems, 30(3–4), 289–308.

monte-carlo-simulation discrete-event-simulation queueing-simulation bayesian-monte-carlo-simulation stochastic-queueing-simulation markov-model


Bayesian Random Effects Model

Bayesian hierarchical model
Bayesian mixed effects model
Bayesian multilevel model
BREM

Lindley & Smith (1972); extended by Gelman, Rubin and colleagues

The Bayesian random effects model combines panel-data random effects with a Bayesian prior framework, allowing unit-specific effects to be treated as draws from a population distribution whose hyperparameters are estimated from the data. This produces regularised, uncertainty-quantified estimates that borrow strength across units — particularly valuable for short panels, sparse groups, or settings where frequentist variance-component estimation is unstable.

Panel / longitudinal / time-series cross-section data

Hsiao, C. (2014). Analysis of Panel Data (3rd ed.). Cambridge University Press.

panel-random-effects panel-fixed-effects hierarchical-linear-model bayesian-var mixed-effects-model ols-regression


Bayesian Random Forest

Bayesian Random Forest (Bayesian Ensemble of Decision Trees)

Bayesian Forest
BRF
Empirical Bayesian Forest
posterior random forest

Bayesian Random Forest extends the classical random forest by placing a prior distribution over tree structures and leaf parameters, then sampling or approximating the posterior over that ensemble. The result is a set of predictions accompanied by calibrated uncertainty estimates — a capability standard random forests lack — making it valuable when knowing how confident the model is matters as much as the prediction itself.

Tabular (continuous, categorical, mixed)

Taddy, M., Chen, C., Yu, J., & Wyle, M. (2015). Bayesian and Empirical Bayesian Forests. Proceedings of the 32nd International Conference on Machine Learning (ICML 2015), PMLR 37, 967–976.

Lakshminarayanan, B., Roy, D. M., & Teh, Y. W. (2016). Mondrian Forests for Large-Scale Regression when Uncertainty Matters. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS 2016), PMLR 51, 1478–1487.

random-forest bayesian-decision-tree gaussian-process bayesian-gradient-boosting bayesian-active-learning bayesian-semi-supervised-learning


Bayesian Randomized Clinical Trial

Bayesian RCT
Bayesian adaptive trial
Bayesian clinical trial design
BRCT

1980s–2000s (formal methodology consolidated ~2004–2006)

Donald A. Berry and David J. Spiegelhalter (applied Bayesian inference formally to RCT design)

A Bayesian randomized clinical trial (Bayesian RCT) combines the rigour of random treatment allocation with Bayesian statistical inference, allowing researchers to incorporate prior evidence and update beliefs continuously as trial data accumulate. Unlike the classical frequentist RCT, it yields direct probability statements about treatment effects and supports pre-specified adaptive stopping rules based on posterior probabilities.

Randomized experimental study with Bayesian inference

Individual patient-level outcome data from randomized arms; prior distributions from historical studies or expert elicitation

Spiegelhalter, D. J., Abrams, K. R., & Myles, J. P. (2004). Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Wiley.

Berry, D. A. (2006). Bayesian clinical trials. Nature Reviews Drug Discovery, 5(1), 27–36.

randomized-clinical-trial adaptive-randomized-clinical-trial bayesian-survival-analysis bayesian-diagnostic-accuracy-study pragmatic-randomized-clinical-trial phase-ii-clinical-trial


Bayesian RDD
Bayesian RD
Bayes RDD
Bayesian regression-discontinuity

Bayesian Regression Discontinuity Design (Bayesian RDD) embeds the classical RD framework — which estimates a local causal effect at a known assignment cutoff — within a Bayesian inferential engine. Prior distributions are placed on the regression functions on either side of the cutoff and on the treatment-effect parameter, yielding a full posterior distribution over the causal estimand rather than a single point estimate with a frequentist p-value.

Bayesian causal inference / quasi-experimental

Cross-sectional or panel data with a continuous running variable and a known cutoff

Karabatsos, G., & Walker, S. G. (2004). Coherent inference in regression discontinuity designs with a Bayesian nonparametric approach. Journal of the American Statistical Association, 99(468), 1121-1131.

Chib, S., & Jacobi, L. (2016). Bayesian fuzzy regression discontinuity analysis and returns to compulsory schooling. Journal of Applied Econometrics, 31(6), 1026-1047.

regression-discontinuity-design fuzzy-regression-discontinuity bayesian-difference-in-differences instrumental-variables local-average-treatment-effect propensity-score-matching


Bayesian Regression

bayesian linear regression
probabilistic regression
bayesian regresyon

Bayesian regression is a probabilistic version of linear regression that treats the model parameters as uncertain quantities. Instead of returning a single best-fit estimate, it combines prior knowledge with the observed data to produce a full posterior probability distribution for each parameter, from which credible intervals and predictions are read off.

posterior distributions / credible intervals

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.


Bayesian Ridge Regression

Bayesian Ridge Regression (MacKay Probabilistic Regularisation)

Bayesian linear regression with automatic relevance determination

Bayesian Ridge Regression is a probabilistic formulation of ridge regression, introduced by David J. C. MacKay in 1992, in which the regularisation strength and noise precision are not fixed by the analyst but are instead estimated automatically by maximising the marginal likelihood (evidence) of the observed data. The result is a full posterior distribution over the regression weights together with calibrated predictive uncertainty.

Regression with automatic regularisation

MacKay, D. J. C. (1992). Bayesian Interpolation. Neural Computation, 4(3), 415–447.

Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Ch. 3). Springer.

ridge-regression lasso-regression elastic-net gaussian-process-regression linear-regression ard-regression


Bayesian Differential Expression Analysis of RNA Sequencing Data

Bayesian DE analysis
Bayesian RNA-seq DE
baySeq
EBSeq

Kendziorski et al. (EBSeq); Hardcastle & Kelly (baySeq)

Bayesian RNA-seq differential expression analysis applies hierarchical Bayesian models to RNA sequencing read-count data to identify genes whose expression levels differ significantly between biological conditions. Rather than relying solely on p-values, these methods quantify the posterior probability that a gene is differentially expressed, borrowing statistical strength across genes and naturally accommodating low sample sizes common in genomics experiments.

RNA-seq read count matrices (integer counts per gene/transcript per sample)

Leng, N., Dawson, J. A., Thomson, J. A., Ruotti, V., Rissman, A. I., Smits, B. M., Haag, J. D., Gould, M. N., Stewart, R. M., & Kendziorski, C. (2013). EBSeq: An empirical Bayes hierarchical model for inference in RNA-seq experiments. Bioinformatics, 29(8), 1035–1043.

Hardcastle, T. J., & Kelly, K. A. (2010). baySeq: Empirical Bayesian methods for identifying differential expression in sequence count data. BMC Bioinformatics, 11, 422.

rna-seq-differential-expression gene-set-enrichment-analysis pathway-enrichment-analysis single-cell-rna-seq-analysis bayesian-gwas variant-calling


Bayesian Robust Regression

Bayesian heavy-tailed regression
Bayesian Student-t regression
robust Bayesian linear model
BRR

Bayesian Robust Regression replaces the Gaussian error assumption of ordinary linear regression with a heavy-tailed distribution — most commonly the Student-t — and estimates all parameters in a Bayesian framework. The heavier tails give outliers less influence on the fitted line, yielding stable coefficient estimates and honest uncertainty intervals even when the data contain unusual observations.

Bayesian regression with heavy-tailed errors

Continuous outcome, continuous or categorical predictors; outlier-prone data

Geweke, J. (1993). Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics, 8(S1), S19–S40.

robust-regression bayesian-multiple-linear-regression bayesian-quantile-regression quantile-regression ols-regression bayesian-generalized-linear-model


Bayesian Root Cause Analysis

Bayesian Network-Based Root Cause Analysis

Bayesian RCA
Bayesian causal analysis
probabilistic root cause analysis
BN-RCA

Rooted in Pearl's Bayesian network theory (Judea Pearl, 1988); applied to RCA in process/reliability engineering from the 1990s onward

Bayesian Root Cause Analysis (Bayesian RCA) integrates Bayesian network theory with structured root cause investigation to quantify the probability that each candidate cause is responsible for an observed failure or undesired event. Unlike deterministic RCA methods, it propagates uncertainty through the causal graph, updates beliefs as evidence accumulates, and ranks competing hypotheses by posterior probability — providing a principled, auditable basis for corrective action.

Failure event records, expert probability estimates, historical process data

Pourret, O., Naim, P., & Marcot, B. (Eds.). (2008). Bayesian Networks: A Practical Guide to Applications. Wiley.

Weber, P., Medina-Oliva, G., Simon, C., & Iung, B. (2012). Overview on Bayesian networks applications for dependability, risk analysis and maintenance areas. Engineering Applications of Artificial Intelligence, 25(4), 671–682.


Bayesian SARIMA Model

Bayesian Seasonal Autoregressive Integrated Moving Average Model

Bayesian SARIMA
Bayesian seasonal ARIMA
BSARIMA
Bayesian seasonal time-series model

Box & Jenkins (classical SARIMA); Bayesian extensions developed through Zellner, Geweke, and later MCMC-era researchers

The Bayesian SARIMA model combines the classical Box-Jenkins Seasonal ARIMA framework with Bayesian inference to handle seasonal time-series data. Rather than producing a single point estimate, it yields a full posterior distribution over model parameters, propagating parameter uncertainty directly into forecasts and enabling principled incorporation of prior knowledge.

Univariate seasonal time-series (equally spaced observations)

Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time Series Analysis: Forecasting and Control (5th ed.). Wiley.

Geweke, J., & Whiteman, C. (2006). Bayesian forecasting. In G. Elliott, C. W. J. Granger, & A. Timmermann (Eds.), Handbook of Economic Forecasting (Vol. 1, pp. 3–80). Elsevier.

arima-model sarima-model bayesian-var-model state-space-model exponential-smoothing dynamic-linear-model


Bayesian Scale Development

Bayesian psychometric scale construction

Harold Jeffreys, expanded into psychometrics by Mislevy and colleagues

Bayesian scale development applies Bayesian statistical inference to the construction and evaluation of psychometric scales. Rather than relying on single point estimates of item and person parameters, it produces full posterior distributions that quantify uncertainty, incorporate prior knowledge, and support principled decisions about item retention, reliability, and validity in small or complex samples.

Bayesian probabilistic scale construction

De Ayala, R. J. (2009). The Theory and Practice of Item Response Theory. Guilford Press.

Levy, R., & Mislevy, R. J. (2016). Bayesian Psychometric Modeling. CRC Press / Chapman & Hall.

confirmatory-factor-analysis exploratory-factor-analysis item-response-theory scale-development bayesian-confirmatory-factor-analysis bayesian-item-response-theory


Bayesian Scenario Analysis

Bayesian Scenario Analysis — Probabilistic scenario weighting via Bayesian inference

BSA
Bayesian scenario planning
probabilistic scenario analysis
Bayesian-weighted scenario analysis

Developed iteratively across Bayesian statistics and scenario planning communities; formalized in risk and decision analysis (Aven, Lempert et al., 2000s)

Bayesian Scenario Analysis (BSA) combines structured scenario planning with Bayesian probability theory, assigning explicit prior probabilities to alternative futures and updating them as new evidence or expert judgments become available. The result is a probability-weighted distribution of outcomes across scenarios rather than a set of equally-weighted or arbitrarily-weighted futures.

Probabilistic hybrid — Bayesian inference integrated with structured scenario analysis

Qualitative scenarios with quantitative prior/likelihood inputs; expert elicitation; historical frequency data

Aven, T., & Reniers, G. (2013). How to define and interpret a probability in a risk and safety setting. Safety Science, 51(1), 223–231.

Lempert, R. J., Popper, S. W., & Bankes, S. C. (2003). Shaping the Next One Hundred Years: New Methods for Quantitative, Long-Term Policy Analysis. RAND Corporation.

scenario-analysis bayesian-sensitivity-analysis stochastic-scenario-analysis monte-carlo-simulation robust-scenario-analysis markov-model


Bayesian Screening Test Evaluation

Bayesian diagnostic test evaluation
Bayesian predictive value analysis
posterior predictive value approach
Bayes theorem screening

1763 (theorem); clinical screening application formalized ~1959–1970s

Thomas Bayes (theorem, 1763); applied to clinical screening by Ledley & Lusted (1959)

Bayesian screening test evaluation applies Bayes' theorem to quantify how a screening test result changes the probability that an individual truly has a disease. Rather than reporting sensitivity and specificity in isolation, the approach centres on predictive values — the probability of disease given a positive or negative test — which depend critically on disease prevalence in the population being screened. The framework allows systematic updating of pre-test probability to post-test probability and supports decision-making under uncertainty.

Bayesian analytical framework for test evaluation

Binary or ordinal test results, disease status (2×2 contingency tables), prevalence estimates

Fletcher, R. H., Fletcher, S. W., & Fletcher, G. S. (2014). Clinical Epidemiology: The Essentials (5th ed.). Lippincott Williams & Wilkins.

Altman, D. G., & Bland, J. M. (1994). Diagnostic tests 2: Predictive values. BMJ, 309(6947), 102.

diagnostic-accuracy-study screening-test-evaluation bayesian-diagnostic-accuracy-study cohort-study case-control-study dose-response-analysis


Bayesian SEM

BSEM
Bayesian latent variable model
approximate zero constraints SEM
Bayesçi Yapısal Eşitlik Modeli

Bayesian SEM, introduced by Muthén and Asparouhov in 2012, extends classical structural equation modeling by placing prior distributions on factor loadings, path coefficients, and covariances. Instead of returning a single maximum-likelihood estimate, it uses Markov chain Monte Carlo to produce a full posterior distribution for every parameter, enabling principled uncertainty quantification in models with latent variables.

posterior distributions / credible intervals / PPP fit index

Muthén, B. & Asparouhov, T. (2012). Bayesian SEM: A More Flexible Representation of Substantive Theory. Psychological Methods, 17(3), 313–335.

sem cfa bayesian-regression bayesian-hierarchical-model latent-growth-curve mcmc


Bayesian Semi-supervised Learning

Bayesian Semi-supervised Learning (Probabilistic Inference with Labeled and Unlabeled Data)

Bayesian SSL
probabilistic semi-supervised learning
generative semi-supervised model
Bayesian transductive learning

Chapelle, Scholkopf & Zien; Zhu, Ghahramani & Lafferty

Bayesian semi-supervised learning is a probabilistic framework that uses both a small labeled dataset and a larger pool of unlabeled observations to infer model parameters and make predictions. By treating missing labels as latent variables and placing priors over parameters, it naturally quantifies uncertainty while leveraging unlabeled data to improve generalization.

Labeled and unlabeled tabular, text, or image data

Chapelle, O., Scholkopf, B., & Zien, A. (Eds.). (2006). Semi-Supervised Learning. MIT Press.

Zhu, X., Ghahramani, Z., & Lafferty, J. (2003). Semi-supervised learning using Gaussian fields and harmonic functions. Proceedings of the 20th International Conference on Machine Learning (ICML), 912–919.

semi-supervised-learning bayesian-active-learning gaussian-process bayesian-gaussian-mixture-model transfer-learning few-shot-learning


Bayesian Sensitivity Analysis for Causality

Bayesian Sensitivity Analysis for Unmeasured Confounding in Causal Inference

probabilistic sensitivity analysis for confounding

Bayesian unmeasured confounding analysis

McCandless, Gustafson & Austin (2007); Gustafson (2015)

Bayesian sensitivity analysis for causality quantifies how much an unmeasured confounder would need to influence both treatment assignment and outcome to overturn a causal conclusion. Rather than testing a single worst-case scenario, it places prior distributions over the strength of hidden confounding, propagates uncertainty through a full Bayesian model, and reports a posterior distribution for the causal effect that honestly reflects what is and is not identified from observed data.

Observational data with binary or continuous treatment and outcome

McCandless, L. C., Gustafson, P., & Austin, P. C. (2007). Bayesian propensity score analysis for observational data. Statistics in Medicine, 26(8), 1704-1718.

Gustafson, P. (2015). Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data. CRC Press / Chapman & Hall.

sensitivity-analysis-for-causality bayesian-difference-in-differences propensity-score-matching instrumental-variables doubly-robust-estimation marginal-structural-model


Bayesian Sensitivity Analysis

Bayesian Sensitivity Analysis — Prior-informed uncertainty propagation and output sensitivity assessment

BSA
Bayesian SA
Bayesian robustness analysis
prior sensitivity analysis

Berger, J. O. (Bayesian robustness); Saltelli et al. (global SA integration)

Bayesian Sensitivity Analysis (BSA) combines Bayesian inference with sensitivity analysis to systematically quantify how uncertain model inputs — expressed as prior probability distributions — propagate through a model and influence outputs. It identifies which parameters most drive output variability, supporting robust conclusions under genuine uncertainty.

Uncertainty propagation and sensitivity quantification

Probabilistic inputs, prior distributions, model parameters

Berger, J. O. (1994). An overview of robust Bayesian analysis. Test, 3(1), 5–124.

Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., & Tarantola, S. (2008). Global Sensitivity Analysis: The Primer. Wiley.

monte-carlo-simulation stochastic-sensitivity-analysis bayesian-markov-model scenario-analysis markov-model bayesian-dynamic-programming


Bayesian Sequence Alignment

Bayesian Probabilistic Sequence Alignment

Bayesian MSA
probabilistic sequence alignment
statistical alignment
BAli-Phy alignment

Ian Holmes & William J. Bruno; Benjamin Redelings & Marc Suchard

Bayesian sequence alignment treats the alignment of biological sequences (DNA, RNA, or protein) as a probabilistic inference problem rather than a deterministic optimization. Instead of returning a single best alignment, it samples from a posterior distribution over all plausible alignments given a substitution model and gap penalty priors, thereby quantifying alignment uncertainty. It is particularly valuable when downstream analyses such as phylogenetic inference or functional annotation are sensitive to alignment error.

Nucleotide or protein sequence data (FASTA/aligned FASTA)

Redelings, B. D., & Suchard, M. A. (2005). Joint Bayesian estimation of alignment and phylogeny. Systematic Biology, 54(3), 401–418.

Holmes, I., & Bruno, W. J. (2001). Evolutionary HMMs: a Bayesian approach to multiple alignment. Bioinformatics, 17(9), 803–820.

sequence-alignment phylogenetic-analysis bayesian-phylogenetic-analysis variant-calling rna-seq-differential-expression hidden-markov-model


Bayesian Simple linear regression

Bayesian SLR
Bayesian univariate regression
probabilistic simple linear regression
Bayesian linear model

Early 19th century; textbook synthesis 2013

Laplace, P.-S. (early 19th c.); modern treatment: Gelman et al.

Bayesian Simple Linear Regression models the relationship between a continuous outcome and a single predictor by combining a Gaussian likelihood with prior distributions over the intercept, slope, and error variance. The result is a full posterior distribution over all parameters, providing probabilistic uncertainty quantification rather than a single point estimate.

Continuous outcome, single continuous predictor

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and Stan (2nd ed.). CRC Press.

ols-regression bayesian-multiple-linear-regression bayesian-generalized-linear-model bayesian-robust-regression simple-linear-regression bayesian-quantile-regression


Bayesian Simulated Annealing

Bayesian Simulated Annealing — Probabilistic global optimization with Bayesian priors on the energy landscape

BSA
Bayesian SA
Bayesian Stochastic Annealing
Bayesian Thermodynamic Optimization

Geman, S. & Geman, D. (Bayesian framing); Kirkpatrick, S. et al. (SA foundation)

Bayesian Simulated Annealing (BSA) integrates Bayesian prior knowledge about the objective landscape into the simulated annealing search process. By encoding beliefs about promising regions as prior distributions and updating them as the search progresses, BSA focuses computational effort on high-probability areas of the solution space, accelerating convergence and improving solution quality compared to uninformed SA.

Probabilistic metaheuristic with Bayesian inference

Continuous or discrete parameter spaces; prior distributions over parameters

Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.

Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(6), 721–741.

simulated-annealing bayesian-optimization markov-chain-monte-carlo stochastic-simulated-annealing genetic-algorithm bayesian-genetic-algorithm


Bayesian single-cell RNA-seq analysis

Bayesian Probabilistic Analysis of Single-Cell RNA Sequencing Data

2018 (scVI landmark); Bayesian scRNA-seq approaches emerged 2015-2018

Romain Lopez, Nir Yosef and Michael I. Jordan (scVI framework); preceded by Bayesian single-cell methods from Kharchenko, Markowetz, and others

Bayesian single-cell RNA-seq analysis applies probabilistic generative models to the sparse, overdispersed count matrices produced by single-cell RNA sequencing. By placing prior distributions over latent biological variables — cell state, batch effects, dropout — the framework propagates uncertainty through every downstream inference step. Tools such as scVI, SCVI-tools, and BayesPrism implement this paradigm, enabling principled cell clustering, differential expression testing, and batch integration that explicitly models technical noise rather than ignoring it.

Probabilistic generative modeling pipeline

Raw or normalized UMI count matrices from single-cell RNA sequencing experiments

Lopez, R., Regier, J., Cole, M. B., Jordan, M. I., & Yosef, N. (2018). Deep generative modeling for single-cell transcriptomics. Nature Methods, 15(12), 1053-1058.

Eraslan, G., Simon, L. M., Mircea, M., Mueller, N. S., & Theis, F. J. (2019). Single-cell RNA-seq denoising using a deep count autoencoder. Nature Communications, 10(1), 390.

single-cell-rna-seq variational-autoencoder negative-binomial-regression latent-dirichlet-allocation gaussian-process-regression dimensionality-reduction


Bayesian Six Sigma DMAIC

Bayesian Six Sigma Define-Measure-Analyze-Improve-Control

Bayesian DMAIC
Bayesian Six Sigma
B-DMAIC
Probabilistic Six Sigma DMAIC

1986 (DMAIC); Bayesian integration circa 1995–2010

Six Sigma: Bill Smith / Mikel Harry at Motorola (1986); Bayesian integration developed in quality literature through 1990s–2000s

Bayesian Six Sigma DMAIC integrates Bayesian statistical inference into the classical Define-Measure-Analyze-Improve-Control quality-improvement framework. Rather than relying solely on frequentist hypothesis tests and point estimates, it incorporates prior knowledge — from expert judgment, historical production data, or pilot studies — and updates beliefs about process parameters as new data arrive. The result is a more adaptive, uncertainty-aware approach to reducing defects and improving process capability, particularly valuable when sample sizes are small or prior domain knowledge is rich.

Process measurements, defect counts, historical data, prior expert knowledge

Pan, J.-N. (2007). Bayesian approach to estimation of process capability indices in process quality assurance. Quality and Reliability Engineering International, 23(1), 3–14.

six-sigma-dmaic bayesian-statistical-process-control bayesian-design-of-experiments bayesian-process-capability-analysis robust-six-sigma-dmaic statistical-process-control


Bayesian Social Network Analysis

Bayesian Social Network Analysis (Probabilistic Network Modeling)

Bayesian SNA
Bayesian network modeling
probabilistic social network analysis
Bayesian relational modeling

Hoff, P. D.; Raftery, A. E.; Handcock, M. S.

Bayesian Social Network Analysis applies Bayesian probabilistic inference to relational data, placing prior distributions over network parameters and updating them with observed tie data to yield full posterior distributions over structural features, tie probabilities, and latent actor positions. It enables principled uncertainty quantification in network models, making it especially valuable when data are sparse, partially observed, or subject to measurement error.

Adjacency matrices, dyadic relational data, network edge lists

Hoff, P. D., Raftery, A. E., & Handcock, M. S. (2002). Latent space approaches to social network analysis. Journal of the American Statistical Association, 97(460), 1090–1098.

exponential-random-graph-model social-network-analysis bayesian-exponential-random-graph-model stochastic-block-model network-diffusion-analysis multilayer-social-network-analysis


Bayesian Spatial Autocorrelation

Bayesian Spatial Autocorrelation Analysis

Bayesian spatial dependence
Bayesian LISA
Bayesian spatial clustering
BSA

Bayesian Spatial Autocorrelation embeds spatial dependence directly into a Bayesian hierarchical model. A Conditional Autoregressive (CAR) prior encodes the expectation that neighboring areas are more similar than distant ones, and posterior inference is obtained via MCMC. This approach is especially valuable in disease mapping, ecology, and regional science, where small-area estimates need borrowing strength across neighbors.

Areal / lattice data; point-referenced counts or rates

Besag, J., York, J., & Mollie, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics, 43(1), 1–20.

Gelfand, A. E., Diggle, P., Guttorp, P., & Fuentes, M. (Eds.). (2010). Handbook of Spatial Statistics. CRC Press.

spatial-autocorrelation morans-i local-indicators-of-spatial-association bayesian-spatial-regression bayesian-kriging local-spatial-autocorrelation


Bayesian Spatial Durbin Model

Bayesian SDM with spatially lagged covariates

The Bayesian Spatial Durbin Model (BSDM) estimates a spatial regression that simultaneously includes a spatially lagged outcome variable and spatially lagged covariates, using Bayesian inference with Markov Chain Monte Carlo sampling. It captures both endogenous and exogenous spatial spillovers while providing full posterior distributions for all parameters, quantifying uncertainty beyond what classical maximum-likelihood estimation offers.

Georeferenced cross-sectional or panel data with continuous outcome

LeSage, J. P., & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press / Taylor & Francis.

LeSage, J. P. (2014). Spatial Econometric Panel Data Model Comparison Using Heterogeneous Panels with Local Spatial Spillovers. Empirical Economics, 46(1), 193–211.

spatial-durbin-model bayesian-spatial-lag-model bayesian-spatial-error-model geographically-weighted-regression spatial-lag-model spatial-error-model


Bayesian Spatial Error Model

Bayesian spatially correlated error model

1988 (classical SEM); 2009 (Bayesian formulation)

LeSage & Pace (Bayesian treatment); Anselin (classical SEM)

The Bayesian Spatial Error Model (Bayesian SEM) estimates a regression in which spatially correlated disturbances are explicitly modelled through a spatial weights matrix, while all parameters — regression coefficients, spatial error autocorrelation, and error variance — receive full posterior distributions via Bayesian inference rather than point estimates.

Cross-sectional or panel areal / lattice data with spatially correlated errors

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic Publishers.

spatial-error-model bayesian-spatial-lag-model bayesian-spatial-durbin-model geographically-weighted-regression spatial-autocorrelation morans-i


Bayesian Spatial Lag Model

Bayesian Spatial Autoregressive Lag Model

Bayesian SAR model
Bayesian spatial autoregressive model
BSLM
Bayesian SLM

LeSage (1997); fully elaborated in LeSage & Pace (2009)

The Bayesian Spatial Lag Model (BSLM) extends the classical spatial autoregressive (SAR) regression by placing prior distributions over all parameters and recovering full posterior distributions via MCMC sampling. It explicitly accounts for spatial dependence — the outcome in one location is partly driven by outcomes in neighboring locations — and yields uncertainty-quantified estimates of both regression coefficients and the spatial autocorrelation parameter rho.

Cross-sectional or panel areal / lattice data with spatial weights matrix

LeSage, J. P. (1997). Bayesian Estimation of Spatial Autoregressive Models. International Regional Science Review, 20(1-2), 113-129.

spatial-lag-model bayesian-spatial-error-model bayesian-spatial-durbin-model geographically-weighted-regression spatial-autocorrelation panel-spatial-lag-model


Bayesian Spatial Panel Model

Bayesian spatial panel
Bayesian spatial econometrics panel
BSPM
Bayesian panel spatial regression

The Bayesian Spatial Panel Model estimates spatial interaction effects (spatial lag, spatial error, or Durbin) in panel data using Bayesian inference via Markov Chain Monte Carlo (MCMC). It combines the ability to control for unobserved unit- and time-specific heterogeneity with principled uncertainty quantification, making it suitable for georeferenced longitudinal datasets in economics, public health, and regional science.

Georeferenced panel data (multiple units observed over multiple time periods)

Elhorst, J. P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Springer.

spatial-lag-model spatial-error-model spatial-durbin-model panel-spatial-lag-model bayesian-spatial-regression geographically-weighted-regression


Bayesian Spatial Regression

Bayesian hierarchical spatial model
BSR
Bayesian geostatistical regression
Bayesian spatial linear model

Banerjee, Carlin & Gelfand (foundational treatment); building on Besag (1974) for lattice priors

Bayesian Spatial Regression embeds a spatially structured random effect into a regression framework and estimates all parameters — including spatial range and variance — through posterior inference rather than point estimation. It handles spatial autocorrelation, quantifies full predictive uncertainty, and accommodates small or irregular spatial datasets via hierarchical priors.

Georeferenced areal, point-referenced, or lattice data with continuous or discrete outcomes

Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2015). Hierarchical Modeling and Analysis for Spatial Data (2nd ed.). CRC Press.

Cressie, N. A. C. (1993). Statistics for Spatial Data (Revised ed.). Wiley-Interscience.

spatial-lag-model spatial-error-model kriging geographically-weighted-regression car-model multilevel-model


Bayesian Stacking Ensemble

Bayesian Stacking Ensemble (Bayesian Stacking of Predictive Distributions)

Bayesian stacking
Bayesian model stacking
stacking with Bayesian weights
predictive distribution stacking

Yao, Y.; Vehtari, A.; Simpson, D.; Gelman, A.

Bayesian stacking combines the predictive distributions of several base models by finding non-negative weights that maximise the leave-one-out log predictive score of the mixture. Formalised by Yao, Vehtari, Simpson, and Gelman (2018), it yields a single calibrated predictive distribution that is provably at least as good as any single constituent model under cross-validation.

Continuous, binary, or count outcomes; any features

Yao, Y., Vehtari, A., Simpson, D., & Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. Bayesian Analysis, 13(3), 917–1007.

Wolpert, D. H. (1992). Stacked generalization. Neural Networks, 5(2), 241–259.

stacking-ensemble bayesian-model-averaging voting-ensemble bagging boosting gaussian-process


Bayesian Statistical Process Control

Bayesian SPC
Bayesian process monitoring
B-SPC
Bayesian control charting

1950s (foundations); formalized 1990s–2000s

Various (Girshick & Rubin 1952 early signal detection; Menzefricke 2002 Bayesian control chart framework)

Bayesian Statistical Process Control (Bayesian SPC) extends classical SPC by replacing fixed, frequentist control limits with a probabilistic framework that incorporates prior knowledge about the process. Rather than waiting for a run of points to exceed a pre-set 3-sigma boundary, Bayesian SPC continuously updates the probability that the process has shifted given the incoming data, enabling earlier and more informed detection of out-of-control states while formally accounting for uncertainty in process parameters.

Continuous or attribute process measurements over time

Menzefricke, U. (2002). On the evaluation of control chart factors for monitoring the process mean and variance. Journal of Quality Technology, 34(2), 167–178.

statistical-process-control control-chart bayesian-reliability-analysis bayesian-design-of-experiments process-capability-analysis six-sigma-dmaic


Bayesian Statistical Inference

Bayesian Methods in Statistical Analysis

Bayes theorem
Bayesian inference
posterior probability

Bayesian inference is a statistical framework using Bayes' theorem to update beliefs about parameters or hypotheses as data accumulate. Published posthumously in 1763, Thomas Bayes' work lay dormant until the 20th century, when computational advances (Gibbs sampling, Markov Chain Monte Carlo) made Bayesian methods practical. Unlike frequentist inference (which treats parameters as fixed unknowns), Bayesian analysis treats parameters as random variables with probability distributions, enabling direct probability statements about parameters, incorporation of prior knowledge, and sequential updating. Essential in precision medicine, adaptive trials, complex hierarchical models, and any context where prior information enriches inference.

Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society, 53, 370–418.

Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

Kruschke, J. K. (2015). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd ed.). Academic Press.


Bayesian Stochastic Block Model

Bayesian SBM
B-SBM
probabilistic block model
Bayesian community detection model

Nowicki, K. & Snijders, T. A. B.; extended by Peixoto, T. P.

The Bayesian Stochastic Block Model (Bayesian SBM) is a principled probabilistic method for community detection in networks. It treats group membership as a latent variable and uses Bayesian inference to simultaneously recover block structure and select the number of communities, avoiding the resolution-limit bias that plagues modularity-based approaches.

Probabilistic generative model with Bayesian inference

Adjacency matrix (binary or weighted graph)

Peixoto, T. P. (2014). Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models. Physical Review E, 89(1), 012804.

Nowicki, K., & Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96(455), 1077–1087.

stochastic-block-model bayesian-social-network-analysis modularity-analysis exponential-random-graph-model community-detection multilayer-stochastic-block-model


Bayesian Structural Time Series

BSTS
Bayesian Yapısal Zaman Serisi (BSTS)
bayesian state-space model
causal impact model

Scott & Varian (2014); Brodersen et al. (2015)

Bayesian Structural Time Series (BSTS) is a state-space modelling framework, introduced by Scott and Varian (2014), that decomposes a time series into additive components — trend, seasonality, and regression — and estimates them jointly through Bayesian inference. It underpins Google's CausalImpact library and is a powerful tool for both forecasting and counterfactual causal analysis of interventions.

State-space model / Bayesian structural model

posterior component decompositions / credible intervals / causal impact estimates

Scott, S. L. & Varian, H. R. (2014). Predicting the Present with Bayesian Structural Time Series. International Journal of Mathematical Modelling and Numerical Optimisation, 5(1/2), 4–23.

Brodersen, K. H., Gallusser, F., Koehler, J., Remy, N. & Scott, S. L. (2015). Inferring Causal Impact Using Bayesian Structural Time-Series Models. Annals of Applied Statistics, 9(1), 247–274.

bayesian-regression interrupted-time-series arima state-space-model causal-impact mcmc


Bayesian Support Vector Machine

Bayesian Support Vector Machine (Bayesian SVM)

Bayesian SVM
probabilistic SVM
Bayesian kernel machine
BSVM

Polson, N. G. & Scott, S. L.; Tipping, M. E.

Bayesian SVM places a prior distribution over the weight vector of a standard SVM and derives a full posterior, enabling calibrated uncertainty estimates, automatic hyperparameter selection, and probabilistic predictions. It combines the strong margin-based geometric intuition of SVMs with the principled uncertainty quantification of Bayesian inference.

Bayesian probabilistic classifier / regressor

Continuous, binary, multi-class tabular data

Polson, N. G., & Scott, S. L. (2011). Data augmentation for support vector machines. Bayesian Analysis, 6(1), 1–23.

Tipping, M. E. (2001). Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1, 211–244.

support-vector-machine gaussian-process bayesian-logistic-regression relevance-vector-machine bayesian-naive-bayes kernel-methods


Bayesian Survey Research

Bayesian survey analysis
Bayesian survey methodology
Bayesian polling
Bayesian questionnaire analysis

1980s–2000s (modern applied development)

Thomas Bayes (theorem, 1763); applied to survey methodology by Donald Rubin, Andrew Gelman, and others (1980s–2000s)

Bayesian survey research applies Bayesian statistical inference to survey data, combining prior knowledge or beliefs about population parameters with observed questionnaire responses to produce posterior probability distributions. Unlike null-hypothesis significance testing, this approach quantifies uncertainty directly, incorporates prior evidence, and yields probabilistic statements about parameters of interest — making it especially powerful for small samples, sequential data collection, and contexts where substantive prior knowledge exists.

Quantitative observational research design with Bayesian inference

Structured questionnaire responses (Likert, categorical, continuous); prior probability distributions

Gelman, A., & Carlin, J. B. (2007). Some issues on the foundations of statistics. In A. Gelman & J. B. Carlin (Eds.), Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

Lee, M. D., & Wagenmakers, E.-J. (2005). Bayesian statistical inference in psychology: Comment on Trafimow (2003). Psychological Review, 112(3), 662–668.

survey-research bayesian-correlational-research bayesian-longitudinal-research multilevel-modeling structural-equation-modeling confirmatory-research


Bayesian Survival regression

Bayesian time-to-event regression
Bayesian parametric survival model
Bayesian survival analysis
Bayesian accelerated failure time model

Ibrahim, Chen & Sinha (seminal textbook treatment, 2001); broader Bayesian framework: Gelman et al.

Bayesian Survival Regression combines parametric or semiparametric survival models — such as Weibull, log-normal, or Cox proportional hazards — with Bayesian inference. Instead of point estimates, it produces full posterior distributions for regression coefficients and the baseline hazard, naturally handling censored observations and incorporating prior knowledge about event times or covariate effects.

Time-to-event (survival) data with censoring

Ibrahim, J. G., Chen, M.-H., & Sinha, D. (2001). Bayesian Survival Analysis. Springer.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

cox-regression survival-regression bayesian-cox-regression bayesian-mixed-effects-model bayesian-generalized-linear-model multilevel-cox-regression


Bayesian Survival Analysis

bayesian sağkalım analizi
bayesian time-to-event analysis
bayesian hazard model

Bayesian survival analysis applies Bayesian inference to time-to-event models — Cox proportional hazards, parametric (Weibull, exponential), and cure models. Formalised comprehensively by Ibrahim, Chen and Sinha (2001), the approach encodes prior knowledge about hazard rates and regression coefficients, then updates it with censored survival data to yield posterior hazard ratios and credible intervals rather than single point estimates.

posterior hazard ratios / credible intervals / survival curves

Ibrahim, J.G., Chen, M.-H. & Sinha, D. (2001). Bayesian Survival Analysis. Springer.

cox-regression kaplan-meier bayesian-regression parametric-survival weibull-regression


Bayesian SVAR model

Bayesian Structural Vector Autoregression Model

Bayesian SVAR
B-SVAR
Bayesian structural VAR
Bayesian identified VAR

Sims & Zha (1998); Uhlig (2005) for sign-restriction identification

The Bayesian Structural Vector Autoregression model combines the structural identification of SVAR with Bayesian prior distributions over parameters. It estimates causal impulse responses between multiple time series while incorporating prior economic knowledge and producing full posterior uncertainty bands rather than point estimates alone.

Structural multivariate time-series model

Sims, C. A., & Zha, T. (1998). Bayesian methods for dynamic multivariate models. International Economic Review, 39(4), 949–968.

Uhlig, H. (2005). What are the effects of monetary policy on output? Results from an agnostic identification procedure. Journal of Monetary Economics, 52(2), 381–419.

structural-var vector-autoregression bayesian-var-model bayesian-vecm vector-error-correction-model bayesian-ardl-bounds-test