Bayesian epigenome-wide association study

Bayesian Epigenome-Wide Association Study

Bayesian epigenetic association analysis

Multiple groups; Bayesian EWAS framework advanced by S. Richardson, P.-C. Tsai, J. T. Bell and colleagues

A Bayesian EWAS is a genome-scale association analysis that links epigenetic marks — most commonly CpG-site DNA methylation — to a phenotype or trait of interest, replacing or supplementing the classical frequentist p-value framework with a Bayesian probabilistic model. It yields posterior probabilities of association and credible intervals for each CpG site, allowing formal incorporation of prior biological knowledge and more principled handling of the multiple-testing burden intrinsic to testing hundreds of thousands of sites simultaneously.

Genome-wide DNA methylation array data (e.g., Illumina 450K or EPIC), phenotype/trait data, covariates

Richardson, S., Tsai, P. C., Bell, J. T., & Timpson, N. J. (2016). Bayesian approaches to studying associations between epigenetic marks and phenotypes. International Journal of Epidemiology, 45(3), 694–705.

Johansson, A., Enroth, S., & Gyllensten, U. (2013). Continuous aging of the human DNA methylome throughout the human lifespan. PLoS ONE, 8(6), e67378.


Bayesian eQTL analysis

Bayesian Expression Quantitative Trait Loci Analysis

Bayesian QTL mapping for gene expression

Matthew Stephens, David J. Balding (Bayesian framework for genetic association); extended by multiple groups for eQTL context

Bayesian eQTL analysis identifies genetic variants (eQTLs) that regulate gene expression by combining genotype and RNA-seq data within a probabilistic framework. Unlike frequentist approaches that rely on p-value thresholds, the Bayesian formulation produces posterior probabilities of association, enabling principled fine-mapping of causal variants and coherent uncertainty quantification across thousands of gene-SNP pairs simultaneously.

Probabilistic genomic association method

Genotype arrays or whole-genome sequencing + RNA-seq gene expression data

Stephens, M., & Balding, D. J. (2009). Bayesian statistical methods for genetic association studies. Nature Reviews Genetics, 10(10), 681–690.

Guan, Y., & Stephens, M. (2011). Bayesian variable selection regression for genome-wide association studies and other large-scale problems. Annals of Applied Statistics, 5(3), 1780–1815.

eqtl-analysis genome-wide-association-study bayesian-gwas rna-seq-differential-expression pathway-enrichment-analysis single-cell-eqtl-analysis


Bayesian Event Study Design

Bayesian Event Study Design for Causal Inference

Bayesian event study
Bayesian abnormal return estimation
Bayesian pre-post event analysis
BES

Developed from classical event study methodology (Fama et al., 1969) with Bayesian extensions proposed through the 1990s–2010s

Bayesian Event Study Design extends the classical event study framework by replacing frequentist significance testing with a full Bayesian inferential framework. It estimates how an event (policy change, announcement, shock) alters an outcome trajectory by learning a prior model from the estimation window and updating it with observed data, yielding posterior distributions over abnormal effects and cumulative causal impacts with full uncertainty quantification.

Time-series or panel data with discrete event indicators; continuous outcomes (e.g., returns, rates)

Sorescu, A., Warren, N. L., & Ertekin, L. (2017). Event study methodology in the marketing literature: An overview. Journal of the Academy of Marketing Science, 45(2), 186-207.

Glassman, M., & McAfee, R. B. (1996). Bayesian estimation of abnormal stock returns. Journal of Business & Economic Statistics, 10(3), 321-332.

event-study-design difference-in-differences bayesian-difference-in-differences panel-event-study interrupted-time-series causal-impact-analysis


Bayesian Event Tree Analysis

Bayesian ETA
B-ETA
Probabilistic Event Tree Analysis
Bayesian Inductive Risk Model

ETA: 1960s–1970s; Bayesian extension: 1990s–2000s

H.E. Watson (Bell Labs, fault tree); ETA formalized via US Nuclear Regulatory Commission; Bayesian extension developed in reliability and risk engineering communities

Bayesian Event Tree Analysis (B-ETA) is a quantitative risk assessment method that extends classical event tree analysis by incorporating Bayesian inference to assign and update branch probabilities. Starting from an initiating event, it maps sequences of successes and failures through safety barriers, using prior distributions and observed evidence to produce posterior outcome probabilities. Widely used in nuclear safety, process industries, and system reliability engineering.

Probabilistic risk and reliability analysis technique

Failure probability data, expert elicitation, historical incident records, prior distributions

Bearfield, G., & Marsh, W. (2005). Generalising event trees using Bayesian networks with a case study of train derailment. In G. Windeknecht et al. (Eds.), Proceedings of the 13th Safety-Critical Systems Symposium. Springer.


Bayesian Ex Post Facto Design

Bayesian causal-comparative design
Bayesian after-the-fact design
Bayesian observational causal design
Bayesian retrospective causal study

1964 (Kerlinger ex post facto); Bayesian integration from 1990s–2000s onward

Frederick N. Kerlinger (ex post facto framework); Bayesian extension draws on Laplace and modern Bayesian statistics

Bayesian ex post facto design investigates possible causal relationships among variables that have already occurred, without researcher manipulation of those variables, and quantifies uncertainty about those relationships using Bayesian statistical inference. The researcher selects groups that differ on an outcome or a presumed cause after the fact, then uses prior knowledge and observed data together — via Bayes' theorem — to estimate credible effect sizes, group differences, or predictors.

Quantitative observational research design with Bayesian inference

Existing records, archival data, survey data, administrative datasets

Kerlinger, F. N. (1973). Foundations of Behavioral Research (2nd ed.). Holt, Rinehart and Winston.

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

ex-post-facto-design causal-comparative-research bayesian-inference observational-study propensity-score-matching retrospective-cohort-study


Bayesian EFA

Bayesian factor analysis
BEFA
Bayesian common factor model
probabilistic factor analysis

2004 (Bayesian formulation); factor analysis roots: 1904

Lopes & West (seminal Bayesian treatment); roots in classical factor analysis (Spearman, 1904)

Bayesian exploratory factor analysis applies a full probabilistic framework to the common factor model. By placing prior distributions over factor loadings and unique variances, it yields posterior distributions rather than point estimates, quantifies uncertainty around every loading, and can treat the number of factors as an unknown to be inferred from data.

Continuous or ordinal item-level responses

Lopes, H. F. & West, M. (2004). Bayesian model assessment in factor analysis. Statistica Sinica, 14(1), 41–67.

Ghosh, J. & Dunson, D. B. (2009). Default prior distributions and efficient posterior computation in Bayesian factor analysis. Journal of Computational and Graphical Statistics, 18(2), 306–320.

exploratory-factor-analysis confirmatory-factor-analysis bayesian-confirmatory-factor-analysis item-response-theory bayesian-item-response-theory mcdonalds-omega


Bayesian Exponential Random Graph Model

Bayesian Exponential Random Graph Model (Bayesian ERGM)

Bayesian ERGM
Bayesian p-star model
Bayesian p* model
BERGM

The Bayesian Exponential Random Graph Model (Bayesian ERGM or BERGM) extends the classical ERGM framework by placing prior distributions over the model parameters and using Markov chain Monte Carlo methods to obtain full posterior distributions. Introduced by Caimo and Friel (2011), it allows researchers to quantify parameter uncertainty and incorporate prior knowledge when modelling the structural features of social and other complex networks.

Binary or valued adjacency matrices (network edge data)

Caimo, A., & Friel, N. (2011). Bayesian inference for exponential random graph models. Social Networks, 33(1), 41–55.

Exponential random graph models. Wikipedia.

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


Bayesian Factor Analysis

Bayesian EFA
Bayesian CFA
Bayesçi Faktör Analizi
probabilistic factor analysis

Lopes & West (2004) for Bayesian model assessment in factor analysis

Bayesian Factor Analysis is a probabilistic latent-variable method that places prior distributions on the factor loading matrix and the residual variances, then infers a full posterior over these parameters from the observed data. Developed prominently in the Bayesian framework by Lopes and West (2004), it extends classical exploratory and confirmatory factor analysis by quantifying uncertainty in every estimated loading rather than reporting single point estimates.

posterior factor loadings / credible intervals / factor scores

Lopes, H. F. & West, M. (2004). Bayesian Model Assessment in Factor Analysis. Statistica Sinica, 14(1), 41–67.

exploratory-factor-analysis cfa bayesian-regression bayesian-network mcmc pca sem


Bayesian failure mode and effects analysis

Bayesian Failure Mode and Effects Analysis

Bayesian FMEA
probabilistic FMEA
B-FMEA
Bayesian risk priority analysis

Extension of classical FMEA (MIL-STD-1629, 1974) with Bayesian inference formalised in reliability literature from the 1990s onward

Bayesian FMEA extends the classical Failure Mode and Effects Analysis framework by replacing fixed point-estimate risk scores with probability distributions, allowing prior engineering knowledge and observed failure data to be formally combined through Bayes' theorem. The result is a probabilistic Risk Priority Number (RPN) that reflects uncertainty in severity, occurrence, and detectability ratings rather than masking it with single consensus values.

Probabilistic reliability and risk analysis

Expert elicitation, historical failure data, prior distributions, observed counts

Bowles, J. B., & Peláez, C. E. (1995). Fuzzy logic prioritization of failures in a system failure mode, effects and criticality analysis. Reliability Engineering and System Safety, 50(2), 203–213.

Liu, H.-C., Liu, L., & Liu, N. (2013). Risk evaluation approaches in failure mode and effects analysis: A literature review. Expert Systems with Applications, 40(2), 828–838.


Bayesian Fault Tree Analysis

BFTA
Bayesian FTA
Bayesian network fault tree
probabilistic fault tree analysis

2001 (BFTA mapping); Bayesian networks: 1988

Andrea Bobbio, Luca Portinale et al. (mapping FTA to Bayesian networks); Judea Pearl (Bayesian networks)

Bayesian Fault Tree Analysis (BFTA) extends classical fault tree analysis by converting the fault tree structure into an equivalent Bayesian network, enabling probabilistic inference in both forward (prediction) and backward (diagnosis) directions. This integration allows analysts to update failure probability estimates with observed evidence, quantify uncertainty explicitly, and identify the most probable root causes of a top-level system failure.

Probabilistic reliability / safety analysis

Component failure probabilities, conditional probability tables, system logic structure

Bobbio, A., Portinale, L., Minichino, M., & Ciancamerla, E. (2001). Improving the analysis of dependable systems by mapping fault trees into Bayesian networks. Reliability Engineering & System Safety, 71(3), 249–260.

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


Bayesian Federated Learning

Bayesian Federated Learning (Probabilistic Federated Model Aggregation)

Bayesian nonparametric federated learning

Yurochkin, M. et al.; McMahan, H. B. et al. (foundational federated learning)

Bayesian Federated Learning combines federated learning — where model training is distributed across multiple clients without sharing raw data — with Bayesian inference, so that each client maintains a posterior distribution over model parameters rather than a single point estimate. This yields principled uncertainty quantification and more robust model aggregation across heterogeneous, privacy-preserving data silos.

Distributed tabular, image, or text data across silos

Yurochkin, M., Agarwal, M., Ghosh, S., Greenewald, K., Hoang, N., & Khazaeni, Y. (2019). Bayesian Nonparametric Federated Learning of Neural Networks. Proceedings of the 36th International Conference on Machine Learning (ICML 2019), PMLR 97, 7101–7110.

Corinzia, L., & Buhmann, J. M. (2019). Variational Federated Multi-Task Learning. arXiv preprint arXiv:1906.06268.

federated-learning bayesian-transfer-learning gaussian-process semi-supervised-federated-learning bayesian-neural-network bayesian-logistic-regression


Bayesian Few-Shot Learning

Bayesian Few-Shot Learning (Meta-Learning with Bayesian Inference)

Bayesian meta-learning
probabilistic few-shot learning
amortized Bayesian few-shot learning
Bayesian FSL

Bayesian few-shot learning combines Bayesian inference with meta-learning to enable a model to generalize from as few as one to five labeled examples per class. By treating task-specific parameters as random variables and learning an informative prior across many training tasks, the method produces calibrated uncertainty estimates alongside predictions — a key advantage over deterministic few-shot learners.

Labeled episodes with very few examples per class

Gordon, J., Bronskill, J., Bauer, M., Nowozin, S. & Turner, R. E. (2019). Meta-Learning Probabilistic Inference for Prediction. International Conference on Learning Representations (ICLR 2019).

Finn, C., Xu, K. & Levine, S. (2018). Probabilistic Model-Agnostic Meta-Learning. Advances in Neural Information Processing Systems (NeurIPS 2018), 31.

few-shot-learning bayesian-transfer-learning bayesian-meta-learning semi-supervised-few-shot-learning gaussian-process transfer-learning


Bayesian Fisher's exact test

Bayesian Fisher's Exact Test for 2x2 Contingency Tables

Bayesian exact test for independence
Bayesian contingency table test
Bayes factor Fisher test
BFexact

1974 (Bayesian form); 1935 (Fisher's exact test)

Gunel & Dickey (Bayesian form); R. A. Fisher (classical exact test)

The Bayesian Fisher's exact test evaluates independence between two categorical variables in a 2x2 table by computing a Bayes factor rather than a p-value. Using conjugate priors on cell probabilities — most commonly the Gunel-Dickey framework — it quantifies how much the observed data favor an association model over an independence model, providing a continuous scale of evidence in both directions.

Bayesian hypothesis test for independence

Categorical (2x2 contingency table counts)

Gunel, E., & Dickey, J. (1974). Bayes factors for independence in contingency tables. Biometrika, 61(3), 545–557.

Jamil, T., Ly, A., Morey, R. D., Love, J., Marsman, M., & Wagenmakers, E.-J. (2017). Default Gunel and Dickey Bayes factors for contingency tables. Behavior Research Methods, 49(2), 638–652.

fishers-exact-test bayesian-chi-square-test chi-square-test bayesian-independent-samples-t-test mann-whitney-u-test bayesian-pearson-correlation


Bayesian Fixed Effects Model

Bayesian within estimator
Bayesian FE model
Bayesian individual fixed effects
Bayesian least squares dummy variable

The Bayesian fixed effects model applies Bayesian inference to the classical within-group panel estimator. Unit-specific intercepts capture time-invariant unobserved heterogeneity, while prior distributions on all parameters allow probability statements about coefficients and full uncertainty quantification via the posterior distribution.

Balanced or unbalanced panel (cross-section × time)

Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95(2), 391–413.

Chib, S. (2008). Panel data modeling and inference: A Bayesian primer. In L. Mátyás & P. Sevestre (Eds.), The Econometrics of Panel Data (3rd ed., pp. 479–515). Springer.

fixed-effects-model bayesian-random-effects-model panel-fixed-effects-model bayesian-panel-data-analysis random-effects-model bayesian-ols


Bayesian Fractional Factorial Design

Bayesian Fractional Factorial Experimental Design

Bayesian FFD
Bayesian screening design
Bayesian factor-screening experiment
BFF design

Bayesian fractional factorial design integrates Bayesian prior information into the selection and analysis of fractional factorial experiments. Rather than running every combination of factor levels, only a carefully chosen subset of runs is executed, with Bayesian inference used to estimate effects and quantify uncertainty — even when the classical aliasing structure leaves effects confounded.

Continuous or categorical factor levels with quantitative response

DuMouchel, W., & Jones, B. (1994). A simple Bayesian modification of D-optimal designs to reduce dependence on an assumed model. Technometrics, 36(1), 37–47.

Meyer, R. D., & Steinberg, D. M. (1996). Follow-up designs to resolve confounding in multifactor experiments. Technometrics, 38(4), 303–313.

fractional-factorial-design bayesian-design-of-experiments full-factorial-design central-composite-design response-surface-methodology taguchi-method


Bayesian Full Factorial Design

Bayesian Full Factorial Design of Experiments

Bayesian FFD
Bayesian complete factorial experiment
Bayesian full factorial experiment
Bayesian all-combinations design

1990s (Bayesian DOE formalized); factorial design roots in 1920s (Fisher)

Kathryn Chaloner & Isabella Verdinelli (Bayesian experimental design framework); building on Fisher's factorial design principles

Bayesian full factorial design combines the complete combinatorial structure of classical full factorial experiments — running every combination of factor levels — with a Bayesian inferential framework that incorporates prior knowledge about factor effects and yields full posterior distributions over main effects, interactions, and model parameters, rather than point estimates and p-values.

Continuous or categorical response measurements across all factor-level combinations

Chaloner, K., & Verdinelli, I. (1995). Bayesian experimental design: A review. Statistical Science, 10(3), 273–304.

Box, G. E. P., Hunter, J. S., & Hunter, W. G. (2005). Statistics for Experimenters: Design, Innovation, and Discovery (2nd ed.). Wiley-Interscience.

full-factorial-design bayesian-design-of-experiments bayesian-response-surface-methodology fractional-factorial-design bayesian-fractional-factorial-design central-composite-design


Bayesian Fuzzy Regression Discontinuity

Bayesian Fuzzy Regression Discontinuity Design

Bayesian Fuzzy RD
Bayesian Fuzzy RDD
Fuzzy RD with Bayesian Inference

2001 (fuzzy RD identification); 2016 (Bayesian formulation by Chib & Jacobi)

Chib & Jacobi (Bayesian formulation); Hahn, Todd & Van der Klaauw (fuzzy RD identification)

Bayesian Fuzzy Regression Discontinuity (Bayesian Fuzzy RD) combines the quasi-experimental logic of fuzzy regression discontinuity design with full Bayesian inference. It estimates a local average treatment effect at a policy threshold where treatment assignment is probabilistic rather than deterministic, placing prior distributions over all unknowns and recovering a complete posterior distribution of the causal effect rather than a single point estimate.

Bayesian causal inference / quasi-experimental design

Cross-sectional or panel data with a continuous running variable and imperfect threshold compliance

Hahn, J., Todd, P., & Van der Klaauw, W. (2001). Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design. Review of Economic Studies, 68(1), 201-209.

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 instrumental-variables bayesian-instrumental-variables local-average-treatment-effect difference-in-differences


Bayesian GARCH model

Bayesian Generalized Autoregressive Conditional Heteroskedasticity Model

Bayesian GARCH
BGARCH
GARCH with Bayesian inference
Bayesian volatility model

Geweke (1989); further developed by Nakatsuma (2000) and Bauwens & Lubrano (1998)

The Bayesian GARCH model combines the GARCH framework for time-varying volatility with Bayesian posterior inference. Instead of maximising a likelihood, it specifies prior distributions for the GARCH parameters and draws from the resulting posterior — typically via Markov chain Monte Carlo (MCMC) — to quantify both point estimates and full uncertainty about volatility dynamics.

Time-series financial or macroeconomic data with time-varying variance

Geweke, J. (1989). Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics, 40(1), 63–86.

Nakatsuma, T. (2000). Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach. Journal of Econometrics, 95(1), 57–69.

garch-model egarch-model stochastic-volatility-model bayesian-var arch-model markov-switching-model


Bayesian Gaussian Mixture Model

Bayesian Gaussian Mixture Model (Variational Bayes / MCMC Inference)

Bayesian GMM
Variational Gaussian Mixture
VBGMM
Dirichlet Process Gaussian Mixture

The Bayesian Gaussian Mixture Model places prior distributions over all mixture parameters and infers their posteriors — typically via Variational Bayes or MCMC — rather than fitting fixed point estimates. This yields principled uncertainty quantification, automatic selection of the effective number of components, and resistance to overfitting small datasets.

Probabilistic clustering / density estimation

Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Ch. 10). Springer.

Attias, H. (1999). Inferring parameters and structure of latent variable models by variational Bayes. Proceedings of the 15th Conference on Uncertainty in Artificial Intelligence (UAI), 21–30.

gaussian-mixture-model k-means bayesian-k-means variational-autoencoder gaussian-process semi-supervised-gaussian-mixture-model


Bayesian Gaussian Process

Bayesian Gaussian Process Regression and Classification

GP regression
GPR
Gaussian process model
GP classifier

O'Hagan, A.; Neal, R. M.; Rasmussen, C. E. & Williams, C. K. I.

A Bayesian Gaussian Process (GP) places a probability distribution directly over functions, using a kernel to encode similarity between inputs. After observing data, Bayes' rule converts this prior into a posterior that yields not just point predictions but calibrated uncertainty estimates at every new input — making it one of the most principled probabilistic models in machine learning.

Continuous, mixed; tabular or structured

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

Bishop, C. M. (2006). Pattern Recognition and Machine Learning (Ch. 6). Springer.

gaussian-process bayesian-linear-regression support-vector-machine kernel-ridge-regression neural-network bayesian-optimization


Bayesian Geary's C

Bayesian Geary C
Bayesian spatial contiguity statistic
Geary's C (Bayesian)
Bayesian contiguity ratio

Geary (1954); Bayesian extension via hierarchical spatial modeling literature

Bayesian Geary's C embeds the classical Geary contiguity ratio within a Bayesian hierarchical framework. Instead of a single point estimate and asymptotic p-value, it produces a posterior distribution over the statistic (or over spatially structured random effects), quantifying uncertainty about spatial autocorrelation while formally incorporating prior knowledge about the spatial process.

Bayesian spatial autocorrelation statistic

Areal (lattice) spatial data with continuous or count attributes

Geary, R. C. (1954). The contiguity ratio and statistical mapping. The Incorporated Statistician, 5(3), 115–145.

Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC.


Bayesian Gene Set Enrichment Analysis

Bayesian GSEA
BGSEA
Bayesian pathway scoring
probabilistic gene set testing

Michael A. Newton, Frank A. Quintana and colleagues; building on Subramanian et al. GSEA framework

Bayesian gene set enrichment analysis (Bayesian GSEA) applies a probabilistic framework to determine whether predefined sets of genes — representing biological pathways, cellular processes, or functional categories — are collectively more differentially expressed than expected by chance. Unlike classical frequentist GSEA, the Bayesian approach models uncertainty in expression estimates explicitly, incorporates prior biological knowledge, and produces posterior probabilities of enrichment rather than raw p-values, enabling more principled inference especially in small-sample settings.

Probabilistic gene set enrichment method

Gene expression matrices (RNA-seq counts or microarray intensities) with curated gene set annotations (e.g., MSigDB, GO, KEGG)

Subramanian, A., Tamayo, P., Mootha, V. K., Mukherjee, S., Ebert, B. L., Gillette, M. A., ... & Mesirov, J. P. (2005). Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proceedings of the National Academy of Sciences, 102(43), 15545-15550.

Newton, M. A., Quintana, F. A., Den Boon, J. A., Bhattacharya, S., & Ahlquist, P. (2007). Random-set methods identify distinct aspects of the enrichment signal in gene-set analysis. The Annals of Applied Statistics, 1(1), 85-106.


Bayesian Generalized additive model

Bayesian GAM
BGAM
Bayesian semiparametric regression
Bayesian smooth regression

Hastie & Tibshirani (GAM framework, 1990); Bayesian formulation developed through work by Wood, Fahrmeir, Lang, and others

Bayesian Generalized Additive Models extend the frequentist GAM framework by placing prior distributions over the smooth functions and any additional model parameters. This yields full posterior distributions over each smooth effect, enabling principled uncertainty quantification, automatic smoothness selection via hyperpriors, and seamless integration with hierarchical or mixed-effects structures.

Continuous, binary, count, or other exponential-family outcomes with potentially nonlinear predictors

Wood, S. N. (2017). Generalized Additive Models: An Introduction with R (2nd ed.). CRC Press.

Bürkner, P.-C. (2017). brms: An R Package for Bayesian Multilevel Models Using Stan. Journal of Statistical Software, 80(1), 1–28.

generalized-additive-model bayesian-generalized-linear-model bayesian-mixed-effects-model bayesian-multiple-linear-regression gaussian-process-regression spline-based-generalized-additive-model


Bayesian Generalized Linear Model

Bayesian GLM
Bayesian GLIM
Bayesian generalized linear regression
Bayes GLM

McCullagh & Nelder (GLM framework); Bayesian treatment formalized by Gelman et al.

A Bayesian Generalized Linear Model (Bayesian GLM) extends the classical GLM framework by placing prior distributions on the regression coefficients and updating them with data via Bayes' theorem. This yields a full posterior distribution over parameters rather than single point estimates, enabling richer uncertainty quantification and principled incorporation of prior knowledge for any exponential-family outcome.

Continuous, binary, count, or other exponential-family outcomes

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 & Hall.

generalized-linear-model bayesian-logistic-regression bayesian-poisson-regression bayesian-multiple-linear-regression bayesian-negative-binomial-regression bayesian-probit-model


Bayesian Genetic Algorithm

Bayesian Genetic Algorithm — Probabilistic model-guided evolutionary optimization

BGA
Bayesian-guided GA
Probabilistic GA
EDA-GA

A Bayesian Genetic Algorithm (BGA) replaces traditional crossover and mutation operators with a probabilistic Bayesian network learned from selected high-fitness individuals. At each generation the algorithm builds a graphical model of promising solution structure, then samples new offspring from that model, enabling the search to capture and exploit variable dependencies that standard GAs miss.

Evolutionary metaheuristic with Bayesian probabilistic model

Continuous, discrete, or mixed decision variables; fitness function evaluations

Pelikan, M., Goldberg, D. E., & Cantu-Paz, E. (1999). BOA: The Bayesian optimization algorithm. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-1999), pp. 525–532. Morgan Kaufmann.

Larranaga, P., & Lozano, J. A. (Eds.) (2002). Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Boston.

genetic-algorithm bayesian-optimization estimation-of-distribution-algorithm particle-swarm-optimization bayesian-multi-objective-optimization stochastic-genetic-algorithm


Bayesian spatially varying coefficient model

Bayesian Geographically Weighted Regression combines the spatially varying coefficient framework of GWR with Bayesian inference, placing Gaussian process priors on the locally varying regression coefficients. This yields full posterior distributions over each coefficient at every location, providing principled uncertainty quantification rather than only point estimates.

Bayesian spatially varying coefficient regression

Georeferenced / point-referenced or areal cross-sectional data

Finley, A. O. (2011). Comparing spatially-varying coefficients models for analysis of ecological data with non-stationary and anisotropic residual dependence. Methods in Ecology and Evolution, 2(2), 143-154.

Wheeler, D., & Calder, C. (2007). An assessment of coefficient accuracy in linear regression models with spatially varying coefficients. Journal of Geographical Systems, 9(2), 145-166.


Bayesian Goal Programming

BGP
Bayesian GP
Probabilistic Goal Programming
Bayesian Multi-Goal Optimization

Bayesian Goal Programming (BGP) integrates Bayesian statistical inference with classic goal programming to handle uncertainty in targets and parameters. Instead of treating goal thresholds as fixed constants, BGP encodes them as probability distributions, updates beliefs using observed data, and then solves the resulting probabilistic optimization problem to find solutions that satisfy multiple aspirational goals under uncertainty.

Multi-objective optimization under uncertainty

Continuous decision variables, probabilistic goal targets

Rios Insua, D. (1990). Sensitivity Analysis in Multi-objective Decision Making. Springer-Verlag, Berlin.

Charnes, A., Cooper, W. W., & Ferguson, R. O. (1955). Optimal estimation of executive compensation by linear programming. Management Science, 1(2), 138-151.

goal-programming bayesian-multi-objective-optimization stochastic-goal-programming multi-objective-optimization bayesian-dynamic-programming robust-goal-programming


Bayesian Granger Causality

Bayesian Granger test
Bayesian predictive causality
BGC
Bayesian causality in mean

1969 (frequentist); 1984 (Bayesian treatment)

Clive W. J. Granger (frequentist basis, 1969); Bayesian extension by Geweke (1984) and subsequent literature

Bayesian Granger causality tests whether past values of one time series carry predictive information about another, framing the hypothesis through Bayesian inference rather than frequentist p-values. It combines a vector autoregressive (VAR) structure with prior distributions over coefficients and evaluates causal claims via posterior probabilities or Bayes factors, providing a probabilistic and nuanced alternative to the classical Granger test.

Multivariate time series (stationary or cointegrated)

Geweke, J. (1984). Inference and causality in economic time series models. Handbook of Econometrics, 2, 1101-1144. Elsevier.

granger-causality-test bayesian-var-model toda-yamamoto-causality-test vector-autoregression bayesian-vecm panel-granger-causality


Bayesian genome-wide association study in educational research

Bayesian Genome-Wide Association Study Applied to Educational Outcomes

Bayesian GWAS for educational attainment

2013–2018 (educational attainment GWAS); Bayesian GWAS framework ~2001–2010

Social Science Genetic Association Consortium (SSGAC); Bayesian GWAS methods developed by Ter Braak, Meuwissen, and others

Bayesian genome-wide association study (Bayesian GWAS) applies Bayesian statistical models to millions of single-nucleotide polymorphisms (SNPs) to identify genetic variants associated with educational outcomes such as years of schooling or cognitive test scores. Unlike classical frequentist GWAS, Bayesian approaches assign prior distributions over effect sizes, enabling more principled handling of the polygenic architecture typical of educational traits, shrinkage of small effects, and direct posterior probability estimates for variant inclusion.

Genome-wide SNP array data (millions of variants) with continuous or binary educational phenotype

Lee, J. J., Wedow, R., Okbay, A., Kong, E., Maghzian, O., Zacher, M., ... & Cesarini, D. (2018). Gene discovery and polygenic prediction from a genome-wide association study of educational attainment in 1.1 million individuals. Nature Genetics, 50(8), 1112–1121.

Rietveld, C. A., Medland, S. E., Derringer, J., Yang, J., Esko, T., Martin, N. W., ... & Koellinger, P. D. (2013). GWAS of 126,559 individuals identifies genetic variants associated with educational attainment. Science, 340(6139), 1467–1471.

polygenic-score-analysis linear-mixed-model-gwas mendelian-randomization heritability-estimation genome-wide-complex-trait-analysis bayesian-variable-selection


Bayesian GWAS

Bayesian genome-wide association analysis

2007–2009 (formal statistical framework)

Matthew Stephens, David J. Balding, Jon Wakefield (key formalizers ca. 2007–2009)

Bayesian GWAS applies Bayesian statistical inference to genome-wide association studies, replacing classical p-value thresholds with Bayes factors and posterior probabilities. This framework naturally incorporates prior knowledge about effect sizes and variant frequencies, quantifies evidence for association on a continuous scale, and supports principled fine-mapping of causal variants within associated loci. It is widely used in complex trait genetics, population genomics, and translational research where uncertainty quantification and multi-variant modeling matter.

Statistical genetic association analysis

Genome-wide SNP genotype data, phenotype measurements (quantitative or binary), population reference panels

Stephens, M., & Balding, D. J. (2009). Bayesian statistical methods for genetic association studies. Nature Reviews Genetics, 10(10), 681–690.

Wakefield, J. (2009). Bayes factors for genome-wide association studies: comparison with P-values. Genetic Epidemiology, 33(1), 79–86.

genome-wide-association-study bayesian-eqtl-analysis fine-mapping polygenic-risk-score bayesian-single-cell-rna-seq-analysis pathway-enrichment-analysis


Bayesian Hausman Test

Bayesian specification test
Bayesian endogeneity test
Bayesian FE vs RE test
Bayesian Durbin-Wu-Hausman

1978 (classical); Bayesian adaptations 1990s–2000s

Bayesian reformulation of Hausman (1978); developed across Bayesian econometrics literature

The Bayesian Hausman test is a Bayesian reformulation of Hausman's (1978) classical specification test, used to assess endogeneity or to choose between fixed effects and random effects panel models. Instead of a chi-squared test statistic, it uses posterior model probabilities or Bayes factors to compare competing specifications, fully incorporating prior uncertainty about model parameters.

Panel or cross-sectional data with potential endogeneity

Hausman, J. A. (1978). Specification tests in econometrics. Econometrica, 46(6), 1251–1271.

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

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


Bayesian Hierarchical Clustering

BHC
probabilistic hierarchical clustering
Bayesian agglomerative clustering

Bayesian hierarchical clustering is a probabilistic agglomerative algorithm that builds a tree of nested cluster merges using Bayesian model comparison at each step. Rather than minimising a geometric linkage criterion, it evaluates at every candidate merge whether the data from two clusters are better explained by a single combined model or by two separate models, yielding a statistically principled dendrogram.

Probabilistic clustering / model-based hierarchical agglomeration

Continuous or mixed multivariate observations

Heller, K. A. & Ghahramani, Z. (2005). Bayesian hierarchical clustering. In Proceedings of the 22nd International Conference on Machine Learning (ICML 2005), pp. 297–304. ACM.

Murtagh, F. & Legendre, P. (2014). Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion? Journal of Classification, 31(3), 274–295.

hierarchical-clustering bayesian-cluster-analysis bayesian-latent-class-analysis cluster-analysis mixture-modeling bayesian-mixture-modeling


Bayesian Hierarchical Linear Model

Bayesian HLM
Bayesian multilevel linear model
Bayesian random-effects linear model
Bayes hierarchical regression

Gelman & Hill (2006); Raudenbush & Bryk (2002) for frequentist HLM; Bayesian treatment consolidated by Gelman et al.

The Bayesian Hierarchical Linear Model (Bayesian HLM) estimates linear relationships in nested or clustered data by placing prior distributions on all model parameters and updating them with observed data. It simultaneously models variation within groups and between groups, propagating uncertainty fully through posterior distributions rather than relying on asymptotic approximations.

Continuous outcome, grouped / clustered observations

Gelman, A., & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

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

hierarchical-linear-model multilevel-modeling bayesian-mixed-effects-model bayesian-multilevel-modeling mixed-effects-model bayesian-multiple-linear-regression


Bayesian Hierarchical Model with Missing Data Imputation

Bayesian multilevel model with incomplete data

Gelman, Rubin, Little (and collaborators)

A Bayesian hierarchical model with missing data treats unobserved values as additional unknowns and samples them jointly with all model parameters from the posterior. The nested structure of the hierarchy borrows strength across groups, while the Bayesian framework naturally propagates uncertainty from missingness through every estimate and prediction.

Bayesian hierarchical model with missing-data integration

partially observed continuous, binary, or count data nested within groups

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

Little, R. J. A. & Rubin, D. B. (2002). Statistical Analysis with Missing Data (2nd ed.). Wiley.


Bayesian Hierarchical Model

Bayesian Hierarchical (Multilevel) Model

Bayesian Hiyerarşik Model (Çok Düzeyli Bayes)

Gelman & Hill (2006); Bayesian multilevel tradition

Bayesian hierarchical modelling, popularised by Gelman and Hill (2006), is a Bayesian approach to nested data structures — such as students within schools within districts — that estimates separate parameters at each level while allowing those levels to share statistical strength through a mechanism called partial pooling. Where a classical hierarchical linear model treats group means as fixed unknown quantities, the Bayesian version places hyperprior distributions on those group means so that information flows freely across levels, producing more reliable group-level estimates whenever any individual group has few observations.

posterior distributions / credible intervals / group-level estimates

Gelman, A. & Hill, J. (2006). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

bayesian-regression hierarchical-linear-model mcmc mixed-effects-model partial-pooling


Bayesian Hot Spot Analysis

Bayesian spatial cluster detection
Bayesian disease mapping hot spots
empirical Bayesian hot spot analysis
Bayesian spatial smoothing hot spots

Clayton & Kaldor (1987); Lawson (2001 onward)

Bayesian Hot Spot Analysis identifies spatial clusters of elevated risk or intensity by combining observed data with prior beliefs about spatial structure. It uses Bayesian smoothing — pooling information across neighboring areas — to stabilize estimates in small areas and then flags locations where the posterior probability of exceeding a risk threshold is high.

Areal or point-referenced count/rate data

Lawson, A. B. (2018). Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology (3rd ed.). CRC Press.

Clayton, D., & Kaldor, J. (1987). Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics, 43(3), 671-681.


Bayesian Hypothesis Testing Research

Bayesian Hypothesis Testing Research Design

Bayesian significance testing
Bayes factor hypothesis testing
BHT research
Bayesian inference testing

1935–1961 (Jeffreys); extended by Kass & Raftery 1995, Wagenmakers 2007–2010

Harold Jeffreys (formal Bayes factor framework)

Bayesian hypothesis testing research is a quantitative design in which competing hypotheses are evaluated by updating prior beliefs with observed data to produce posterior probabilities and Bayes factors. Unlike frequentist null-hypothesis significance testing, it quantifies the relative evidence for each hypothesis, supports optional stopping, and allows accumulation of evidence across studies without inflating Type I error rates.

Numeric / continuous / categorical measured variables

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

Wagenmakers, E.-J. (2010). A practical solution to the pervasive problems of p values. Psychonomic Bulletin and Review, 14(5), 779–804.

bayesian-confirmatory-research hypothesis-testing-research bayesian-correlational-research confirmatory-research bayesian-model-testing-research bayesian-survey-research


Bayesian Independent Samples t-test

Bayesian two-sample t-test
Bayes factor t-test
JZS t-test
Bayesian unpaired t-test

2009 (modern form); 1961 (Jeffreys prior framework)

Harold Jeffreys (foundational); operationalized by Rouder et al.

The Bayesian independent samples t-test quantifies evidence for or against a mean difference between two independent groups using a Bayes factor rather than a p-value. Rooted in Jeffreys's probability framework and popularized by Rouder et al. (2009), it places a Cauchy prior on the standardized effect size and returns continuous evidence for both the null and alternative hypotheses.

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.


Bayesian Inference with Measurement Error (Errors-in-Variables)

Bayesian errors-in-variables model
Bayesian EIV model
Bayesian measurement error model
Bayesian misclassification model

Richardson & Gilks (Bayesian formulation); Carroll et al. (comprehensive framework)

Bayesian inference with measurement error extends the standard Bayesian framework to situations where one or more covariates or outcomes are observed with noise or misclassification. By treating the true unobserved values as latent variables and assigning them priors, the model jointly estimates the true exposure distribution and the structural parameters of interest, propagating all uncertainty through the posterior.

Continuous or categorical covariates observed with error; replicate or validation data

Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective (2nd ed.). Chapman & Hall/CRC.

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 Inference with Missing Data

Bayesian missing data analysis
Bayesian data augmentation
Bayesian imputation
missing data Bayesian model

Rubin, D. B. (missing-data mechanisms); Tanner & Wong (data augmentation)

Bayesian inference with missing data treats unobserved values as unknown parameters and integrates them out of the posterior distribution. Rather than deleting or ad hoc imputing incomplete records, the method jointly models observed and missing data under an explicit missing-data mechanism, producing fully calibrated posterior uncertainty that honestly reflects what the data cannot tell us.

any data type with partially observed variables

Little, R. J. A. & Rubin, D. B. (2002). Statistical Analysis with Missing Data (2nd ed.). Wiley-Interscience.


Bayesian Inference

Bayes inference
Bayesian statistics
Bayesian updating
posterior inference
Bayesian data analysis

Bayesian inference is a statistical paradigm in which probability represents degrees of belief rather than long-run frequencies. It encodes prior knowledge about parameters in a prior distribution, combines that prior with the likelihood of observed data via Bayes' theorem, and produces a posterior distribution that quantifies updated uncertainty. The foundational theorem was published posthumously by Thomas Bayes in 1763 and subsequently systematized by Pierre-Simon Laplace in his 1812 Théorie analytique des probabilités.

Posterior (conjugate, MCMC, or variational)

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

Laplace, P.-S. (1812). Théorie analytique des probabilités. Courcier, Paris.

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

mcmc-sampling hierarchical-model bayesian-linear-regression prior-predictive-check hypothesis-test-bayesian-factor independent-t-test maximum-likelihood-estimation


Bayesian Information Criterion

BIC
Schwarz criterion
Schwarz information criterion

The Bayesian Information Criterion is an information-theoretic model selection criterion that approximates Bayesian model comparison. Introduced by Gideon Schwarz in 1978, BIC penalizes model complexity more heavily than AIC by using a sample-size-dependent penalty, making it particularly suitable for identifying the true underlying model structure.

Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461-464.

Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.). New York: Springer.

Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773-795.

akaike-information-criterion r-squared adjusted-r-squared mean-squared-error


Bayesian Instrumental Variables

Bayesian Instrumental Variables Estimation

Bayesian IV
Bayesian 2SLS
Bayesian LIML
BayesIV

Kleibergen & Zivot (2003); Lancaster (2004)

Bayesian Instrumental Variables combines the instrumental variable strategy for addressing endogeneity with Bayesian posterior inference. Instead of relying on asymptotic sampling distributions, it places prior distributions over all structural parameters and recovers a full posterior distribution for the causal effect, providing probability statements about the parameter rather than p-values — especially valuable when instruments are weak or the sample is small.

Cross-sectional or panel; continuous or binary endogenous regressor

Kleibergen, F., & Zivot, E. (2003). Bayesian and classical approaches to instrumental variable regression. Journal of Econometrics, 114(1), 29-72.

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

instrumental-variables bayesian-regression two-stage-least-squares difference-in-differences propensity-score-matching bayesian-difference-in-differences


Bayesian Integer Programming

Bayesian Integer Programming — Probabilistic Prior-Guided Combinatorial Optimization

BIP
Bayesian combinatorial optimization
Bayesian discrete optimization
probabilistic integer programming

Baptiste, Lassagne, Nuijten and others in Bayesian optimization community

Bayesian Integer Programming (BIP) integrates Bayesian probabilistic reasoning with integer programming to solve combinatorial optimization problems under uncertainty. Instead of treating parameters as fixed, it encodes prior beliefs about uncertain coefficients and updates them with observed data, producing a posterior-guided search over integer-feasible solutions. The approach is widely used in scheduling, resource allocation, and supply-chain planning where data are incomplete or noisy.

Discrete/integer decision variables with uncertain parameters

Baptiste, P., Lassagne, I., & Nuijten, W. (2001). Bayesian reasoning in mixed integer programming. European Journal of Operational Research, 130(2), 293–313.

bayesian-mixed-integer-programming stochastic-integer-programming bayesian-linear-programming mixed-integer-programming bayesian-multi-objective-optimization robust-integer-programming


Bayesian Inverse Probability Weighting

Bayesian Inverse Probability Weighting Estimator

Bayesian IPW
BIPW
Bayesian propensity-weighted estimation
Bayesian marginal structural weighting

Saarela, Stephens, Moodie & Klein (2015); Liao & Zigler (2020)

Bayesian Inverse Probability Weighting (Bayesian IPW) extends the classical IPW estimator by placing prior distributions over the propensity-score model parameters and propagating that uncertainty into the causal-effect estimate. The result is a posterior distribution for the average treatment effect that fully accounts for both propensity-score estimation uncertainty and outcome-model uncertainty, enabling credible-interval inference rather than relying on asymptotic approximations.

Cross-sectional or panel observational data with binary or multi-valued treatment

Saarela, O., Stephens, D. A., Moodie, E. E. M., & Klein, M. B. (2015). On risk prediction and characterisation of treatment effects in a Bayesian framework using the propensity score. Statistics in Medicine, 34(14), 2170-2185.

Liao, S. X., & Zigler, C. M. (2020). Uncertainty in the design stage of two-stage Bayesian propensity score analysis. Statistics in Medicine, 39(17), 2265-2290.

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


Bayesian Item Analysis

BIA
Bayesian classical item analysis
Bayesian item statistics
Bayesian item-level diagnostics

Originated in Bayesian psychometrics literature, developed extensively by Jean-Paul Fox and colleagues

Bayesian item analysis applies Bayesian inference to estimate item-level statistics — difficulty, discrimination, and distractor effectiveness — by combining observed response data with prior knowledge. It produces full posterior distributions over item parameters rather than single point estimates, providing richer uncertainty information especially with small samples.

Bayesian inference / item-level diagnostics

Dichotomous or polytomous item responses

Fox, J.-P. (2010). Bayesian Item Response Modeling: Theory and Applications. 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.

item-response-theory bayesian-reliability-analysis bayesian-confirmatory-factor-analysis differential-item-functioning cronbachs-alpha scale-development


Bayesian K-means clustering

Bayesian K-means
probabilistic K-means
Dirichlet K-means
BKM

Kulis & Jordan (ICML 2012) formalized the Bayesian nonparametric derivation; Bishop (2006) established the variational Bayesian EM framework for Gaussian mixture models as a probabilistic foundation

Bayesian K-means clustering extends the classical K-means algorithm by placing prior distributions over cluster centroids and mixing proportions. This probabilistic framework provides uncertainty estimates for cluster assignments, allows principled model selection for the number of clusters, and regularises centroid estimation — especially valuable when data are scarce or high-dimensional.

Probabilistic clustering / Bayesian nonparametric

Kulis, B. & Jordan, M. I. (2012). Revisiting k-means: New algorithms via Bayesian nonparametrics. In Proceedings of the 29th International Conference on Machine Learning (ICML), Edinburgh, Scotland, pp. 513–520.

Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer. Chapter 9 (Mixture models and EM) and Chapter 10 (Approximate Inference).

cluster-analysis bayesian-cluster-analysis mixture-modeling bayesian-mixture-modeling bayesian-hierarchical-clustering latent-class-analysis


Bayesian k-nearest neighbors

Bayesian KNN
BKNN
probabilistic k-nearest neighbors
Bayesian nearest-neighbor classifier

Bayesian k-Nearest Neighbors (Bayesian KNN) extends the classical KNN algorithm by placing a prior distribution over the neighborhood size k and combining likelihood evidence from neighbors with that prior to produce calibrated posterior class probabilities. It retains KNN's intuitive instance-based logic while adding principled uncertainty quantification over predictions.

Holmes, C. C., & Adams, N. M. (2002). A probabilistic nearest neighbour method for statistical pattern recognition. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2), 295–306.

K-nearest neighbors algorithm. Wikipedia.

k-nearest-neighbors naive-bayes gaussian-process-classification logistic-regression random-forest support-vector-machine


Bayesian Kaplan-Meier analysis

Bayesian Nonparametric Kaplan-Meier Survival Analysis

Bayesian nonparametric survival analysis

Susarla & Van Ryzin (Bayesian nonparametric survival estimation)

Bayesian Kaplan-Meier analysis extends the classical Kaplan-Meier estimator by placing a prior distribution over the survival function and updating it with observed time-to-event data to obtain a full posterior distribution for the survival curve. This approach, rooted in Susarla and Van Ryzin's 1976 Dirichlet-process framework, yields credible intervals rather than confidence intervals and enables coherent incorporation of prior clinical knowledge, making it particularly valuable in small-sample or early-phase clinical settings.

Time-to-event data with right-censored observations

Susarla, V., & Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations. Journal of the American Statistical Association, 71(356), 897–902.

Diaconis, P., & Freedman, D. (1986). On the consistency of Bayes estimates. The Annals of Statistics, 14(1), 1–26.

kaplan-meier-analysis bayesian-survival-analysis cox-proportional-hazards competing-risks-analysis survival-analysis bayesian-cox-proportional-hazards


Bayesian Kernel Density Estimation

Bayesian nonparametric density estimation

Hjort & Glad (1995); extended by various authors in Bayesian nonparametrics

Bayesian Kernel Density Estimation (BKDE) is a nonparametric method for estimating the probability density function of a spatial or attribute variable by combining a kernel smoother with a Bayesian prior over the bandwidth parameter. The posterior distribution of the bandwidth propagates uncertainty into the final density estimate rather than treating the bandwidth as a fixed tuning constant.

Continuous spatial or attribute data (point patterns, areal counts)

Hjort, N. L., & Glad, I. K. (1995). Nonparametric density estimation with a parametric start. The Annals of Statistics, 23(3), 882–904.

kernel-density-estimation bayesian-spatial-regression hot-spot-analysis spatial-autocorrelation local-kriging bayesian-kriging


Bayesian Knowledge Graph Analysis

Bayesian Knowledge Graph Analysis (Probabilistic Inference over Knowledge Graphs)

Bayesian KG analysis
probabilistic knowledge graph reasoning
Bayesian knowledge base completion
BKGA

Nickel, M.; Murphy, K.; Tresp, V.; Gabrilovich, E. (and related Bayesian KG literature, 2010s)

Bayesian knowledge graph analysis applies probabilistic Bayesian inference to knowledge graphs — structured representations of entities and their relations — to reason under uncertainty, complete missing links, and quantify confidence in inferred facts. It treats unknown graph edges as random variables and updates beliefs about them given observed relational evidence, making it especially suited to incomplete or noisy knowledge bases.

Relational triples (subject, predicate, object); entity-relation graphs

Chen, M., Zhang, W., Zhang, W., Chen, Q., & Chen, H. (2020). Meta Relational Learning for Few-Shot Link Prediction in Knowledge Graphs. Proceedings of EMNLP 2020.

knowledge-graph-analysis bayesian-network exponential-random-graph-model bayesian-exponential-random-graph-model bayesian-stochastic-block-model multilayer-knowledge-graph-analysis


Bayesian Kriging

Bayesian Kriging (Model-Based Geostatistics)

Bayesian geostatistics
model-based geostatistics
Bayesian spatial interpolation
stochastic kriging

Bayesian Kriging embeds classical geostatistical interpolation inside a full probabilistic framework. Instead of treating variogram parameters as fixed point estimates, it places prior distributions on them and updates these priors with observed spatial data to obtain a posterior distribution. Predictions at unsampled locations are then marginalised over this uncertainty, yielding honest predictive intervals that account for both spatial dependence and parameter uncertainty.

Georeferenced continuous observations (point-referenced spatial data)

Diggle, P. J., Tawn, J. A., & Moyeed, R. A. (1998). Model-based geostatistics. Journal of the Royal Statistical Society: Series C (Applied Statistics), 47(3), 299–350.

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

ordinary-kriging universal-kriging co-kriging spatial-autocorrelation gaussian-process-regression bayesian-spatial-regression


Bayesian LASSO Regression

Bayesian Least Absolute Shrinkage and Selection Operator Regression

Bayesian LASSO
Bayesian L1 regression
double-exponential prior regression
Laplace prior regression

Bayesian LASSO regression places double-exponential (Laplace) priors on regression coefficients, which is the Bayesian analogue of the classical LASSO penalty. It simultaneously shrinks small coefficients toward zero and performs soft variable selection, all within a coherent posterior inference framework that naturally quantifies parameter uncertainty through credible intervals.

Continuous outcome, continuous and/or categorical predictors

Park, T., & Casella, G. (2008). The Bayesian Lasso. Journal of the American Statistical Association, 103(482), 681–686.

Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society: Series B, 58(1), 267–288.

lasso-regression bayesian-ridge-regression bayesian-elastic-net-regression bayesian-multiple-linear-regression elastic-net-regression ridge-regression


Bayesian Latent Class Analysis

Bayesian LCA
BLCA
Bayesian mixture of multinomials
Bayesian finite mixture model

Lazarsfeld (classical LCA); Bayesian formulation developed through Cheeseman & Stutz (1996) and Dunson & Xing (2009)

Bayesian latent class analysis extends classical LCA by placing prior distributions on all model parameters and using posterior inference — typically via MCMC — to classify individuals into unobserved categorical groups, quantify uncertainty around class membership, and select the number of classes in a principled, probabilistic way.

Bayesian latent variable / finite mixture model

Multivariate categorical (binary or polytomous) indicators

Dunson, D. B. & Xing, C. (2009). Nonparametric Bayes modeling of multivariate categorical data. Journal of the American Statistical Association, 104(487), 1042–1051.

White, A. & Murphy, T. B. (2016). BayesLCA: An R package for Bayesian latent class analysis. Journal of Statistical Software, 61(13), 1–28.

latent-class-analysis bayesian-mixture-modeling bayesian-confirmatory-factor-analysis mixture-modeling bayesian-cluster-analysis latent-profile-analysis


Bayesian LightGBM

LightGBM with Bayesian Hyperparameter Optimization

Bayesian-tuned LightGBM
LightGBM + Bayesian optimization
BayesOpt LightGBM
LightGBM with BayesOpt

2017 (LightGBM); 2012 (Bayesian optimization)

Ke et al. (LightGBM); Snoek et al. (Bayesian optimization)

Bayesian LightGBM combines LightGBM — a highly efficient histogram-based gradient boosting framework — with Bayesian hyperparameter optimization. Instead of exhaustive grid search or random search, a probabilistic surrogate model guides the search for optimal hyperparameters, dramatically reducing the number of costly model evaluations needed to reach strong predictive performance.

Gradient boosting with Bayesian hyperparameter search

Tabular (continuous, categorical, binary, ordinal)

Ke, G., Meng, Q., Finley, T., Wang, T., Chen, W., Ma, W., Ye, Q., & Liu, T.-Y. (2017). LightGBM: A highly efficient gradient boosting decision tree. In Advances in Neural Information Processing Systems, 30, 3146–3154.

Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian optimization of machine learning algorithms. In Advances in Neural Information Processing Systems, 25, 2951–2959.

lightgbm xgboost bayesian-xgboost gradient-boosting bayesian-gradient-boosting random-forest


Bayesian Linear Programming

Bayesian Linear Programming — Bayesian inference integrated with linear programming under parameter uncertainty

BLP
Bayesian LP
Bayesian stochastic linear programming
prior-posterior LP

Integrated from Dantzig (LP) and Zellner/Bayesian econometrics traditions

Bayesian Linear Programming (BLP) integrates Bayesian statistical inference with classical linear programming to handle uncertainty in model parameters such as objective function coefficients, constraint coefficients, or right-hand-side values. Instead of treating parameters as fixed or governed by worst-case bounds, BLP uses prior beliefs updated by data to form posterior distributions, which then guide the LP formulation and solution, producing decisions that are optimal in a probabilistic, data-informed sense.

Continuous or mixed numerical data with uncertain parameters (costs, RHS, constraint coefficients)

Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton, NJ.

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

stochastic-linear-programming robust-linear-programming bayesian-dynamic-programming bayesian-mixed-integer-programming multi-objective-linear-programming deterministic-linear-programming


Bayesian Linear Regression

bayesian linear model
probabilistic linear regression
Bayesçi Doğrusal Regresyon

2013 (modern reference); foundations 18th–19th century

Thomas Bayes / Pierre-Simon Laplace (foundations); modern workflow codified by Gelman et al.

Bayesian linear regression is a probabilistic extension of the ordinary linear model, introduced through Bayes' rule and formalised in its modern computational workflow by Gelman et al. (2013). Rather than returning a single point estimate for each coefficient, it combines a user-specified prior distribution with the likelihood of the observed data to produce a full posterior distribution over all parameters, from which credible intervals and posterior predictive distributions are derived.

continuous, categorical, binary (outcome continuous)

MCMC / variational inference / conjugate closed-form

posterior distributions, credible intervals, posterior predictive distributions

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-regression ols-regression mcmc hierarchical-bayes bayesian-anova


Bayesian LISA
Bayesian local spatial autocorrelation
Bayesian local Moran
B-LISA

Extension of Anselin (1995) LISA framework within Bayesian hierarchical modeling traditions (Banerjee, Carlin, Gelfand)

Bayesian Local Indicators of Spatial Association extend the classical LISA framework by embedding local spatial association statistics within a Bayesian hierarchical model. Rather than relying on asymptotic permutation-based significance tests, this approach places prior distributions on spatial parameters and derives posterior probabilities that a location is part of a genuine spatial cluster, accounting for uncertainty and borrowing strength across nearby units.

Georeferenced areal or point data with a continuous or count outcome

Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical Analysis, 27(2), 93–115.

Banerjee, S., Carlin, B. P., & Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC.

local-indicators-of-spatial-association local-morans-i bayesian-spatial-autocorrelation local-getis-ord-gi-star local-gearys-c spatial-autocorrelation


Bayesian Logistic Regression

bayesian binary logistic regression
bayesian classification model
Bayesian Lojistik Regresyon

Gelman, Jakulin, Pittau & Su (weakly-informative prior framework, 2008)

Bayesian logistic regression is a classification model that applies Bayesian inference to a logistic (sigmoid) likelihood for binary or multinomial outcomes. Developed within the weakly-informative prior framework formalised by Gelman, Jakulin, Pittau and Su (2008), it places a prior distribution over the coefficients and combines that prior with the data likelihood to yield a full posterior distribution for each parameter — delivering calibrated class probabilities and honest uncertainty even in small samples, rare-event settings, or cases of complete separation where frequentist maximum likelihood estimation collapses.

posterior distributions / credible intervals / class probabilities

Gelman, A., Jakulin, A., Pittau, M. G. & Su, Y.-S. (2008). A Weakly Informative Default Prior Distribution for Logistic and Other Regression Models. Annals of Applied Statistics, 2(4), 1360–1383.

bayesian-regression logistic-regression mcmc hierarchical-bayes probit-regression


Bayesian MA model

Bayesian MA
Bayesian moving average
BMA time series
MA model with Bayesian estimation

Bayesian framework applied to Box-Jenkins MA models; West & Harrison (1997) canonical treatment

The Bayesian MA model estimates a moving average time series model within a fully Bayesian framework, placing prior distributions on the MA parameters and error variance and updating them via Bayes' theorem. This approach yields full posterior distributions over model parameters and produces probabilistic forecasts with coherent uncertainty quantification.

West, M., & Harrison, J. (1997). Bayesian Forecasting and Dynamic Models (2nd ed.). Springer.

Geweke, J., & Meese, R. (1981). Estimating regression models of finite but unknown order. International Economic Review, 22(1), 55–70.

moving-average-model bayesian-arma-model bayesian-arima-model bayesian-var-model arima-model bayesian-ar-model


Bayesian Mann-Whitney U test

Bayesian rank-sum test
Bayesian Wilcoxon rank-sum test
Bayesian nonparametric two-sample test

2020 (Bayesian formulation); 1947 (classical test)

van Doorn, Ly, Marsman, Wagenmakers (building on Mann & Whitney 1947)

The Bayesian Mann-Whitney U test is a nonparametric Bayesian procedure for comparing two independent groups when data are ordinal or non-normal continuous. Instead of a binary reject/fail-to-reject decision, it quantifies the relative evidence for the null and alternative hypotheses through a Bayes factor, allowing researchers to conclude in favour of either hypothesis or express uncertainty.

Ordinal or continuous; two independent groups

van Doorn, J., Ly, A., Marsman, M., & Wagenmakers, E.-J. (2020). Bayesian rank-based hypothesis testing for the rank sum test, the signed rank test, and Spearman's rho. Journal of Applied Statistics, 47(16), 2984–3006.

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 18(1), 50–60.

mann-whitney-u-test bayesian-wilcoxon-signed-rank-test bayesian-independent-samples-t-test bayesian-kruskal-wallis-test wilcoxon-signed-rank-test independent-samples-t-test


Bayesian MANOVA

Bayesian MANOVA
Bayesian multivariate ANOVA
BF-MANOVA
Bayesian multivariate group comparison

Bayesian framework applied to MANOVA; foundational multivariate Bayesian work by Dickey (1974) and Rouder et al. (2012)

Bayesian Multivariate Analysis of Variance (Bayesian MANOVA) extends the classical MANOVA framework by replacing null-hypothesis significance testing with Bayesian inference. It uses prior distributions on multivariate group means and covariance structures, updates them with data to yield posterior distributions, and quantifies evidence through Bayes factors rather than p-values.

Multiple continuous dependent variables, one or more categorical independent variables

Olkin, I., & Rubin, H. (1964). Multivariate beta distributions and independence properties of the Wishart distribution. The Annals of Mathematical Statistics, 35(1), 261–269.

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.

manova bayesian-ancova bayesian-one-way-anova bayesian-repeated-measures-anova bayesian-independent-samples-t-test multivariate-regression


Bayesian Marginal Structural Model

Bayesian Marginal Structural Model with Inverse Probability Weighting

Bayesian MSM
Bayesian MSM-IPW
Bayesian weighted structural model
Bayesian causal MSM

2015 (Bayesian extension); 2000 (MSM foundation)

Saarela, Stephens, Moodie & Klein (Bayesian extension); Robins, Hernan & Brumback (original MSM)

Bayesian Marginal Structural Model (Bayesian MSM) combines the causal identification power of inverse-probability-weighted marginal structural models with Bayesian posterior inference. Rather than relying on point estimates and asymptotic standard errors, it propagates uncertainty through a full posterior distribution over causal effect parameters, offering coherent uncertainty quantification for causal effects of time-varying treatments.

Causal inference / Bayesian weighted regression

Longitudinal or panel data with time-varying treatments

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.

Robins, J. M., Hernan, M. A., & Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, 11(5), 550-560.

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


Bayesian Markov Model

Bayesian Markov Model — State-Transition Modeling with Bayesian Parameter Estimation

Bayesian Markov Chain Model
Bayesian State-Transition Model
BMM
Bayesian Cohort Simulation

Briggs, A.; Sculpher, M.; and broader Bayesian statistics community

A Bayesian Markov model is a state-transition simulation method that combines Markov chain cohort modeling with Bayesian statistical inference. By placing prior distributions on transition probabilities and updating them with observed data, the approach propagates full parameter uncertainty through the simulation, yielding posterior distributions over outcomes such as costs, life-years, or quality-adjusted life-years rather than single-point estimates.

Transition probabilities, event counts, costs, utilities, prior distributions

Briggs, A., Sculpher, M., Claxton, K. (2006). Decision Modelling for Health Economic Evaluation. Oxford University Press, Oxford.

Jackson, C. H., Sharples, L. D., Thompson, S. G. (2010). Structural and parameter uncertainty in Bayesian cost-effectiveness models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 59(2), 233-253.

markov-model monte-carlo-simulation bayesian-sensitivity-analysis stochastic-markov-model health-economic-modeling probabilistic-sensitivity-analysis


Bayesian Matching Estimator

Bayesian Matching Estimator for Average Treatment Effects

Bayesian matching
Bayesian nonparametric matching
Bayes-ATE matching
posterior matching estimator

Donald B. Rubin (Bayesian causal framework); extended by Heckman, Ichimura & Todd (matching estimator formalization)

The Bayesian Matching Estimator estimates average treatment effects in observational studies by combining classical nearest-neighbour or kernel matching with a Bayesian posterior over the treatment effect. It inherits matching's covariate-balancing logic while propagating uncertainty through a full posterior distribution rather than relying on asymptotic standard errors, yielding credible intervals that reflect both sampling variability and prior knowledge.

Bayesian causal inference / nonparametric matching

Observational cross-sectional or panel data with binary treatment

Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. The Annals of Statistics, 6(1), 34-58.

Heckman, J. J., Ichimura, H., & Todd, P. (1998). Matching as an econometric evaluation estimator. Review of Economic Studies, 65(2), 261-294.

matching-estimator propensity-score-matching bayesian-propensity-score-matching bayesian-difference-in-differences doubly-robust-estimation entropy-balancing


Bayesian McDonald's omega

Bayesian Estimation of McDonald's Omega Reliability Coefficient

Bayesian omega
Bayesian composite reliability
posterior omega
Bayesian omega total

1999 (omega); 2010s (Bayesian estimation)

R. P. McDonald (omega); Bayesian extension developed by Kelley, Pornprasertmanit, and others

Bayesian McDonald's omega applies Bayesian statistical estimation to the omega reliability coefficient, yielding a full posterior distribution over omega rather than a single point estimate. This provides credible intervals and probabilistic uncertainty quantification for the reliability of a composite or scale score, making it especially useful for small samples and complex factor structures.

Reliability / internal consistency estimation

Ordinal or continuous item-level responses

Kelley, K. & Pornprasertmanit, S. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for composite measures. Psychological Methods, 21(1), 69–92.

McDonald, R. P. (1999). Test theory: A unified treatment. Lawrence Erlbaum Associates.

mcdonalds-omega cronbachs-alpha bayesian-cronbachs-alpha bayesian-confirmatory-factor-analysis bayesian-reliability-analysis confirmatory-factor-analysis


Bayesian Measurement Invariance

Bayesian MI
approximate measurement invariance
Bayesian multigroup CFA invariance
BSEM measurement invariance

Bengt Muthen, Tihomir Asparouhov, Rens Van de Schoot

Bayesian measurement invariance testing evaluates whether a scale's factor loadings and item intercepts are equivalent across groups, using a Bayesian framework that allows parameters to deviate from strict equality by a small, probabilistically specified amount rather than imposing an exact constraint.

Bayesian multigroup latent variable test

Ordinal or continuous item-level data across groups

Van de Schoot, R., Kluytmans, A., Tummers, L., Lugtig, P., Hox, J., & Muthen, B. (2013). Facing off with Scylla and Charybdis: a comparison of scalar, partial, and the novel possibility of approximate measurement invariance. Frontiers in Psychology, 4, 770.

Muthen, B., & Asparouhov, T. (2013). BSEM measurement invariance analysis. Mplus Web Notes: No. 17.


Bayesian Metabolomics Analysis

Bayesian Statistical Methods for Metabolomics Data Analysis

Bayesian metabolomics
probabilistic metabolomics
Bayesian metabolite profiling
Bayesian metabolic flux analysis

Simon Rogers, Mark Girolami and colleagues (Bayesian NMR metabolomics framework, ~2009); broader Bayesian metabolomics developed through 2000s–2010s

Bayesian metabolomics analysis applies probabilistic inference to metabolite abundance data — typically from mass spectrometry or NMR spectroscopy — to identify differentially abundant metabolites, annotate spectral features, and integrate pathway knowledge. By encoding prior biological knowledge into prior distributions and propagating uncertainty throughout the analysis, it yields more calibrated probability statements about metabolic differences than classical frequentist testing alone.

Mass spectrometry (LC-MS, GC-MS) or NMR spectroscopy peak tables; metabolite intensity matrices

Rogers, S., Scheltema, R. A., & Girolami, M. A. (2009). Bayesian analysis of metabolomic NMR data. Bioinformatics, 25(14), 1809-1815.

Saccenti, E., Hoefsloot, H. C., Smilde, A. K., Westerhuis, J. A., & Hendriks, M. M. (2014). Reflections on univariate and multivariate analysis of metabolomics data. Metabolomics, 10(3), 361-374.

metabolomics-analysis bayesian-pathway-enrichment-analysis bayesian-gene-set-enrichment-analysis multi-omics-metabolomics-analysis pathway-enrichment-analysis rna-seq-differential-expression


Bayesian Metric Learning

Bayesian Metric Learning (Probabilistic Distance Function Learning)

BML
probabilistic metric learning
Bayesian distance metric learning
Bayesian similarity learning

Multiple (Xing et al. 2002; Weinberger & Saul 2009; probabilistic extensions by various authors ~2010s)

Bayesian Metric Learning frames the problem of learning a task-adapted distance function as probabilistic inference. Rather than producing a single optimal metric matrix, it places a prior over metrics, updates it with pairwise similarity or label constraints, and yields a posterior distribution that quantifies uncertainty about which metric best captures the true structure of the data.

Labeled or pairwise-constrained numerical / embedding data

Weinberger, K. Q., & Saul, L. K. (2009). Distance metric learning for large margin nearest neighbor classification. Journal of Machine Learning Research, 10, 207–244.

metric-learning gaussian-process bayesian-gaussian-process k-nearest-neighbors few-shot-learning bayesian-few-shot-learning


Bayesian Microbiome Diversity Analysis

Bayesian Statistical Analysis of Microbiome Diversity

2010s (Dirichlet-Multinomial approach formalized ~2012; extensions ongoing)

Ian Holmes, Katie Harris, Christopher Quince (Dirichlet-Multinomial Mixture framework, 2012); broader Bayesian microbiome modeling community

Bayesian microbiome diversity analysis applies probabilistic models — chiefly Dirichlet-Multinomial and related hierarchical frameworks — to 16S rRNA or shotgun metagenomic count data to estimate alpha-diversity (within-sample richness and evenness) and beta-diversity (between-sample compositional differences) while propagating uncertainty through the entire inference chain. Unlike frequentist rarefaction-based approaches, Bayesian methods treat taxon counts as draws from a latent composition, enabling credible intervals on diversity metrics and principled comparison across groups with unequal sequencing depth.

Probabilistic/Bayesian pipeline for compositional count data

16S rRNA amplicon OTU/ASV count tables, shotgun metagenomics count tables, phylogenetic trees

Holmes, I., Harris, K., & Quince, C. (2012). Dirichlet Multinomial Mixtures: Generative Models for Microbial Metagenomics. PLOS ONE, 7(2), e30126.

La Rosa, P. S., Brooks, J. P., Deych, E., Boone, E. L., Edwards, D. J., Wang, Q., Sodergren, E., Weinstock, G., & Shannon, W. D. (2012). Hypothesis Testing and Power Calculations for Taxonomic-Based Human Microbiome Data. PLOS ONE, 7(12), e52078.


Bayesian Microsimulation

Bayesian Microsimulation — Probabilistic individual-level simulation with Bayesian parameter estimation

Bayesian micro-simulation
BMS
Bayesian individual-level simulation
Probabilistic microsimulation

Williamson, P.; Birkin, M.; Rees, P. H. and related health-economics researchers

Bayesian Microsimulation combines individual-level simulation of heterogeneous populations with Bayesian statistical inference. Each synthetic individual follows a probabilistic life path, while model parameters are governed by prior beliefs updated with observed data. This approach is widely used in health technology assessment, public policy costing, and demographic projection, where uncertainty in both model inputs and structural assumptions must be formally quantified and propagated through to output estimates.

Individual-level probabilistic simulation with Bayesian updating

Individual or household microdata; prior distributions; survey or administrative records

Williamson, P., Birkin, M., & Rees, P. H. (2000). The estimation of population microdata by using data from small area statistics and samples of anonymised records. Environment and Planning A, 30(5), 785-816.

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

microsimulation bayesian-inference monte-carlo-simulation markov-model stochastic-microsimulation agent-based-microsimulation


Bayesian Mixed Effects Model

Bayesian multilevel model
Bayesian random effects model
Bayesian LME
Bayesian hierarchical mixed model

Gelman, Hill, and the broader Bayesian hierarchical modeling tradition

The Bayesian mixed effects model extends the classical mixed effects framework by placing prior distributions on all parameters — fixed effects, random effect variances, and residual variance — and updating them with data to produce full posterior distributions. This provides coherent uncertainty quantification for both population-level and group-level effects simultaneously.

Continuous, grouped, or longitudinal data with fixed and random effects

Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1–48.

mixed-effects-model hierarchical-linear-model multilevel-modeling bayesian-hierarchical-linear-model bayesian-multilevel-modeling bayesian-generalized-linear-model


Bayesian Mixed-Integer Programming

Bayesian Mixed-Integer Programming — Surrogate-Assisted Optimization over Mixed-Integer Search Spaces

Bayesian MIP
BO-MIP
Bayesian Combinatorial Optimization
Mixed-Integer Bayesian Optimization

2018 (surrogate-BO-MIP synthesis); MIP foundations 1958

Baptista, R. & Poloczek, M. (formal Bayesian-BO-MIP formulation); mixed-integer programming roots in Gomory (1958)

Bayesian Mixed-Integer Programming (BO-MIP) couples a probabilistic surrogate model — typically a Gaussian process — with a mixed-integer programming solver to efficiently optimize expensive black-box objectives defined over spaces that contain both continuous and discrete or integer-valued decision variables. It is especially valuable when each function evaluation is costly and exhaustive search is infeasible.

Objective function evaluations (possibly black-box), integer and continuous decision variables, optional constraint matrices

Baptista, R., Poloczek, M. (2018). Bayesian Optimization of Combinatorial Structures. Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR 80:462–471.

Bonami, P., Biegler, L. T., Conn, A. R., Cornuejols, G., Grossmann, I. E., Laird, C. D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A. (2008). An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optimization, 5(2), 186–204.


Bayesian Mixture Modeling

Bayesian mixture model
BMM
Bayesian model-based clustering
Bayesian finite mixture

1997 (Richardson & Green Bayesian formulation)

Richardson & Green (seminal Bayesian treatment, 1997); broader Bayesian mixture roots trace to Dempster, Laird & Rubin (EM, 1977) and Titterington, Smith & Makov (1985)

Bayesian mixture modeling represents the population as a weighted sum of K component distributions and estimates all unknowns — mixing weights, component parameters, and even the number of components — through posterior inference. It extends classical mixture analysis by placing priors on every parameter and quantifying uncertainty over latent group assignments rather than treating them as fixed.

Continuous, binary, count, or mixed multivariate observations

Fruhwirth-Schnatter, S., Celeux, G. & Robert, C. P. (Eds.) (2019). Handbook of Mixture Analysis. CRC Press / Chapman & Hall.

Richardson, S. & Green, P. J. (1997). On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society: Series B, 59(4), 731–792.

mixture-modeling latent-class-analysis bayesian-latent-class-analysis bayesian-latent-profile-analysis bayesian-cluster-analysis bayesian-structural-equation-modeling


Bayesian Model Averaging with Measurement Error Correction

Bayesian model averaging errors-in-covariates

Hoeting, Madigan, Raftery, Volinsky (BMA); Carroll, Stefanski and colleagues (ME correction)

Bayesian model averaging with measurement error (BMA-ME) combines two probabilistic ideas: it averages predictions across competing regression models weighted by each model's posterior probability, while simultaneously accounting for the fact that one or more predictors are observed with random error rather than exactly. The result is a posterior that propagates both model uncertainty and covariate measurement noise into every inference and prediction.

Bayesian ensemble model with covariate error correction

continuous or binary outcomes with mismeasured predictors

Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14(4), 382-417.

Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective (2nd ed.). CRC Press.

bayesian-model-averaging bayesian-regression errors-in-variables-regression hierarchical-bayes mcmc bayesian-variable-selection


Bayesian model averaging with missing data

Bayesian Model Averaging with Missing Data

Bayesian model averaging under missingness

1999 (BMA seminal); 2000s (missing-data extensions)

Hoeting, Madigan, Raftery, Volinsky (BMA); extended to missing data by Raftery, Madigan and others

Bayesian Model Averaging with missing data (BMA-MD) simultaneously addresses two sources of uncertainty: which model best describes the data, and what the unobserved values are. Rather than selecting a single imputed dataset and a single model, the approach averages predictions across the full space of candidate models and plausible completions of the missing values, propagating both sources of uncertainty into every estimate and prediction.

Bayesian ensemble inference under incomplete data

Continuous, binary, or categorical outcomes with partially observed predictors or outcomes

Hoeting, J. A., Madigan, D., Raftery, A. E. & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14(4), 382-417.

Rubin, D. B. (1987). Multiple Imputation for Nonresponse in Surveys. John Wiley & Sons, New York.


Bayesian Model Averaging

BMA
Bayesian model combination
Bayesian Model Ortalaması (BMA)

Bayesian Model Averaging (BMA), formalised as a tutorial by Hoeting, Madigan, Raftery and Volinsky in 1999, addresses model uncertainty by averaging over all plausible model specifications rather than selecting a single best model. Each candidate model receives a posterior probability that reflects how well it fits the data given a prior, and predictions or coefficient estimates are formed as weighted averages across the entire model space. This approach reduces the bias and overconfidence that arise when a single selected model is treated as the true one.

posterior inclusion probabilities / model-averaged coefficients / BMA predictions

Hoeting, J. A., Madigan, D., Raftery, A. E. & Volinsky, C. T. (1999). Bayesian Model Averaging: A Tutorial. Statistical Science, 14(4), 382–401.

Zeugner, S. & Feldkircher, M. (2015). Bayesian Model Averaging Employing Fixed and Flexible Priors: The BMS Package for R. Journal of Statistical Software, 68(4), 1–37.

bayesian-regression lasso-regression elastic-net bayesian-hierarchical-model mcmc


Bayesian Model Testing Research

Bayesian hypothesis testing
Bayesian model comparison
Bayes factor analysis
BMT

1935 (Jeffreys); widely adopted in social and behavioral sciences from the 1990s onward

Harold Jeffreys; formalized for applied sciences by Robert Kass and Adrian Raftery

Bayesian model testing research is a quantitative design in which competing theoretical models or hypotheses are evaluated by comparing their marginal likelihoods given observed data. The central tool is the Bayes factor — a ratio that quantifies how much more likely the data are under one model than under another. Unlike null-hypothesis significance testing, Bayesian model testing yields direct evidence for or against specific hypotheses, incorporates prior knowledge, and can support a null hypothesis rather than merely failing to reject it.

Quantitative inferential research design

Continuous, categorical, or count outcome data; requires specification of competing theoretical models

Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795.

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

bayesian-inference structural-equation-modeling confirmatory-factor-analysis multilevel-modeling meta-analysis null-hypothesis-significance-testing


Bayesian Moderated Mediation

Bayesian conditional process analysis
Bayesian mediated moderation
Bayesian PROCESS model
Bayesian conditional indirect effect

Yuan & MacKinnon (Bayesian mediation); Hayes (conditional process framework)

Bayesian moderated mediation estimates how a mediator transmits the effect of a predictor onto an outcome, and whether that indirect effect varies in size depending on a moderator variable — all within a Bayesian framework that quantifies uncertainty via posterior distributions rather than p-values and confidence intervals.

Yuan, Y. & MacKinnon, D. P. (2009). Bayesian mediation analysis. Psychological Methods, 14(4), 301–322.

Hayes, A. F. (2013). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. Guilford Press.

moderated-mediation bayesian-mediation-analysis bayesian-moderation-analysis structural-equation-modeling bayesian-structural-equation-modeling path-analysis


Bayesian Moderation Analysis

Bayesian interaction analysis
Bayesian moderated regression
Bayesian moderator testing
BMA

Bayesian framework applied to moderation by Kruschke, Gelman and colleagues

Bayesian moderation analysis tests whether the relationship between a predictor and an outcome changes depending on the value of a third variable (the moderator). By placing prior distributions on all model parameters and updating them with observed data, it yields full posterior distributions for the interaction effect — enabling direct probability statements about the moderation rather than binary significance decisions.

Continuous, ordinal, or categorical predictors and outcomes

Hayes, A. F. (2018). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach (2nd ed.). Guilford Press.

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

bayesian-mediation-analysis bayesian-structural-equation-modeling bayesian-path-analysis moderated-mediation bayesian-moderated-mediation multilevel-moderation-analysis


Bayesian Monte Carlo Simulation

Bayesian Monte Carlo Simulation — Prior-informed stochastic sampling for uncertainty quantification

Bayesian MC
BMC simulation
Bayesian stochastic simulation
Bayesian uncertainty propagation

Bayesian Monte Carlo Simulation integrates Bayesian statistical inference with Monte Carlo sampling to propagate uncertainty through complex models. Instead of drawing samples from arbitrary distributions, it conditions sampling on observed data and expert prior knowledge via Bayes' theorem, yielding posterior-based uncertainty estimates that are both statistically coherent and interpretable in probabilistic terms.

Continuous numerical data, expert elicited priors, probability distributions

O'Hagan, A., Buck, C. E., Daneshkhah, A., Eiser, J. R., Garthwaite, P. H., Jenkinson, D. J., Oakley, J. E., & Rakow, T. (2006). Uncertain Judgements: Eliciting Experts' Probabilities. Wiley.

O'Hagan, A. (1987). Monte Carlo is fundamentally unsound. The Statistician, 36(2-3), 247-249.

monte-carlo-simulation stochastic-monte-carlo-simulation markov-chain-monte-carlo bayesian-sensitivity-analysis bayesian-system-dynamics probabilistic-risk-assessment


Bayesian Moran's I

Bayesian Moran's I Spatial Autocorrelation Test

Bayesian spatial autocorrelation test
Bayesian Moran statistic
Moran's I under Bayesian inference
Bayesian global spatial association

Moran (1950), Bayesian extension developed in spatial statistics literature (late 1990s–2000s)

Bayesian Moran's I embeds the classical Moran's I spatial autocorrelation test within a Bayesian probabilistic framework. Rather than producing a single p-value, it yields a posterior distribution over the spatial autocorrelation parameter, enabling uncertainty quantification, incorporation of prior knowledge, and more principled inference in small or irregular spatial datasets.

Georeferenced areal or point data with a spatial weights matrix

Haining, R. (2003). Spatial Data Analysis: Theory and Practice. Cambridge University Press.

Lee, S.-I. (2001). Developing a bivariate spatial association measure: an integration of Pearson's r and Moran's I. Journal of Geographical Systems, 3(4), 369–385.

morans-i local-morans-i bayesian-spatial-autocorrelation gearys-c bayesian-spatial-regression local-indicators-of-spatial-association


Bayesian Multi-Objective Optimization

Bayesian Multi-Objective Optimization (BMOO) — Surrogate-assisted Pareto frontier exploration under uncertainty

BMOO
Bayesian MOO
Multi-objective Bayesian optimization
MOBO

Emmerich, M.; Svenson, J.; and related Gaussian process optimization community

Bayesian Multi-Objective Optimization (BMOO/MOBO) uses Gaussian process surrogate models to approximate multiple expensive objective functions and guides the search toward the Pareto frontier with minimal real evaluations. By quantifying prediction uncertainty at each candidate point, it balances exploration of unknown regions against exploitation of promising solutions, making it especially powerful when each function evaluation is computationally or experimentally costly.

Continuous or mixed numerical parameters; expensive black-box objective functions

Svenson, J., Santner, T. (2016). Multiobjective optimization of expensive-to-evaluate deterministic computer simulator models. Computational Statistics & Data Analysis, 94, 250-264.

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

multi-objective-optimization bayesian-optimization nsga-ii gaussian-process-regression pareto-optimization stochastic-multi-objective-optimization


Bayesian Multidimensional Scaling

Bayesian MDS
BMDS
probabilistic MDS
Bayesian proximity scaling

Bayesian Multidimensional Scaling places objects in a low-dimensional latent space so that inter-object distances reproduce observed dissimilarities, while a full Bayesian treatment quantifies uncertainty in the coordinates, handles missing proximities naturally, and selects the number of dimensions via model comparison rather than heuristic inspection.

Bayesian latent-space dimensionality reduction

Dissimilarity or distance matrices; proximity data

Oh, M.-S. & Raftery, A. E. (2001). Bayesian multidimensional scaling and choice of dimension. Journal of the American Statistical Association, 96(455), 1031–1044.

multidimensional-scaling bayesian-cluster-analysis bayesian-principal-component-analysis bayesian-confirmatory-factor-analysis bayesian-latent-class-analysis bayesian-exploratory-factor-analysis


Bayesian polytomous logistic regression
Bayesian multinomial logit
Bayesian softmax regression
Bayesian nominal logistic regression

1966 (classical); Bayesian extensions established by 1990s

Gelman et al. (Bayesian treatment); classical multinomial logit by Cox (1966)

Bayesian Multinomial Logistic Regression models a nominal outcome with three or more unordered categories by placing prior distributions over the regression coefficients and updating them with data via Bayes' theorem. The result is a full posterior distribution over category probabilities for each observation, enabling principled uncertainty quantification and regularization through the prior.

Nominal categorical outcome (3+ unordered classes), continuous or categorical predictors

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

Agresti, A. (2002). Categorical Data Analysis (2nd ed.). Wiley-Interscience.

multinomial-logistic-regression bayesian-ordinal-logistic-regression bayesian-logistic-regression ordinal-logistic-regression bayesian-generalized-linear-model dirichlet-regression


Bayesian multiway correspondence analysis

Bayesian categorical dimension reduction

Extension of MCA (Benzecri, 1973) with Bayesian inference

Bayesian Multiple Correspondence Analysis extends classical MCA by embedding the geometric decomposition of categorical data tables within a Bayesian probabilistic framework, enabling principled uncertainty quantification around category coordinates, dimension selection via marginal likelihood, and incorporation of prior knowledge about variable relationships.

Bayesian dimension reduction for categorical data

Nominal or ordinal categorical variables (multiple)

Greenacre, M. & Blasius, J. (Eds.) (2006). Multiple Correspondence Analysis and Related Methods. Chapman & Hall/CRC.

Delattre, M., Lavielle, M. & Poursat, M.-A. (2014). A note on BIC in mixed-effects models. Electronic Journal of Statistics, 8(1), 456–475.

multiple-correspondence-analysis bayesian-correspondence-analysis bayesian-latent-class-analysis bayesian-cluster-analysis latent-class-analysis correspondence-analysis