Bai-Perron Test

Bai-Perron Multiple Structural Break Test

Bai-Perron Multiple Break Test
Multiple Structural Change Test
Sequential Structural Break Test
Çoklu Yapısal Kırılma Testi

The Bai-Perron test, introduced by Jushan Bai and Pierre Perron in their landmark 1998 Econometrica paper, is a least-squares-based procedure for detecting, estimating, and testing the number of structural breaks in a linear regression model estimated on time-series data. Unlike single-break tests, it simultaneously identifies multiple change-points in a sample, providing economists and empirical researchers with a rigorous, data-driven way to locate parameter instability across time.

Sequential hypothesis test for multiple structural breaks

Bai, J., & Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47–78.


Balanced Accuracy

Average Recall
Equal-weight Average Sensitivity

Balanced accuracy is the average of recall values computed for each class separately. It corrects for class imbalance by giving equal weight to the performance on each class, regardless of class frequency in the dataset.

Brodersen, K. H., Ong, C. S., Stephan, K. E., & Buhmann, J. M. (2010). The balanced accuracy and its posterior distribution. 20th International Conference on Pattern Recognition (ICPR), 3121-3124.

Powers, D. M. (2011). Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness and Correlation. Journal of Machine Learning Technologies, 2(1), 37-63.

accuracy recall specificity f1-score matthews-correlation-coefficient


Balanced Scorecard in Healthcare

Balanced Scorecard Framework for Healthcare Performance Management

Healthcare BSC
Balanced Scorecard Healthcare

Strategic management, Performance measurement

The Balanced Scorecard is a strategic performance management framework that translates an organization's mission and strategy into a comprehensive set of performance measures across four perspectives: financial, customer, internal processes, and learning and growth. Developed by Kaplan and Norton in 1992 for general business, it has been extensively adapted for healthcare organizations to align hospital operations with strategic objectives.

Strategic planning and management framework

Kaplan, R. S., & Norton, D. P. (1992). The balanced scorecard: Measures that drive performance. Harvard Business Review, 70(1), 71–79.

Kaplan, R. S., & Norton, D. P. (2001). The Strategy-Focused Organization: How Balanced Scorecard Companies Thrive in the New Business Environment. Harvard Business School Press.

Niven, P. R. (2008). Balanced Scorecard Step-by-Step for Government and Nonprofit Agencies (2nd ed.). John Wiley & Sons.

dea-hospital-efficiency lean-healthcare clinical-audit cost-effectiveness-analysis-hta staffing-ratio-analysis


Balanced Scorecard Performance Measure

Balanced Scorecard (BSC) Strategy Measurement and Management System

BSC
Balanced Scorecard Framework
Kaplan-Norton Scorecard

The Balanced Scorecard (BSC) is a strategic management system that translates organizational strategy into a coherent set of performance measures across four perspectives: Financial, Customer, Internal Process, and Learning and Growth. Developed by Kaplan and Norton (1992) in Harvard Business Review, the BSC addresses a fundamental management gap: most organizations measure what is easy to measure (financial results) while neglecting what drives results (customer satisfaction, operational efficiency, employee capability). By balancing financial outcomes with non-financial drivers, the BSC enables organizations to understand and manage strategy execution, identify causal relationships between performance drivers, and align organizational actions with strategic objectives.

Organizational performance measurement and management system

Kaplan, R. S., & Norton, D. P. (1996). The balanced scorecard: Translating strategy into action. Harvard Business School Press.

Niven, P. R. (2002). Balanced scorecard step-by-step: Maximizing performance and maintaining results. John Wiley & Sons.

strategic-orientation-scale corporate-governance-questionnaire knowledge-management-scale organizational-resilience-scale supply-chain-integration-scale


BALANCED-SPOTIS

BALANCED-SPOTIS (Balanced SPOTIS — Balanced Stable Preference Ordering Towards Ideal Solution) is a ranking multi-criteria decision-making (MCDM) method introduced by Shekhovtsov, A., Dezert, J., Sałabun, W. in 2025. It turns a decision matrix of alternatives scored on multiple criteria into a structured, reproducible result.

Distance-to-ideal with blended ISP/ESP reference

Shekhovtsov, A., Dezert, J., Sałabun, W. (2025). Enhancing Personalized Decision-Making with the Balanced SPOTIS Algorithm. 17th International Conference on Agents and Artificial Intelligence (ICAART 2025)


Banister TRIMP

Training Impulse and Fitness-Fatigue Modeling

TRIMP
training impulse
fitness-fatigue model

The Training Impulse (TRIMP) model, developed by Eric Banister and colleagues (1975), quantifies the physiological stimulus of a training session by combining duration and intensity. The Banister fitness-fatigue model proposes that training effects on performance follow two opposing dynamics: fitness (beneficial) accumulates with time constant tau_f (~42 days) and fatigue (temporary decrement) accumulates faster but decays quickly (tau_d ~5-10 days). By tracking TRIMP and modeling these two processes, coaches can predict performance trajectories and optimize training load. Although superseded by newer frameworks, the Banister model remains influential and intuitive.

Banister, E. W., Calvert, T. W., Savage, M. V., & Bach, T. (1975). A systems model of training responses and its relationship to muscular strength. Transactions of the ASME, 97(3), 177-183.

Morton, R. H., Frick, U., & Bazalgette, D. (2005). Modeling human performance in running. Journal of Applied Physiology, 92(6), 2393-2402.

Clarke, D. C., & Skiba, P. F. (2013). Rationale and resources for teaching the mathematical modeling of athletic training and performance. Advances in Physiology Education, 37(2), 134-142.


Bark and Mel Scales

Perceptual Frequency Scales for Audio Analysis and Perception

bark scale
mel scale
critical bandwidth
perceptual frequency

Bark and Mel scales are perceptual frequency scales that map physical frequency (Hz) to perceived pitch and auditory perception. Formalized by Zwicker (Bark, 1961) and Stevens (Mel, 1937), these non-linear scales reflect how the human ear processes sound. Bark scale divides hearing into 24 critical bands; Mel scale models pitch perception. Both are essential for audio feature extraction, speech processing, and designing audio systems that align with human hearing.

Zwicker, E. (1961). Subdivision of the audible frequency range into critical bands. Journal of the Acoustical Society of America, 33(2), 248–248.

Stevens, S. S., Volkmann, J., & Newman, E. B. (1937). A scale for the measurement of the psychological magnitude pitch. Journal of the Acoustical Society of America, 8(3), 185–190.

Moore, B. C. J. (2012). An Introduction to the Psychology of Hearing (6th ed.). Academic Press.

psychoacoustic-masking linear-predictive-coding cepstral-analysis speech-intelligibility fxlms-active-noise-control


Barkley Adult ADHD Rating Scale

neurodevelopmental and executive function

The Barkley Adult ADHD Rating Scale (BAARS-IV) is a 27-item self- or observer-report measure of ADHD symptoms and executive function deficits in adolescents and adults. Developed by Russell Barkley and colleagues, the BAARS operationalizes ADHD beyond the traditional inattention and hyperactivity domains to include executive function deficits (working memory, organization, time management, emotional regulation) that are prominent in adolescent and adult ADHD. It is widely used in clinical and research settings for screening, diagnosis, and outcome measurement in ADHD treatment.

Self-report and observer-report rating scale

Barkley, R. A., & Murphy, K. R. (2011). The nature of executive function deficits in adults with ADHD and their relationship to symptoms and impairment. Journal of Attention Disorders, 15(1), 56–71.

Barkley, R. A., DuPaul, G. J., & Costello, A. (1993). Stimulants. In J. S. Werry & M. C. Aman (Eds.), Practitioners guide to psychoactive drugs for children and adolescents (pp. 205–237). Plenum Press.


Barnard's Exact Test

Barnard's Unconditional Exact Test for 2×2 Tables

Barnard test
unconditional exact test
CSM test
Barnard's exact unconditional test
Barnard's test of homogeneity

Barnard's exact test is an unconditional exact hypothesis test for comparing two independent proportions in a 2×2 contingency table, proposed by George A. Barnard in 1945. Unlike Fisher's exact test, it does not condition on both margins being fixed, and is generally more powerful when column totals are not predetermined by the study design.

Unconditional exact test for 2×2 contingency tables

exact (maximized over nuisance parameter)

Barnard, G. A. (1945). A new test for 2×2 tables. Nature, 156(3954), 177.

Suissa, S., & Shuster, J. J. (1985). Exact unconditional sample sizes for the 2×2 binomial trial. Journal of the Royal Statistical Society, Series A, 148(4), 317–327.

Lydersen, S., Fagerland, M. W., & Laake, P. (2009). Recommended tests for association in 2×2 tables. Statistics in Medicine, 28(7), 1159–1175.

fisher-s-exact-test chi-squared-test mcnemar-s-test boschloo-s-test z-test-for-two-proportions


Barnes-Cressman Analysis

Barnes analysis
Cressman analysis
Objective analysis
Grid interpolation

Barnes-Cressman analysis is an objective interpolation method that creates gridded meteorological fields from irregularly spaced observations (station data, radiosonde profiles, buoys). It is widely used for synoptic analysis, quality control, and initialization of numerical weather models.

Barnes, S. L. (1964). A Technique for Maximizing Details in Numerical Weather Map Analysis. Journal of Applied Meteorology, 3(4), 396-409.

Cressman, G. P. (1959). An Operational Objective Analysis System. Monthly Weather Review, 87(10), 367-374.


Karen Sechrist, Susan Noble Walker, and Nola J. Pender

The Barriers to Physical Activity Questionnaire (BPA) is a scale designed to identify and measure perceived obstacles to exercise engagement. Rooted in the Health Belief Model and Health Promotion Model, the BPA assesses multiple categories of barriers—time constraints, lack of motivation, physical discomfort, cost, lack of facilities, social/family factors, and weather—that individuals perceive as preventing or limiting physical activity. Understanding which barriers are most salient for a given individual or population enables targeted intervention design, such as time management coaching, facility access solutions, or social support programs. The BPA is widely used in primary care, community health, occupational health, and exercise research to segment populations and tailor physical activity prescriptions.

Sechrist, K. R., Walker, S. N., & Pender, N. J. (1987). Development and psychometric evaluation of the Exercise Benefits/Barriers Scale. Research in Nursing & Health, 10(6), 357-365.


Barthel ADL Index

Barthel Index of Activities of Daily Living

The Barthel Index is a brief, observer-rated scale measuring independence in activities of daily living (ADL) in patients with disability, stroke, and neurological conditions. Developed by Barthel and Mahoney in 1965, it has become a widely used outcome measure in rehabilitation, stroke care, and geriatrics for assessing functional independence and predicting discharge placement and long-term outcomes.

Barthel, D. W. (1965). Functional evaluation: the Barthel Index. Maryland State Medical Journal, 14(5), 61–65.

Mahoney, F. I., & Barthel, D. W. (1965). Functional evaluation: the Barthel Index. Maryland State Medical Journal, 14(2), 61–65.

Collin, C., Wade, D. T., Davies, S., & Horne, V. (1988). The Barthel ADL Index: a reliability study. International Disability Studies, 10(2), 61–63.

fim-functional-independence mif-motor-index functional-reach-test tug-test


Barthel Index

BI
Barthel ADL Index
Functional Independence Index

Functional assessment and activities of daily living

The Barthel Index (BI) is one of the most widely used functional assessment tools measuring independence in activities of daily living. Developed by Florence I. Mahoney and Dorothea W. Barthel in 1965, the Barthel Index evaluates a patient's ability to perform ten essential self-care and mobility activities. Its longevity and widespread adoption across rehabilitation, geriatric, and acute care settings reflect its reliability, simplicity, and clinical utility for assessing functional status and predicting rehabilitation outcomes.

Mahoney, F. I., & Barthel, D. W. (1965). Functional evaluation: The Barthel Index. Maryland State Medical Journal, 14, 61-65.

care-dependency-scale braden-scale patient-fall-risk-assessment nursing-sensitive-indicators


Bartlett's Test

Bartlett's Test for Homogeneity of Variances

Bartlett's Chi-Square Test
Test for Equality of Variances
Bartlett's Homogeneity Test
Varyans Homojenliği Testi

Bartlett's Test is a classical parametric procedure for testing whether two or more independent groups share a common population variance. Introduced by Maurice Stevenson Bartlett in 1937, it formalises the null hypothesis that all group variances are equal by constructing a chi-square statistic from the ratio of pooled to individual group variances. It is a standard pre-analysis step before applying ANOVA or other procedures whose validity depends on the homoscedasticity assumption.

All group population variances are equal

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London. Series A, 160(901), 268–282.


Baryon Acoustic Oscillations

Baryon Acoustic Oscillations for Cosmological Distance Measurements

BAO
Baryon Oscillations
Standard Ruler Method

Baryon Acoustic Oscillations are imprints of sound waves in the early universe that appear as a characteristic scale in the large-scale distribution of galaxies today. First predicted theoretically by Piet Peebles and Joseph Yu in 1970, and detected observationally by the Sloan Digital Sky Survey in 2005, BAO provides a standard ruler for measuring cosmic distances and constraining the expansion history of the universe.

Peebles, P. J. E., & Yu, J. T. (1970). Primeval adiabatic perturbation in an expanding universe. Astrophysical Journal, 162, 815-836.

Eisenstein, D. J., et al. (2005). Detection of the baryon acoustic peak in the correlation function of SDSS luminous red galaxies. Astrophysical Journal, 633(2), 560-574.

Ross, A. J., et al. (2015). The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey. Monthly Notices of the Royal Astronomical Society, 449(1), 835-847.


The BASDAI is a patient-reported outcome measure of disease activity in ankylosing spondylitis (AS), a chronic inflammatory arthropathy affecting the spine and axial skeleton. Introduced by Garrett et al. in 1994, BASDAI uses six simple patient self-report items focused on the cardinal symptoms of AS: fatigue, spinal pain, peripheral joint involvement, and morning stiffness. As a PRO measure, BASDAI is practical for routine monitoring, responsive to treatment, and strongly associated with clinical outcomes and spinal damage progression, making it a cornerstone outcome in AS management and clinical trials.

Garrett S, Jenkinson T, Kennedy LG, Whitelock H, Gaisford P, Calin A. A new approach to defining disease status in ankylosing spondylitis: the Bath Ankylosing Spondylitis Disease Activity Index. The Journal of Rheumatology. 1994;21(12):2286-2291.


The BASFI is a patient-reported outcome measure of functional disability in ankylosing spondylitis (AS), assessing physical limitations in 10 common daily activities. Introduced by Calin et al. in 1994, BASFI measures the impact of AS on quality of life and functional capacity, complementing BASDAI (disease activity). While BASDAI reflects inflammatory burden, BASFI reflects the consequences of inflammation and spinal damage on a patient's ability to perform daily tasks. BASFI is used alongside BASDAI in AS management to capture both disease activity and its functional impact.

Calin A, Garrett S, Whitelock H, Kennedy LG, O'Hea J, Mallorie P, Jenkinson T. A new approach to defining functional ability in ankylosing spondylitis: the development of the Bath Ankylosing Spondylitis Functional Index. The Journal of Rheumatology. 1994;21(12):2281-2285.


Basin Subsidence Analysis

tectonic subsidence
backstripping
thermal history analysis

Basin subsidence analysis is the quantitative study of how sedimentary basins deepen over geological time, driven by tectonics, isostasy, and load. Formalized by McKenzie (1978) and Sclater and Christie (1980), this method reveals the mechanical causes of basin development, predicts subsurface temperature and pressure histories, and constrains petroleum generation. Analysis integrates well stratigraphy, seismic geometry, gravity data, and thermal models to reconstruct basin evolution.

Sclater, J. G., & Christie, P. A. F. (1980). Continental stretching: An explanation of the post-mid-Cretaceous subsidence of the Central North Sea Basin. Journal of Geophysical Research, 85(B7), 3711–3739.

Allen, P. A., & Allen, J. R. (1995). Geology of Deltas. Ellis Horwood Limited.

McKenzie, D. (1978). Some remarks on the development of sedimentary basins. Earth and Planetary Science Letters, 40(1), 25–32.

stratigraphic-correlation geologic-mapping basin-modelling geochronological-dating geophysical-inversion


Bat Algorithm

BA
Bat-Inspired Algorithm
Echolocation-Based Optimization
Yarasa Algoritması

The Bat Algorithm (BA) is a nature-inspired metaheuristic optimization method proposed by Xin-She Yang in 2010. It mimics the echolocation behavior of microbats to balance global exploration and local exploitation. Each artificial bat adjusts its position, velocity, and emission frequency, with loudness and pulse rate dynamically controlling the transition from broad search to refined local tuning. BA is suited to continuous and combinatorial optimization problems across engineering, scheduling, and machine learning domains.

Yang, X.-S. (2010). A new metaheuristic bat-inspired algorithm. Nature Inspired Cooperative Strategies for Optimization (NICSO), 65–74.


Batch Normalization

Batch Normalization (Normalizing Layer Activations per Mini-Batch)

BatchNorm
BN
batch norm
mini-batch normalization
internal covariate shift reduction

Batch Normalization is a training technique introduced by Sergey Ioffe and Christian Szegedy in 2015 that normalizes the pre-activation outputs of each layer using the mean and variance computed over the current mini-batch. By stabilizing the input distribution to each layer throughout training, it substantially reduces internal covariate shift, enabling the use of higher learning rates and making deep networks train faster and more reliably.

Normalization technique (applied per mini-batch during training)

Stabilizing and accelerating training of deep neural networks

Any differentiable layer in a neural network

Ioffe, S. & Szegedy, C. (2015). Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. Proceedings of the 32nd International Conference on Machine Learning (ICML), PMLR 37, 448–456.

Goodfellow, I., Bengio, Y. & Courville, A. (2016). Deep Learning (Ch. 8). MIT Press.

Ioffe, S. & Szegedy, C. (2015). Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. arXiv preprint arXiv:1502.03167.

layer-normalization dropout residual-network adam-optimizer weight-initialization group-normalization


Bates Model

Bates Stochastic Volatility Jump Diffusion Model

The Bates model (1996) combines stochastic volatility and jump diffusion to capture both the volatility smile and the implied volatility skew observed in equity and currency option markets. It extends the Heston model by adding a Poisson jump component to returns, making it suitable for pricing options when sudden price moves are expected.

Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1), 69-107.

Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144.

sabr-model local-volatility hull-white-model risk-neutral-valuation


Battery Equivalent Circuit Model

Battery Equivalent Circuit Model for Electrochemical Analysis

The Battery Equivalent Circuit Model (ECM) represents battery electrochemical behavior using an electrical circuit analogy. It includes an ideal voltage source (open-circuit voltage dependent on state of charge), internal resistance(s) for ohmic losses, and capacitive/resistive elements for transient response. ECM enables rapid simulation of battery behavior in electric vehicles, renewable energy systems, and portable devices without solving complex electrochemical equations.

Seaman, C. V., Strutt, A. S., & Murray, A. (2014). Portable and plug-in hybrid electric vehicle battery electric range impacts on U.S. gasoline consumption. Journal of Power Sources, 243, 773-783.

Plett, G. L. (2004). Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs. Journal of Power Sources, 134(2), 252-261.


Bayes Factor Test

bayes factor
BF10
Bayesian hypothesis test
Bayes Faktörü — Hipotez Testi

The Bayes factor test, formalised by Harold Jeffreys in 1961, is a Bayesian method for comparing two competing hypotheses. Rather than returning a binary reject/retain verdict, it produces a continuous ratio BF₁₀ that quantifies how much more (or less) probable the data are under the alternative hypothesis H₁ than under the null hypothesis H₀.

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

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

bayesian-regression independent-t-test mcmc hierarchical-bayes


Bayesian Active Learning

Bayesian Active Learning (Query-by-Committee and BALD)

Bayesian optimal experimental design for ML

BALD (Bayesian Active Learning by Disagreement)

Bayesian Active Learning (BAL) combines a probabilistic model with an active query strategy to identify the unlabeled examples that, once labeled, would most reduce model uncertainty. Instead of labeling data at random, BAL guides an oracle — typically a human annotator — toward the points where labeling will provide the greatest information gain, making it highly label-efficient.

Active learning with Bayesian uncertainty

Labeled and unlabeled tabular, image, or text data

Houlsby, N., Huszár, F., Ghahramani, Z., & Lengyel, M. (2011). Bayesian Active Learning for Classification and Preference Learning. arXiv preprint arXiv:1112.5745.

Settles, B. (2012). Active Learning. Synthesis Lectures on Artificial Intelligence and Machine Learning, 6(1), 1–114. Morgan & Claypool.

active-learning gaussian-process bayesian-optimization semi-supervised-learning few-shot-learning bayesian-logistic-regression


Bayesian ADF unit root test

Bayesian Augmented Dickey-Fuller Unit Root Test

Bayesian ADF test
Bayesian unit root test
Bayesian Dickey-Fuller
BADF

Sims & Uhlig (1991); Koop, Osiewalski & Steel (1992)

The Bayesian Augmented Dickey-Fuller (BADF) unit root test re-frames the classical ADF test within a Bayesian framework. Rather than computing a frequentist p-value, it quantifies evidence for or against a unit root by comparing posterior probabilities or Bayes factors under the null (unit root) and alternative (stationarity) hypotheses, incorporating prior beliefs about the autoregressive parameter.

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

Koop, G., Osiewalski, J., & Steel, M. F. J. (1992). Bayesian analysis of long-run multipliers in cointegrating models. Journal of Econometrics, 54(1–3), 27–44.

augmented-dickey-fuller-unit-root-test phillips-perron-unit-root-test bayesian-ardl-bounds-test bayesian-var-model bayesian-vecm zivot-andrews-structural-break-test


Bayesian Agent-Based Modeling

Bayesian Agent-Based Modeling — Parameter Estimation and Uncertainty Quantification for Agent-Based Models

Bayesian ABM
ABC-ABM
Bayesian Calibration of ABM
Bayesian Agent Simulation

Sunnaker et al. / Grazzini & Richiardi (among key contributors)

Bayesian Agent-Based Modeling integrates Bayesian statistical inference with agent-based simulation to calibrate model parameters and quantify uncertainty. Rather than fixing agent rules and parameters by assumption, this approach treats unknown parameters as probability distributions and updates them systematically against observed data, yielding a full posterior over plausible model configurations.

Simulation calibration and inference framework

Observed aggregate or micro-level data; agent behavior logs

Sunnaker, M., Busetto, A. G., Numminen, E., Corander, J., Foll, M., Dessimoz, C. (2013). Approximate Bayesian Computation. PLOS Computational Biology, 9(1), e1002803.

Grazzini, J., Richiardi, M. (2015). Estimation of agent-based models by simulated minimum distance. Journal of Economic Dynamics and Control, 51, 148-165.

agent-based-modeling approximate-bayesian-computation monte-carlo-simulation bayesian-markov-model stochastic-agent-based-modeling bayesian-microsimulation


Bayesian ANCOVA

Bayesian covariate-adjusted group comparison

2012 (formalized; Bayesian general linear models since 1960s)

Building on Jeffreys (1961) and developed formally for regression/ANCOVA by Rouder & Morey (2012)

Bayesian Analysis of Covariance (Bayesian ANCOVA) extends classical ANCOVA by placing prior distributions on group effects and covariate slopes, then updating them with observed data to obtain posterior distributions and Bayes factors. It quantifies evidence for group differences on a continuous outcome after statistically adjusting for one or more continuous covariates, without relying on p-value thresholds.

Bayesian parametric covariate-adjusted group comparison

Continuous outcome, categorical group factor, continuous covariate(s)

Rouder, J. N., & Morey, R. D. (2012). Default Bayes factors for model selection in regression. Multivariate Behavioral Research, 47(6), 877–903.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2014). Bayesian Data Analysis (3rd ed.). CRC Press.

ancova bayesian-one-way-anova bayesian-mancova bayesian-linear-regression robust-ancova bayesian-manova


Bayesian ANOVA

Bayesian Analysis of Variance with Bayes Factors

Bayesçi ANOVA — Bayes Faktörü ile Grup Karşılaştırması

Bayesian ANOVA, formalised by Rouder, Morey, Speckman and Province (2012), tests whether group means differ by quantifying the evidence for the alternative hypothesis relative to the null using the Bayes Factor (BF₁₀). Unlike classical ANOVA, it can also measure evidence in favour of the null hypothesis, making it equally informative when groups do not differ.

Bayesian hypothesis test / group comparison

Cauchy (r = 0.707, Rouder et al. standard)

Bayes Factor (BF₁₀ / BF₀₁), posterior distributions, credible intervals

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.

bayesian-regression one-way-anova bayes-factor-test mcmc hierarchical-bayes


Bayesian Ant Colony Optimization

Bayesian Ant Colony Optimization — ACO with Bayesian probabilistic parameter learning

BACO
Bayesian ACO
Bayesian-guided ACO
Probabilistic ACO

Dorigo, M. et al. (ACO); Bayesian extensions by multiple researchers in the 2000s–2010s

Bayesian Ant Colony Optimization (BACO) is a hybrid metaheuristic that embeds Bayesian inference into the Ant Colony Optimization framework. By treating pheromone intensities or algorithm parameters as probability distributions updated with collected evidence, BACO improves convergence reliability and robustness compared to classical ACO on noisy or uncertain combinatorial optimization problems.

Metaheuristic with Bayesian probabilistic learning

Combinatorial or continuous optimization problem instances

Dorigo, M., Maniezzo, V., Colorni, A. (1996). Ant system: optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 26(1), 29–41.

Ant colony optimization algorithms. Wikipedia.

ant-colony-optimization bayesian-genetic-algorithm bayesian-particle-swarm-optimization stochastic-ant-colony-optimization multi-objective-ant-colony-optimization bayesian-simulated-annealing


Bayesian AR model

Bayesian autoregressive model
BAR model
Bayesian AR
Bayesian time-series autoregression

Arnold Zellner; foundational Bayesian time-series work by West & Harrison

The Bayesian AR model estimates an autoregressive time-series process by combining a likelihood derived from the AR structure with prior distributions over the lag coefficients and error variance. Rather than producing single point estimates, it yields full posterior distributions, enabling principled uncertainty quantification and probabilistic forecasting.

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

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

autoregressive-model bayesian-var-model bayesian-arma-model bayesian-arima-model arma-model vector-autoregression


Bayesian ARCH model

Bayesian Autoregressive Conditional Heteroskedasticity Model

Bayesian conditional heteroskedasticity model

Robert F. Engle (ARCH, 1982); Bayesian treatment: John Geweke (1989)

The Bayesian ARCH model estimates Engle's Autoregressive Conditional Heteroskedasticity specification within a Bayesian framework. Instead of maximising a likelihood, it combines a prior distribution over the volatility parameters with the data likelihood to obtain a full posterior distribution, providing richer uncertainty quantification than classical maximum-likelihood ARCH.

Volatility model with Bayesian inference

Time series (financial returns, macroeconomic series with time-varying variance)

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987–1007.

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

arch-model garch-model bayesian-garch-model bayesian-egarch bayesian-tgarch dcc-garch-model


Bayesian ARDL Bounds Test

Bayesian Autoregressive Distributed Lag Bounds Test

Bayesian ARDL
Bayesian bounds testing approach
Bayes ARDL cointegration
Bayesian PSS bounds test

Pesaran, Shin & Smith (ARDL framework, 2001); Bayesian adaptation by subsequent literature

The Bayesian ARDL Bounds Test extends the classical Pesaran-Shin-Smith (2001) bounds testing approach to cointegration by embedding it within a Bayesian inferential framework. Instead of relying on frequentist F- and t-statistics with tabulated critical values, the researcher specifies prior distributions on the model parameters and derives posterior evidence of a long-run level relationship between variables that may be integrated of order zero or one.

Time-series (levels and differences of I(0)/I(1) variables)

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.

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

ardl-bounds-test nonlinear-ardl bayesian-vecm bayesian-var-model johansen-cointegration-test engle-granger-cointegration-test


Bayesian ARIMA model

Bayesian Autoregressive Integrated Moving Average Model

Bayesian ARIMA
BARIMA
Bayesian Box-Jenkins model
Bayesian integrated time series model

1970s (ARIMA); Bayesian extension prominent from 1990s

Pole, West & Harrison (Bayesian treatment); Box & Jenkins (ARIMA foundation)

The Bayesian ARIMA model combines the classical Box-Jenkins ARIMA framework with Bayesian inference. Instead of obtaining single point estimates for autoregressive and moving average parameters, it places prior distributions over them and uses observed data to update beliefs into a full posterior distribution, enabling coherent uncertainty quantification and probabilistic forecasting.

Univariate time series (continuous, regularly spaced)

Pole, A., West, M., & Harrison, J. (1994). Applied Bayesian Forecasting and Time Series Analysis. Chapman & Hall.

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

arima-model bayesian-var-model bayesian-sarima-model bayesian-ardl-bounds-test sarima-model vector-autoregression


Bayesian ARMA model

Bayesian Autoregressive Moving Average Model

Bayesian ARMA
B-ARMA
Bayesian autoregressive moving average
ARMA with Bayesian inference

Box & Jenkins (classical ARMA); Bayesian treatment developed through work of Zellner, Geweke, and others in 1970s–1980s

The Bayesian ARMA model applies Bayesian inference to the classical autoregressive moving average framework for stationary univariate time series. Rather than producing single point estimates for the AR and MA parameters, it yields full posterior distributions, naturally incorporating prior knowledge and providing coherent uncertainty quantification over forecasts and impulse responses.

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

arma-model arima-model bayesian-var-model bayesian-arima-model vector-autoregression bayesian-ols


Bayesian Association Rules

Bayesian rule learning
probabilistic association rules
Bayesian itemset mining
BAR

Heckerman, D. et al.; Agrawal, R. & Srikant, R.

Bayesian Association Rules extend classical association rule mining by placing a prior probability distribution over rules and scoring them by their posterior probability given the data. Rather than thresholding on raw support and confidence counts, this Bayesian framework naturally penalises complexity, corrects for multiple comparisons, and produces calibrated probabilistic rule strengths across transactional or categorical datasets.

Transactional / categorical tabular data

Heckerman, D., Geiger, D., & Chickering, D. M. (1995). Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning, 20(3), 197–243.

Agrawal, R., & Srikant, R. (1994). Fast algorithms for mining association rules. In Proceedings of the 20th International Conference on Very Large Data Bases (VLDB), 1215, 487–499.

association-rules apriori-algorithm bayesian-naive-bayes bayesian-gaussian-mixture-model semi-supervised-association-rules fp-growth


Bayesian Autoencoder Anomaly Detection

Bayesian Autoencoder Anomaly Detection (Probabilistic Reconstruction-Error Framework)

probabilistic autoencoder anomaly detection

variational autoencoder anomaly detection

Kingma, D. P. & Welling, M.; applied to anomaly detection by An & Cho

Bayesian Autoencoder Anomaly Detection uses a Variational Autoencoder — a probabilistic generative model trained on normal data — to flag anomalies by their high reconstruction error or low likelihood under the learned distribution. By treating the latent space as a probability distribution rather than a fixed point, it delivers principled uncertainty estimates alongside each anomaly score, making it especially valuable in high-stakes detection tasks.

Probabilistic generative model for unsupervised anomaly detection

Continuous, high-dimensional tabular or image data

Kingma, D. P. & Welling, M. (2014). Auto-Encoding Variational Bayes. Proceedings of the 2nd International Conference on Learning Representations (ICLR 2014).

An, J. & Cho, S. (2015). Variational Autoencoder based Anomaly Detection using Reconstruction Probability. ICDM Workshop on Data Mining in Networks.


Bayesian Bagging

Bayesian Bagging (Bootstrap Aggregation with Bayesian Bootstrap)

Bayesian bootstrap aggregation
BB-ensemble
Bayesian model averaging via bootstrap
Bayesian bagged ensemble

Clyde, M. & Lee, H. (building on Rubin's Bayesian bootstrap, 1981)

Bayesian Bagging replaces the classical bootstrap with the Bayesian bootstrap — drawing Dirichlet-distributed weights over training observations rather than sampling with replacement — and trains an ensemble of base learners under those weights. The result is a principled ensemble that approximates a Bayesian posterior over predictions, yielding calibrated uncertainty estimates alongside strong predictive accuracy.

Ensemble (Bayesian bootstrap aggregation)

Tabular (continuous, categorical, mixed)

Clyde, M. & Lee, H. (2001). Bagging and the Bayesian bootstrap. In T. Richardson & T. Jaakkola (Eds.), Proceedings of the Eighth International Workshop on Artificial Intelligence and Statistics (AISTATS 2001).

Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9(1), 130–134.

random-forest bayesian-random-forest boosting bayesian-boosting voting-ensemble semi-supervised-bagging


Bayesian Betweenness Centrality

Bayesian Betweenness Centrality (Probabilistic Inference of Shortest-Path Centrality)

Brandes, U. (betweenness); Bayesian extension developed by multiple authors (2010s)

Bayesian Betweenness Centrality estimates how often a node lies on shortest paths in a network while explicitly quantifying uncertainty arising from incomplete, sampled, or noisy edge observations. Rather than producing a single point estimate, it yields a posterior distribution over betweenness scores, enabling credible intervals and probabilistic comparisons between nodes.

Probabilistic network centrality measure

Network adjacency data (observed or sampled edges, possibly with uncertainty)

Newman, M.E.J. (2010). Networks: An Introduction. Oxford University Press.

Fortunato, S., Bergstrom, C.T., Borner, K., Evans, J.A., Helbing, D., Milojevi, S., Petersen, A.M., Radicchi, F., Sinatra, R., Uzzi, B., Vespignani, A., Waltman, L., Wang, D. & Barabasi, A.-L. (2018). Science of science. Science, 359(6379), eaao0185.

betweenness-centrality bayesian-social-network-analysis bayesian-degree-centrality bayesian-closeness-centrality weighted-betweenness-centrality exponential-random-graph-model


Bayesian Boosting

Bayesian Boosting (Probabilistic Ensemble Learning)

Bayesian ensemble boosting
probabilistic boosting
Bayesian additive model
Bayesian boosted ensemble

Bayesian boosting integrates probabilistic Bayesian inference with boosting ensemble techniques, combining multiple weak learners while maintaining full uncertainty quantification over predictions. Unlike standard gradient boosting that produces a single point estimate, Bayesian boosting yields a posterior distribution over the ensemble output, enabling calibrated confidence intervals alongside predictions.

Probabilistic ensemble (Bayesian interpretation of boosting)

Ridgeway, G. (1999). The state of boosting. Computing Science and Statistics, 31, 172–181.

Chipman, H. A., George, E. I., & McCulloch, R. E. (2010). BART: Bayesian additive regression trees. Annals of Applied Statistics, 4(1), 266–298.

boosting gradient-boosting bayesian-random-forest xgboost semi-supervised-boosting bayesian-gradient-boosting


Bayesian Bootstrap

Bayesian Bootstrap (Rubin)
Rubin bootstrap
Dirichlet-weighted bootstrap

Rubin (1981); large-sample theory by Lo (1987)

The Bayesian Bootstrap, introduced by Donald B. Rubin in 1981, is a resampling method that produces a Bayesian counterpart to the frequentist bootstrap by assigning each observation a random weight drawn from a Dirichlet distribution. It yields a full posterior distribution for a statistic and allows prior information to be incorporated.

Dirichlet-weighted resampling of the statistic

Rubin, D. B. (1981). The Bayesian Bootstrap. The Annals of Statistics, 9(1), 130-134.

Lo, A. Y. (1987). A Large Sample Study of the Bayesian Bootstrap. The Annals of Statistics, 15(1), 360-375.

bootstrap-inference block-bootstrap randomization-inference permutation-test jackknife


Bayesian Box-Behnken Design

Bayesian Box-Behnken Design for Response Surface Optimization

Bayesian BBD
Bayesian RSM Box-Behnken
Bayesian three-level design
BBD with Bayesian optimization

1960 (BBD); Bayesian integration ~1990s–2000s

Box & Behnken (classical BBD, 1960); Bayesian extension developed by multiple authors in response surface literature

Bayesian Box-Behnken Design combines the classical Box-Behnken three-level design structure with Bayesian statistical inference to fit and optimize response surface models. It uses mid-edge and center points to efficiently estimate a second-order polynomial response surface while incorporating prior knowledge about model parameters and propagating uncertainty through to predictions and optimal factor settings. The approach is widely applied in engineering process optimization and formulation studies.

Bayesian response surface experimental design

Continuous factor-response data from physical or computational experiments

Box, G. E. P., & Behnken, D. W. (1960). Some new three level designs for the study of quantitative variables. Technometrics, 2(4), 455–475.

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

box-behnken-design central-composite-design response-surface-methodology bayesian-optimization d-optimal-design factorial-design


Bayesian Canonical Correlation Analysis

Francis Bach & Michael Jordan (probabilistic formulation, 2005); Klami, Virtanen & Kaski (fully Bayesian treatment, 2013)

Bayesian canonical correlation analysis is a probabilistic generative model that identifies shared latent structure between two or more sets of observed variables. It extends classical CCA by placing priors on model parameters, enabling principled uncertainty quantification, automatic determination of the number of shared dimensions, and robustness when sample sizes are small relative to dimensionality.

Latent variable model / dimensionality reduction

Two or more sets of continuous multivariate observations

Bach, F. R. & Jordan, M. I. (2005). A probabilistic interpretation of canonical correlation analysis. Technical Report 688, Department of Statistics, University of California, Berkeley.

Klami, A., Virtanen, S. & Kaski, S. (2013). Bayesian canonical correlation analysis. Journal of Machine Learning Research, 14, 965-1003.


Bayesian Case-Control Study

Bayesian Case-Control Epidemiological Study

Bayesian logistic regression case-control

1990s–2000s (systematic application); Bayesian inference foundations: Bayes/Laplace 18th–19th c.

Sander Greenland (Bayesian epidemiology formalization); earlier Bayesian logistic methods: Leonard (1972)

A Bayesian case-control study applies Bayesian statistical inference to the classic case-control epidemiological design, formally combining prior knowledge about exposure-disease associations with observed case and control data to estimate posterior odds ratios and credible intervals. Rather than relying solely on observed data, the Bayesian framework allows investigators to incorporate external evidence — from prior studies, expert knowledge, or mechanistic understanding — into the analysis, yielding probability statements about effect sizes that are often more interpretable than classical p-values and confidence intervals.

Observational analytic study with Bayesian inference

Binary outcome (case/control status), exposure variables, prior distributions

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

Gustafson, P. (2004). Measurement Error and Misclassification in Statistics and Epidemiology: Impacts and Bayesian Adjustments. Chapman and Hall/CRC.

case-control-study nested-case-control bayesian-cohort-study bayesian-randomized-clinical-trial matched-case-control-study logistic-regression


Bayesian Case-Crossover Design

Bayesian case-crossover
BCCO
Bayesian self-matched design
Bayesian within-person crossover

1991 (case-crossover); Bayesian extension ~2000s

Malcolm Maclure (case-crossover); Bayesian extension developed by Lumley, Sheppard, and colleagues

The Bayesian case-crossover design is a self-matched epidemiological method that estimates the transient effect of a time-varying exposure on the risk of an acute event. Each case serves as their own control, eliminating confounding by time-stable individual characteristics. Bayesian inference replaces or supplements the classical conditional logistic regression, enabling the incorporation of prior knowledge, more stable estimation in sparse data, and full uncertainty quantification via posterior distributions.

Self-matched observational study design with Bayesian inference

Individual-level time-stamped event data with time-varying exposure measurements

Maclure, M. (1991). The case-crossover design: a method for studying transient effects on the risk of acute events. American Journal of Epidemiology, 133(2), 144–153.

Janes, H., Sheppard, L., & Lumley, T. (2005). Case-crossover analyses of air pollution exposure data: referent selection strategies and their implications for bias. Epidemiology, 16(6), 717–726.

case-crossover-design conditional-logistic-regression bayesian-hierarchical-model time-series-analysis matched-case-control self-controlled-case-series


Bayesian Case Series

Bayesian case-series
BCS analysis
Bayesian self-controlled case series
Bayesian SCCS

1995–2006 (core SCCS method 1995; Bayesian extensions throughout 2000s)

Farrington, Whitaker, and colleagues (self-controlled case series); Bayesian extension by multiple authors in pharmacovigilance

Bayesian case series is an observational epidemiological method that applies Bayesian inference to case series data — typically records of patients who experienced both a drug or vaccine exposure and an adverse health event. By incorporating prior evidence and computing posterior estimates of the incidence rate ratio within pre-specified risk windows, the method quantifies the strength of a temporal association between an exposure and an outcome while controlling for fixed individual-level confounding.

Observational analytical study design with Bayesian inference

Individual-level longitudinal event data (hospitalisation records, adverse event reports, electronic health records)

Strom, B. L. (Ed.). (2001). Pharmacoepidemiology (3rd ed.). Wiley. [Chapter on case series and signal detection]

Whitaker, H. J., Farrington, C. P., Spiessens, B., & Musonda, P. (2006). Tutorial in biostatistics: The self-controlled case series method. Statistics in Medicine, 25(10), 1768–1797.

self-controlled-case-series bayesian-network-meta-analysis case-crossover signal-detection pharmacovigilance cohort-study


Bayesian Causal Impact Analysis

Bayesian Causal Impact Analysis via Structural Time Series

Bayesian structural time series causal inference

Brodersen, Gallusser, Koehler, Remy & Scott (Google)

Bayesian Causal Impact Analysis uses a Bayesian structural time series (BSTS) model to estimate the causal effect of an intervention on a time series outcome. Developed by Brodersen and colleagues at Google in 2015, it builds a probabilistic counterfactual — what the series would have looked like without the intervention — from pre-intervention data and optional control covariates, then compares it with the observed post-intervention values to produce a fully Bayesian posterior over the causal effect.

Univariate or multivariate time series with a clear pre- and post-intervention period

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

Scott, S. L., & Varian, H. R. (2014). Predicting the present with Bayesian structural time series. International Journal of Mathematical Modelling and Numerical Optimisation, 5(1-2), 4-23.

causal-impact-analysis interrupted-time-series difference-in-differences synthetic-control-method bayesian-interrupted-time-series event-study-design


Bayesian Cellular Automata

Bayesian Cellular Automata — Probabilistic calibration of transition rules via Bayesian inference

Probabilistic Cellular Automata (Bayesian)

Multiple contributors (Bayesian calibration of CA emerged in spatial / land-use modeling literature, 2000s–2010s)

Bayesian Cellular Automata (BCA) couples the local-rule spatial dynamics of classical cellular automata with Bayesian inference to learn or calibrate transition probabilities from observed data. Rather than fixing rules by hand, the analyst encodes prior knowledge about how cells change state and updates those beliefs with empirical evidence, producing a posterior distribution over rule parameters that drives principled uncertainty-aware simulation.

Simulation — probabilistic rule inference

Spatial grid / lattice data; prior beliefs; observed state transitions

Hosseinali, F., Alesheikh, A. A., Nourian, F. (2013). Agent-based modeling of urban land-use development, case study: Simulating future scenarios of Qazvin city. Cities, 31, 105-113.

stochastic-cellular-automata monte-carlo-simulation bayesian-markov-model agent-based-cellular-automata bayesian-agent-based-modeling markov-model


Bayesian chi-square test

Bayesian Chi-Square Test of Independence

Bayesian contingency table test
Bayes factor chi-square
Bayesian goodness-of-fit test
Bayesian association test

I. J. Good; extended by Gunel, Dickey, and Wagenmakers et al.

The Bayesian chi-square test evaluates independence or goodness-of-fit in frequency tables using Bayes factors rather than classical p-values. It quantifies evidence for or against an association between categorical variables, updating prior beliefs with observed counts and delivering an odds-like ratio that distinguishes 'no evidence' from 'evidence of no effect'.

Categorical (nominal) frequency counts in a contingency table

Good, I. J. (1967). A Bayesian significance test for multinomial distributions. Journal of the Royal Statistical Society: Series B (Methodological), 29(3), 399–418.

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.

chi-square-test fisher-exact-test bayesian-fishers-exact-test bayesian-independent-samples-t-test bayesian-pearson-correlation goodness-of-fit-test


Bayesian ChIP-seq peak calling

Bayesian Chromatin Immunoprecipitation Sequencing Peak Calling

Bayesian ChIP-seq analysis
probabilistic peak detection
Bayesian peak caller
ChIP-seq Bayesian enrichment calling

Spyrou et al. (BayesPeak, 2009); broader Bayesian ChIP-seq framework developed across multiple groups ~2008–2012

Bayesian ChIP-seq peak calling applies probabilistic models — typically Poisson, negative binomial, or hidden Markov models with Bayesian inference — to detect genomic regions enriched for a protein of interest in chromatin immunoprecipitation followed by sequencing experiments. By explicitly modelling read-count noise and incorporating prior distributions, Bayesian callers yield posterior probabilities of enrichment rather than simple p-values, providing a principled framework for uncertainty quantification across the genome.

Aligned short-read sequencing data (ChIP and matched input/IgG control BAM files)

Zhang, Y., Liu, T., Meyer, C. A., Eeckhoute, J., Johnson, D. S., Bernstein, B. E., Nusbaum, C., Myers, R. M., Brown, M., Li, W., & Liu, X. S. (2008). Model-based analysis of ChIP-Seq (MACS). Genome Biology, 9(9), R137.

Spyrou, C., Stark, R., Lynch, A. G., & Tavare, S. (2009). BayesPeak: Bayesian analysis of ChIP-seq data. BMC Bioinformatics, 10, 299.


Bayesian Cluster Analysis

BCA
Bayesian clustering
probabilistic cluster analysis
Bayesian model-based clustering

Fraley & Raftery (model-based); Dirichlet process formulations by Ferguson (1973) and Antoniak (1974)

Bayesian cluster analysis assigns observations to latent groups by combining a probabilistic model of within-cluster data with prior beliefs about cluster parameters and the number of clusters. It yields posterior probabilities of cluster membership and principled uncertainty estimates, making it more transparent than classical distance-based clustering algorithms.

Continuous, binary, or mixed multivariate data

Fraley, C. & Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97(458), 611–631.

Lau, J. W. & Green, P. J. (2007). Bayesian model-based clustering procedures. Journal of Computational and Graphical Statistics, 16(3), 526–558.

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


Bayesian Co-Kriging

Bayesian Co-Kriging Spatial Interpolation

Bayesian cokriging
Bayesian co-regionalization
BCK
Bayesian multivariate kriging

Gelfand, Banerjee & colleagues; building on Matheron's cokriging framework

Bayesian Co-Kriging is a multivariate geostatistical method that uses auxiliary spatially correlated variables to improve predictions of a primary variable of interest. By placing Bayesian priors on cross-covariance parameters, it propagates all uncertainty — including parameter uncertainty — into the prediction intervals, yielding fully probabilistic maps with calibrated uncertainty bounds.

Georeferenced multivariate continuous data

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

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

co-kriging ordinary-kriging bayesian-kriging bayesian-spatial-regression bayesian-universal-kriging kriging


Bayesian Coarsened Exact Matching

Bayesian Coarsened Exact Matching Estimator

Bayesian monotonic imbalance bounding matching

Iacus, King & Porro (CEM framework, 2012); Bayesian extensions by Hill and subsequent authors

Bayesian Coarsened Exact Matching (Bayesian CEM) combines the coarsening-and-exact-matching framework of Iacus, King, and Porro with Bayesian posterior inference. Covariates are discretised into coarser bins so that treated and control units can be matched exactly within those bins, and Bayesian priors are then placed on the treatment-effect parameters to produce full posterior distributions over the causal estimand rather than a single point estimate.

Quasi-experimental matching with Bayesian inference

Observational cross-sectional or panel data with continuous, binary, or categorical covariates and continuous or binary outcome

Iacus, S. M., King, G., & Porro, G. (2012). Causal Inference without Balance Checking: Coarsened Exact Matching. Political Analysis, 20(1), 1-24.

Hill, J. L. (2011). Bayesian Nonparametric Modeling for Causal Inference. Journal of Computational and Graphical Statistics, 20(1), 217-240.

coarsened-exact-matching bayesian-propensity-score-matching propensity-score-matching entropy-balancing bayesian-matching-estimator matching-estimator


Bayesian Cohort Research

Bayesian cohort study
Bayesian prospective cohort
Bayesian longitudinal cohort analysis
Bayesian follow-up study

Formalised in health research from the 1990s onward

Synthesis of cohort epidemiology (Doll & Hill, 1950s) with Bayesian inference (Bayes, Laplace, Jeffreys)

Bayesian cohort research follows a defined group of individuals over time to track outcomes, and uses Bayesian statistical inference to update beliefs about risk, incidence, or causal effects as follow-up data accumulate. Prior knowledge — from earlier studies, registries, or expert judgment — is formalised into a prior distribution and combined with the cohort's likelihood to yield a posterior distribution that quantifies uncertainty in a directly interpretable way.

Quantitative longitudinal observational design

Time-to-event data, repeated continuous/binary measurements, registry data

Ibrahim, J. G., & Chen, M. H. (2000). Power prior distributions for regression models. Statistical Science, 15(1), 46–60.

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

cohort-research longitudinal-research bayesian-longitudinal-research bayesian-survey-research panel-research survival-analysis


Bayesian Cohort Study

Bayesian longitudinal cohort
Bayesian prospective cohort
Bayesian cohort analysis
Bayesian follow-up study

1990s–2000s (widespread adoption in epidemiology)

Bayesian framework: Thomas Bayes / Pierre-Simon Laplace; applied to cohort epidemiology from the 1990s onward

A Bayesian cohort study follows a defined group of individuals over time to estimate incidence, risk, or rate of outcomes, while using Bayesian statistical inference to incorporate prior knowledge and quantify uncertainty through posterior probability distributions rather than classical p-values and confidence intervals. It combines the longitudinal observational design of a cohort study with the probability-updating logic of Bayesian analysis, allowing richer uncertainty quantification and sequential updating as data accumulate.

Observational longitudinal study with Bayesian inference

Longitudinal individual-level follow-up data (time-to-event, counts, continuous outcomes)

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

cohort-study prospective-cohort-study bayesian-randomized-clinical-trial survival-analysis bayesian-survival-analysis nested-case-control


Bayesian Community Detection

Bayesian Community Detection in Networks

Bayesian stochastic block model community detection

Nowicki, K. & Snijders, T. A. B. (formal Bayesian framing); extended by Peixoto, T. P.

Bayesian community detection infers latent group structure in networks by treating community membership as unobserved variables and using Bayesian inference — typically via Markov chain Monte Carlo or variational methods — to compute a posterior distribution over all plausible partitions. Unlike modularity optimisation, it selects the number of communities from data and provides principled uncertainty estimates for every node assignment.

Probabilistic generative model / inference

Adjacency matrix or edge list (binary or weighted networks)

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

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

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


Bayesian Competing Risks Analysis

Bayesian Competing Risks Survival Analysis

Bayesian cause-specific hazard model
Bayesian subdistribution hazard model
BCRA
Bayesian cumulative incidence analysis

1980s–2000s (classical CR: 1970s; Bayesian extension: 1990s–2000s)

Various; Bayesian formulation advanced by Gelfand, Dey, Larson, and Dinse among others

Bayesian competing risks analysis is a time-to-event method for settings where subjects can fail from more than one mutually exclusive cause — such as death from cancer versus death from cardiovascular disease — and prior knowledge or small-sample uncertainty makes a Bayesian framework advantageous. It extends classical competing risks models (cause-specific hazards and cumulative incidence functions) by placing probability distributions over unknown parameters and updating those distributions with observed data, yielding full posterior inference for each failure type.

Time-to-event data with multiple mutually exclusive failure types (right-censored)

Larson, M. G., & Dinse, G. E. (1985). A mixture model for the regression analysis of competing risks data. Applied Statistics, 34(3), 201–211.

Crowder, M. J. (2001). Classical Competing Risks. Chapman and Hall/CRC.

cause-specific-hazard-model fine-gray-subdistribution-hazard bayesian-survival-analysis cox-proportional-hazards kaplan-meier-estimator multi-state-model


Bayesian Confirmatory Factor Analysis

Bayesian structural equation measurement model

Sik-Yum Lee; Bengt Muthén and Tihomir Asparouhov

Bayesian confirmatory factor analysis tests a pre-specified factor structure using Bayesian inference. Instead of point estimates with p-values, it produces full posterior distributions for loadings, factor correlations, and residual variances, allowing the researcher to incorporate prior knowledge and propagate parameter uncertainty naturally.

Lee, S.-Y. (2007). Structural Equation Modeling: A Bayesian Approach. Wiley.

Muthén, B. & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible representation of substantive theory. Psychological Methods, 17(3), 313–335.

confirmatory-factor-analysis bayesian-exploratory-factor-analysis bayesian-item-response-theory bayesian-measurement-invariance exploratory-factor-analysis bayesian-reliability-analysis


Bayesian Confirmatory Research

Bayesian hypothesis testing
confirmatory Bayesian analysis
Bayes factor hypothesis testing
BCR

1961 (Jeffreys); 2009–2018 (contemporary confirmatory formulation)

Harold Jeffreys (theoretical foundation); Jeffrey Rouder, Eric-Jan Wagenmakers (applied confirmatory framework)

Bayesian confirmatory research is a quantitative framework that tests pre-specified hypotheses by computing the Bayes factor — a ratio expressing how much more likely the observed data are under one hypothesis than another. Unlike classical null-hypothesis significance testing (NHST), it provides direct evidence for both the alternative and the null hypothesis, supports optional stopping rules under certain conditions, and updates prior beliefs with observed data through Bayes' theorem.

Continuous, ordinal, or categorical data from pre-planned confirmatory studies

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.

Wagenmakers, E.-J., Marsman, M., Jamil, T., Ly, A., Verhagen, A. J., Love, J., Selker, R., Gronau, Q. F., Smira, M., Epskamp, S., Matzke, D., Rouder, J. N., & Morey, R. D. (2018). Bayesian inference for psychology. Part I: Theoretical advantages and practical ramifications. Psychonomic Bulletin & Review, 25(1), 35–57.

null-hypothesis-significance-testing bayes-factor-design-analysis sequential-bayes-factor-testing power-analysis meta-analysis pre-registration


Bayesian Conjoint Analysis

Bayesian CA
hierarchical Bayes conjoint
HB conjoint
Bayesian preference modeling

Allenby & Ginter (hierarchical Bayes formulation); conjoint roots in Luce & Tukey (1964)

Bayesian conjoint analysis estimates individual-level consumer preference weights for product attributes by combining conjoint choice tasks with a hierarchical Bayesian model. It yields part-worth utilities for each respondent rather than only group averages, enabling precise market simulation and segment discovery even from small per-person choice sets.

Preference measurement / Bayesian hierarchical model

Choice / ranking / rating data from designed attribute profiles

Allenby, G. M. & Ginter, J. L. (1995). Using extremes to design products and segment markets. Journal of Marketing Research, 32(4), 392–403.

Rossi, P. E., Allenby, G. M. & McCulloch, R. (2005). Bayesian Statistics and Marketing. John Wiley & Sons.

conjoint-analysis latent-class-analysis bayesian-latent-class-analysis mixture-modeling bayesian-mixture-modeling structural-equation-modeling


Bayesian Construct Validity

Bayesian validity analysis
Bayesian CFA-based validity
Bayesian structural validity
posterior construct validity

Cronbach & Meehl (validity framework); Muthén & Asparouhov (Bayesian SEM extension)

Bayesian construct validity assessment uses Bayesian confirmatory factor analysis and related Bayesian structural equation models to evaluate whether a scale or test measures the intended latent construct. It yields full posterior distributions for factor loadings, structural coefficients, and model-fit indices rather than single point estimates, enabling more nuanced and uncertainty-aware validity conclusions.

Validity assessment / Bayesian inference

Cronbach, L. J. & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52(4), 281–302.

construct-validity bayesian-confirmatory-factor-analysis confirmatory-factor-analysis convergent-validity discriminant-validity bayesian-measurement-invariance


Bayesian Control Chart

Bayesian Statistical Process Control Chart

Bayesian SPC chart
Bayesian monitoring chart
posterior control chart
Bayesian Shewhart chart

Formally developed in the 1990s–2000s; roots in Shewhart (1924)

Ulrich Menzefricke and others building on Shewhart (1924) and Bayesian inference (Bayes, 1763)

A Bayesian control chart integrates prior knowledge about a process — such as historical mean and variance — with incoming measurement data to produce dynamically updated control limits. Unlike classical Shewhart charts that fix limits from a Phase-I baseline, Bayesian charts update the posterior distribution of process parameters after each sample, yielding limits that adapt to accumulated evidence and are better calibrated under small sample sizes or non-stationary processes.

Statistical process monitoring / quality control

Continuous or attribute process measurements (time-series)

Menzefricke, U. (2002). On the evaluation of control chart limits based on predictive distributions. Communications in Statistics — Theory and Methods, 31(8), 1423–1440.

control-chart statistical-process-control bayesian-statistical-process-control process-capability-analysis failure-mode-and-effects-analysis six-sigma-dmaic


Bayesian Convergent Validity

Bayesian convergent validity analysis
Bayesian MTMM convergent validity
Bayesian multitrait convergent validity
BCV

Building on Campbell & Fiske (1959) convergent validity; Bayesian extension developed in modern psychometrics literature

Bayesian convergent validity applies Bayesian statistical inference to assess whether different measures of the same construct converge as theory predicts. Rather than a single-point correlation estimate, it yields a full posterior distribution over the convergent correlation, enabling probability statements about the magnitude of shared variance between theoretically related measures.

Ordinal or continuous item scores from multiple constructs or methods

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

Van de Schoot, R., Depaoli, S., King, R., Kramer, B., Märtens, K., Tadesse, M. G., Vannucci, M., Gelman, A., Veen, D., Willemsen, J. & Yau, C. (2021). Bayesian statistics and modelling. Nature Reviews Methods Primers, 1(1), 1.

convergent-validity discriminant-validity bayesian-confirmatory-factor-analysis bayesian-measurement-invariance construct-validity confirmatory-factor-analysis


Bayesian Copy Number Variation Analysis

Bayesian CNV analysis
Bayesian CNV calling
probabilistic CNV detection
Bayesian HMM-CNV

Colella et al. (QuantiSNP); Fridlyand et al. (HMM-based Bayesian CNV)

Bayesian copy number variation (CNV) analysis is a probabilistic framework for detecting genomic segments where an individual's DNA copy count deviates from the diploid norm. By placing prior distributions over copy-number states and updating them with array CGH, SNP array, or sequencing read-depth evidence, the approach yields posterior probabilities for each copy-number state along the genome, providing statistically principled uncertainty quantification that frequentist segmentation methods lack.

Array CGH, SNP array, whole-genome sequencing read-depth data

Colella, S., Yau, C., Taylor, J. M., Mirza, G., Butler, H., Clouston, P., Bassett, A. S., Seller, A., Holmes, C. C., & Ragoussis, J. (2007). QuantiSNP: an Objective Bayes Hidden-Markov Model to detect and accurately map copy number variation using SNP genotyping data. Nucleic Acids Research, 35(6), 2013–2025.

Fridlyand, J., Snijders, A. M., Pinkel, D., Albertson, D. G., & Jain, A. N. (2004). Hidden Markov models approach to the analysis of array CGH data. Journal of Multivariate Analysis, 90(1), 132–153.


Bayesian structural time-series causal inference

2015 (canonical implementation); Rubin potential outcomes: 1974-2005

Brodersen, Gallusser, Koehler, Remy & Scott; Rubin potential outcomes framework

Bayesian Counterfactual Impact Evaluation estimates the causal effect of an intervention by constructing a Bayesian posterior distribution over the counterfactual outcome — what would have happened without treatment. The method, popularized by Brodersen et al. (2015) through the CausalImpact framework, uses Bayesian structural time-series models fitted on the pre-intervention period to predict the counterfactual trajectory, then compares observed post-intervention outcomes to that prediction.

Bayesian causal inference / counterfactual estimation

Time-series or panel data with pre- and post-intervention observations

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

Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association, 100(469), 322-331.

causal-impact-analysis bayesian-interrupted-time-series difference-in-differences synthetic-control-method counterfactual-impact-evaluation bayesian-difference-in-differences


Bayesian Cox Proportional Hazards

Bayesian partial likelihood survival model

1972 (Cox); Bayesian formulation developed through the 1990s

D. R. Cox (frequentist CPH, 1972); Bayesian extensions by Joseph Ibrahim, Ming-Hui Chen, Debajyoti Sinha (1990s–2001)

The Bayesian Cox proportional hazards model combines Cox's classical semiparametric survival regression with Bayesian inference, replacing point estimates and p-values with full posterior distributions over regression coefficients. It handles right-censored time-to-event outcomes, quantifies uncertainty about hazard ratios in probabilistic terms, and allows the incorporation of prior clinical or historical knowledge directly into the analysis.

Bayesian semiparametric survival regression

Time-to-event (survival) data with covariates; right-censored or interval-censored observations

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

Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B, 34(2), 187–220.

cox-proportional-hazards bayesian-survival-analysis survival-analysis kaplan-meier-analysis competing-risks-analysis bayesian-randomized-clinical-trial


Bayesian Cox Regression

Bayesian Cox Proportional Hazards Regression

Bayesian Cox PH model
Bayesian proportional hazards model
Bayesian survival regression
BCox

1972 (Cox PH); 2001 (Bayesian treatment)

Cox (1972) for the base model; Bayesian formulation by Sinha, Chen & Ghosh (1990s); comprehensive treatment by Ibrahim, Chen & Sinha (2001)

Bayesian Cox regression combines the Cox proportional hazards model for time-to-event data with Bayesian inference. Instead of point estimates, it produces full posterior distributions over the hazard ratios, naturally incorporating prior knowledge and providing coherent uncertainty quantification even with small samples or informative censoring.

Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, Series B, 34(2), 187–220.

cox-regression bayesian-survival-regression bayesian-mixed-effects-model survival-regression bayesian-generalized-linear-model zero-inflated-model


Bayesian Cronbach's alpha

Bayesian alpha
Bayesian internal consistency
Bayes-alpha
posterior alpha

2011 (Bayesian form); 1951 (classical alpha)

Padilla & Zhang (Bayesian adaptation); Cronbach (classical alpha, 1951)

Bayesian Cronbach's alpha applies Bayesian inference to estimate the classical internal-consistency coefficient, yielding a full posterior distribution over alpha rather than a single point estimate. This allows researchers to quantify uncertainty with credible intervals and incorporate prior knowledge, making reliability assessment more informative — especially with small or skewed samples.

Ordinal or continuous item scores from a single test administration

Padilla, M. A., & Zhang, G. (2011). Estimating internal consistency using Bayesian methods. Journal of Modern Applied Statistical Methods, 10(1), 277–286.

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.

cronbachs-alpha mcdonalds-omega bayesian-reliability-analysis bayesian-confirmatory-factor-analysis bayesian-item-response-theory generalizability-theory


Bayesian cross-tabulation analysis

Bayesian chi-square test
Bayesian contingency table test
Bayes factor association test
Bayesian crosstab analysis

Bayesian cross-tabulation analysis tests whether two categorical variables are associated by computing a Bayes factor that quantifies the evidence for an association model against an independence model. Unlike classical chi-square testing, it provides a continuous measure of evidence, supports the null hypothesis directly, and updates naturally with prior knowledge about the cell probabilities.

Categorical (nominal/ordinal) counts in contingency tables

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.

cross-tabulation-analysis bayesian-chi-square-test bayesian-fisher-exact-test bayesian-independent-samples-t-test fisher-exact-test chi-square-test


Bayesian DCC-GARCH

Bayesian Dynamic Conditional Correlation GARCH Model

Engle (2002) for DCC; Bayesian extension via MCMC literature (2000s onwards)

Bayesian DCC-GARCH estimates time-varying correlations across multiple financial or economic series by combining Engle's DCC-GARCH structure with Bayesian inference. Rather than maximising a likelihood, it places prior distributions over all parameters and uses Markov Chain Monte Carlo (MCMC) sampling to produce full posterior distributions, yielding richer uncertainty quantification than classical DCC-GARCH.

Multivariate time series of financial or macroeconomic returns

Engle, R. F. (2002). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics, 20(3), 339-350.

Virbickaite, A., Ausin, M. C., & Galeano, P. (2015). Bayesian inference methods for univariate and multivariate GARCH models: A survey. Journal of Economic Surveys, 29(1), 76-96.

dcc-garch-model bayesian-garch-model bayesian-egarch bayesian-tgarch vector-autoregression bayesian-var-model


Bayesian Decision Tree

Bayesian CART
BCART
Bayesian tree induction
probabilistic decision tree

Chipman, H. A.; George, E. I.; McCulloch, R. E.

Bayesian Decision Tree (Bayesian CART) places a prior distribution over tree structures and leaf parameters, then uses Markov chain Monte Carlo to explore the posterior distribution of trees given data. Instead of a single best tree, it produces a distribution of plausible trees whose predictions are averaged, yielding calibrated uncertainty estimates alongside point predictions.

Tabular (continuous and categorical features)

Chipman, H. A., George, E. I., & McCulloch, R. E. (1998). Bayesian CART model search. Journal of the American Statistical Association, 93(443), 935–948.

Denison, D. G. T., Mallick, B. K., & Smith, A. F. M. (1998). A Bayesian CART algorithm. Biometrika, 85(2), 363–377.

decision-tree random-forest bayesian-random-forest gaussian-process bayesian-gradient-boosting regularized-decision-tree


Bayesian descriptive statistics

Bayesian summaries
posterior descriptives
Bayesian parameter estimation
credible-interval summaries

Bayesian descriptive statistics summarizes data by combining observed information with prior knowledge through Bayes' theorem, yielding posterior distributions over parameters such as the mean and variance. Instead of point estimates and p-values, results are expressed as posterior means, medians, and credible intervals that carry a direct probability interpretation.

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

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

bayesian-chi-square-test bayesian-independent-samples-t-test bayesian-pearson-correlation robust-descriptive-statistics power-analysis effect-size-analysis


Bayesian Design of Experiments

Bayesian DOE
Bayesian optimal design
Bayesian experimental design
BDE

1956 (foundational); formalized 1970s–1990s

Lindley (1956); Chaloner & Verdinelli (1995) landmark review

Bayesian design of experiments selects experimental runs by maximising a utility function — typically the expected information gain — computed over prior beliefs about model parameters. Unlike classical design, which optimizes algebraic criteria such as D-optimality under fixed assumptions, Bayesian DOE incorporates prior knowledge and uncertainty about the system, yielding designs that are optimal in expectation across all plausible parameter values.

Continuous and categorical experimental factor settings; prior distributions over model parameters

Chaloner, K., & Verdinelli, I. (1995). Bayesian Experimental Design: A Review. Statistical Science, 10(3), 273–304.

Ryan, E. G., Drovandi, C. C., McGree, J. M., & Pettitt, A. N. (2016). A Review of Modern Computational Algorithms for Bayesian Optimal Design. International Statistical Review, 84(1), 128–154.

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


Bayesian Diagnostic Accuracy Study

Bayesian DTA study
Bayesian test evaluation
Bayesian diagnostic test accuracy
BDAS

Joseph, Gyorkos & Coupal; Dendukuri & Joseph (formal Bayesian DTA framework)

A Bayesian diagnostic accuracy study evaluates how well a medical test distinguishes between people who have a condition and those who do not, using Bayesian statistical methods that formally incorporate prior knowledge into the estimation of sensitivity, specificity, and related measures. Unlike classical approaches that rely solely on the observed sample, Bayesian inference combines a likelihood model of the data with prior probability distributions to produce posterior estimates with intuitive credible intervals.

Binary or ordinal test results, reference-standard outcomes, prior information

Dendukuri, N., & Joseph, L. (2001). Bayesian approaches to modeling the conditional dependence between multiple diagnostic tests. Biometrics, 57(1), 158–167.

Gatsonis, C., & Paliwal, P. (2006). Meta-analysis of diagnostic and screening test accuracy evaluations: Methodologic primer. American Journal of Roentgenology, 187(2), 271–281.

diagnostic-accuracy-study bayesian-randomized-clinical-trial bayesian-cohort-study screening-test-evaluation meta-analytic-diagnostic-accuracy-study bayesian-case-control-study


Bayesian Difference GMM

Bayesian Difference Generalized Method of Moments

Bayesian Arellano-Bond estimator
Bayesian difference GMM
quasi-Bayesian difference GMM
Bayesian first-difference GMM

Arellano & Bond (1991) for Difference GMM; Chernozhukov & Hong (2003) for Bayesian GMM framework

Bayesian Difference GMM combines the Arellano-Bond first-differencing strategy for dynamic panel data with a Bayesian inference framework. By treating the GMM moment conditions as a quasi-likelihood and placing priors on parameters, the approach produces a full posterior distribution over coefficients rather than a single point estimate with asymptotic standard errors.

Balanced or unbalanced panel data with a lagged dependent variable

Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Economic Studies, 58(2), 277-297.

Chernozhukov, V., & Hong, H. (2003). An MCMC approach to classical estimation. Journal of Econometrics, 115(2), 293-346.

difference-gmm bayesian-system-gmm bayesian-arellano-bond-gmm system-gmm bayesian-dynamic-panel-data-model panel-dynamic-panel-data-model


Bayesian Difference-in-Differences

Bayesian DiD
Bayes DiD
Bayesian diff-in-diff
Bayesian panel causal estimator

Li & Marchand (formal Bayesian DiD framework); Brodersen et al. (Bayesian causal inference in time series)

Bayesian Difference-in-Differences applies Bayesian statistical inference to the classic DiD design, replacing frequentist point estimates with full posterior distributions over the treatment effect. This yields not only an estimate of the causal effect but also a coherent probability statement about its magnitude and uncertainty, making it especially useful when sample sizes are modest or informative prior knowledge is available.

Bayesian causal inference / panel regression

Panel data or repeated cross-sections with pre/post periods

Li, F., & Marchand, J. (2023). Bayesian inference for difference-in-differences. Econometrics Journal, 26(3), 509-529.

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

difference-in-differences dynamic-difference-in-differences bayesian-interrupted-time-series synthetic-control-method panel-fixed-effects causal-impact-analysis


Bayesian Differential Item Functioning

Bayesian Differential Item Functioning Analysis

Bayesian DIF
Bayesian DIF analysis
Bayesian item bias detection
BDIF

H. Swaminathan & H. J. Rogers (classical DIF); Bayesian extensions developed through Markov chain Monte Carlo IRT methods in the 1990s–2000s

Bayesian differential item functioning analysis detects whether a test item behaves differently across demographic or cultural groups — such as males vs. females — after accounting for the underlying ability or trait being measured. It applies Bayesian IRT estimation to obtain posterior distributions of item parameters separately per group, then evaluates group differences with posterior credibility intervals or Bayes factors rather than classical p-values.

Item bias detection / Bayesian inference

Ordinal or binary item responses across two or more groups

Swaminathan, H., & Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27(4), 361–370.

Bolt, D. M. (2002). A Monte Carlo comparison of parametric and nonparametric polytomous DIF detection methods. Applied Measurement in Education, 15(2), 113–141.


Bayesian Discrete-Event Simulation

Bayesian Discrete-Event Simulation — Posterior-informed stochastic process modeling

Developed across operations research and Bayesian statistics communities; prominently formalized in health economic simulation in the 2000s–2010s

Bayesian Discrete-Event Simulation (BDES) integrates Bayesian statistical inference with discrete-event simulation. Prior beliefs about system parameters — such as service rates, arrival times, or failure probabilities — are updated with observed data via Bayes' theorem, and the resulting posterior distributions directly drive the simulation engine. This coupling allows modelers to propagate both aleatory and epistemic uncertainty through event-driven process models.

Event logs, time-stamped process records, count and duration data

Onggo, B. S., & Kunc, M. (2016). Combining discrete-event simulation and Bayesian updating for incorporating evidence from real-world data. Journal of Simulation, 10(1), 1-12.

Pidd, M. (2004). Computer Simulation in Management Science (5th ed.). Wiley.

discrete-event-simulation bayesian-markov-model monte-carlo-simulation stochastic-discrete-event-simulation agent-based-discrete-event-simulation bayesian-agent-based-modeling


Bayesian Discriminant Analysis

Bayesian quadratic discriminant analysis

Bayesian discriminant analysis assigns observations to predefined groups by combining a multivariate Gaussian likelihood for each class with prior distributions over the class means and covariance matrices. Posterior predictive probabilities replace point-estimate decision boundaries, providing principled uncertainty quantification for classification in small or high-dimensional samples.

Supervised classification / Bayesian inference

Continuous multivariate observations with group labels

Geisser, S. (1964). Posterior odds for multivariate normal classifications. Journal of the Royal Statistical Society, Series B, 26(1), 69–76.

Minka, T. P. (2000). Bayesian linear regression. Technical Report, MIT Media Lab.

discriminant-analysis bayesian-cluster-analysis bayesian-confirmatory-factor-analysis bayesian-structural-equation-modeling bayesian-latent-class-analysis principal-component-analysis


Bayesian Discriminant Validity

Bayesian Discriminant Validity Assessment

Bayesian HTMT
Bayesian HTMTb
Bayesian discriminant evidence
Bayesian CFA discriminant validity

2020 (Bayesian HTMT formalization); 1959 (discriminant validity concept)

Adaptation of Campbell & Fiske (1959) discriminant validity into Bayesian CFA framework; Bayesian HTMT formalization by Garnier-Villarreal & Jorgensen (2020)

Bayesian discriminant validity assessment evaluates whether two theoretically distinct latent constructs are empirically separable, using posterior distributions and credible intervals rather than single-point null-hypothesis tests. It is applied within Bayesian confirmatory factor analysis or via the Bayesian heterotrait-monotrait ratio (HTMTb) to determine whether constructs measuring different traits are sufficiently differentiated.

Ordinal or continuous Likert-type item scores from multi-factor questionnaires

Garnier-Villarreal, M. & Jorgensen, T. D. (2020). Adapting fit indices for Bayesian structural equation modeling: Comparison to maximum likelihood. Psychological Methods, 25(1), 46–70.

Campbell, D. T. & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56(2), 81–105.

discriminant-validity convergent-validity bayesian-confirmatory-factor-analysis confirmatory-factor-analysis construct-validity bayesian-measurement-invariance


Bayesian Dose-Response Analysis

Bayesian DRA
Bayesian dose-response modeling
Bayesian benchmark dose analysis
BDR

Developed from classical frequentist dose-response traditions; Bayesian formulations advanced by Dempster, Gelman, and colleagues

Bayesian dose-response analysis models the relationship between the level of exposure (dose) to a substance and the magnitude or probability of a biological response, embedding that model in a Bayesian probabilistic framework. Unlike frequentist approaches that yield a single point estimate with confidence intervals, the Bayesian framework produces a full posterior distribution over model parameters, allowing explicit quantification of uncertainty, incorporation of prior scientific knowledge, and principled model averaging. It is widely applied in toxicology, pharmacology, environmental risk assessment, and clinical dose-finding studies.

Dose (exposure) levels and quantitative or binary response 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.

Piegorsch, W. W., & Bailer, A. J. (2005). Analyzing Environmental Data. Wiley.

benchmark-dose-modeling pharmacokinetic-modeling meta-analysis bayesian-network survival-analysis logistic-regression


Bayesian Doubly Robust Estimation

Bayesian Doubly Robust Estimation of Average Treatment Effects

Bayesian augmented inverse probability weighting

Bayesian semiparametric causal estimation

Bang & Robins (2005); Bayesian extensions by Scharfstein, Kennedy, and others

Bayesian Doubly Robust Estimation combines the classical doubly robust (DR) augmented inverse probability weighting framework with Bayesian inference. It simultaneously models the propensity score and the outcome regression, placing prior distributions over both, and derives a posterior distribution over the average treatment effect that remains consistent even if one of the two component models is misspecified.

Semiparametric causal estimation with Bayesian inference

Observational panel or cross-sectional data with binary or continuous treatment and outcome

Bang, H., & Robins, J. M. (2005). Doubly robust estimation in missing data and causal inference models. Biometrics, 61(4), 962-973.

Scharfstein, D., Nabi, R., Kennedy, E. H., Huang, M.-Y., Bonvini, M., & Smid, M. (2021). Semiparametric sensitivity analysis: Unmeasured confounding in observational studies. arXiv:1910.14694.

doubly-robust-estimation bayesian-propensity-score-matching inverse-probability-weighting marginal-structural-model targeted-maximum-likelihood-estimation bayesian-causal-impact-analysis


Bayesian Dynamic Panel Data Model

Bayesian lagged dependent variable panel model

Hsiao, Pesaran, Tahmiscioglu; Arellano & Bonhomme

The Bayesian dynamic panel data model extends standard dynamic panel models — which include a lagged dependent variable to capture state dependence — by estimating all parameters within a Bayesian framework. Prior distributions are combined with the likelihood to yield a full posterior distribution over model parameters, enabling probabilistic inference and coherent uncertainty quantification even in short panels.

Balanced or unbalanced panel data with lagged dependent variable

Hsiao, C., Pesaran, M. H., & Tahmiscioglu, A. K. (2002). Maximum likelihood estimation of fixed effects dynamic panel data models covering short time periods. Journal of Econometrics, 109(1), 107–150.

Arellano, M., & Bonhomme, S. (2007). Robust priors in nonlinear panel data models. Econometrica, 77(2), 489–536.

dynamic-panel-data-model bayesian-panel-data-analysis arellano-bond-gmm-estimator bayesian-var-model panel-fixed-effects-model bayesian-random-effects-model


Bayesian Dynamic Programming

Bayesian Dynamic Programming — Sequential decision optimization under uncertainty with Bayesian belief updating

BDP
Bayesian DP
Bayesian sequential optimization
Bayesian stochastic control

1957 (Bellman DP); Bayesian extensions 1990s–2000s

Bellman, R.; extended by Bayesian frameworks (Duff, Bertsekas)

Bayesian Dynamic Programming (BDP) combines Bellman's dynamic programming framework with Bayesian inference to optimize sequential decisions when transition probabilities or reward structures are unknown. At each stage, the agent updates beliefs about the environment using observed outcomes, then computes an optimal policy that explicitly accounts for both immediate rewards and the value of information gained through exploration.

Sequential optimization with Bayesian belief updating

Sequential decision data, prior distributions, observed outcomes

Bertsekas, D. P. (1995). Dynamic Programming and Optimal Control. Athena Scientific, Belmont, MA.

Duff, M. O. (2002). Optimal Learning: Computational procedures for Bayes-adaptive Markov decision processes. PhD Dissertation, University of Massachusetts Amherst.

dynamic-programming stochastic-dynamic-programming markov-decision-process bayesian-markov-model reinforcement-learning stochastic-monte-carlo-simulation


Bayesian Ecological Study

Bayesian ecological analysis
Bayesian disease mapping
Bayesian ecological regression
Bayesian spatial ecological study

1991–2000s (Besag 1991 for spatial priors; Lawson 2001 for disease mapping framework)

Andrew Lawson; Julian Besag (spatial Bayesian foundations)

A Bayesian ecological study combines the group-level observational design of classical ecological epidemiology with Bayesian hierarchical modelling. Rather than treating disease rates as fixed quantities, it places prior distributions over latent spatial or temporal effects — commonly using the Besag-York-Mollié (BYM) convolution prior — and updates beliefs from aggregate data to produce posterior maps of disease risk, smoothed rate estimates, and credible intervals for ecological associations between exposures and outcomes.

Observational epidemiological design with Bayesian statistical framework

Aggregate (group-level) count or rate data; spatial or temporal covariate data

Lawson, A. B. (2013). Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology (2nd ed.). CRC Press.

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

ecological-study bayesian-survival-analysis bayesian-cohort-study multilevel-modeling spatial-analysis disease-mapping


Bayesian EGARCH

Bayesian Exponential Generalized Autoregressive Conditional Heteroscedasticity Model

Bayesian EGARCH model
Bayesian Exponential GARCH
EGARCH with Bayesian estimation
B-EGARCH

1991 (EGARCH); 2000s (Bayesian estimation)

Nelson (1991) for EGARCH; Bayesian inference via MCMC developed from early 2000s

The Bayesian EGARCH model combines Nelson's (1991) Exponential GARCH specification — which models the log of conditional variance and captures the leverage effect — with Bayesian posterior inference via Markov Chain Monte Carlo (MCMC). This allows full uncertainty quantification of all volatility parameters, including the asymmetry coefficient, without requiring large-sample normality of the estimates.

Volatility model with Bayesian inference

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347–370.

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

egarch-model bayesian-garch-model bayesian-tgarch bayesian-dcc-garch arch-model bayesian-var-model


Bayesian Ego Network Analysis

Bayesian Ego Network Analysis (Probabilistic Inference on Personal Networks)

Bayesian personal network analysis
Bayesian egocentric network analysis
probabilistic ego network modeling
Bayesian egonet

Various (Bayesian SNA tradition; Krivitsky, Kolaczyk, Handcock among key contributors)

Bayesian ego network analysis applies probabilistic inference to ego-centered (personal) network data, combining a likelihood model for the ego's local network with prior distributions over network parameters. The result is a full posterior distribution that quantifies uncertainty about structural features such as alter composition, tie density, and network size — rather than producing point estimates alone.

Ego-centered network data (alter lists, tie attributes, composition variables)

Krivitsky, P. N., & Kolaczyk, E. D. (2015). On the question of effective sample size in network modeling: An asymptotic inquiry. Statistical Science, 30(2), 184–198.

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


Bayesian Entropy Balancing

Bayesian Entropy Balancing for Causal Inference

entropy balancing with Bayesian inference

Hainmueller (2012, entropy balancing foundation); Bayesian extension developed in subsequent causal inference literature

Bayesian Entropy Balancing extends the classical entropy balancing approach — which reweights control units so that their covariate moments match the treated group exactly — by embedding this reweighting within a Bayesian framework. This allows researchers to incorporate prior beliefs about treatment propensities, propagate parameter uncertainty into the final causal estimate, and obtain credible intervals rather than only classical confidence intervals.

Weighting-based causal estimator with Bayesian uncertainty quantification

Observational cross-sectional or panel data with binary or multi-valued treatment

Hainmueller, J. (2012). Entropy balancing for causal effects: A multivariate reweighting method to produce balanced samples in observational studies. Political Analysis, 20(1), 25-46.

Vegetabile, B. G., Griffin, B. A., Coffman, D. L., Cefalu, M., Robbins, M. W., & McCaffrey, D. F. (2021). Nonparametric estimation of population average dose-response curves using entropy balancing weights for continuous exposures. Health Services and Outcomes Research Methodology, 21(1), 69-110.

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


Bayesian epigenome-wide association study in educational research

Bayesian Epigenome-Wide Association Study Applied to Educational Research Outcomes

Bayesian methylation-wide association study

EWAS framework ~2010–2011; Bayesian EWAS variants ~2013–2017; educational applications ~2015–present

Rakyan, Down, Balding, and Beck (conceptual EWAS framework); Bayesian extensions by multiple groups including Teschendorff and colleagues

A Bayesian epigenome-wide association study (Bayesian EWAS) scans hundreds of thousands of DNA methylation sites across the genome to identify those statistically associated with an educational outcome — such as cognitive ability, attainment, or socioeconomic exposure during schooling. Unlike classical frequentist EWAS, the Bayesian framework incorporates prior biological knowledge to compute posterior probabilities of association, improving power and reducing false discoveries when applied to complex educational phenotypes.

Genomic association study with Bayesian inference

DNA methylation array data (e.g., Illumina 450K or EPIC array), phenotypic/educational outcome variables

Rakyan, V. K., Down, T. A., Balding, D. J., & Beck, S. (2011). Epigenome-wide association studies for common human diseases. Nature Reviews Genetics, 12(8), 529–541.

Ligthart, S., Marzi, C., Aslibekyan, S., Mendelson, M. M., Conneely, K. N., Tanaka, T., ... & Dehghan, A. (2016). DNA methylation signatures of chronic low-grade inflammation are associated with complex diseases. Genome Biology, 17(1), 255.

genome-wide-association-study dna-methylation-analysis epigenetic-clock mendelian-randomization bayesian-network-analysis mediation-analysis