Modeling Expert Judgments for Bayesian Updating
Genest, C.; Schervish, M. J.. 1985. The Annals of Statistics, 13(3), 1198–1212.
Bayesian predictive synthesis: tutorial
Bayesian predictive synthesis combines predictive distributions from models or agents in a Bayesian analysis. The synthesis model represents the relationship between the outcome and the agents’ forecasts, allowing for biases, calibration, and dependencies. Dynamic models allow these relationships to change over time.
The collection includes papers by Ken McAlinn and other authors, with links to papers and code. Related work includes Bayesian causal synthesis, predictive decision synthesis, and the foundations of model combination.
Reading
The foundations develop Bayesian analysis of expert opinion and predictive distributions. Dynamic Bayesian predictive synthesis in time series develops the framework for forecasting; the multivariate paper extends the analysis to several outcomes.
The causal inference and decision synthesis papers extend the framework to treatment effects and decisions. The extensions and applications develop predictive synthesis for different data and decision problems. The code provides examples of the models and computations.
Foundations
Modelling Agent Forecast Distributions
West, M.. 1992. Journal of the Royal Statistical Society: Series B (Methodological), 54(2), 553–567.
Modelling Probabilistic Agent Opinion
West, M.; Crosse, J.. 1992. Journal of the Royal Statistical Society: Series B (Methodological), 54(1), 285–299.
Dynamic Bayesian predictive synthesis in time series forecasting
McAlinn, K.; West, M. 2019. Journal of Econometrics, 210(1), 155–169.
Dynamic models
Multivariate Bayesian Predictive Synthesis in Macroeconomic Forecasting
McAlinn, K.; Aastveit, K. A.; Nakajima, J.; West, M. 2020. Journal of the American Statistical Association, 115(531), 1092–1110.
Mixed-Frequency Bayesian Predictive Synthesis for Economic Nowcasting
McAlinn, K. 2021. Journal of the Royal Statistical Society Series C: Applied Statistics, 70(5), 1143–1163.
Earlier title: Dynamic Mixed Frequency Synthesis for Economic Nowcasting.
Sequential Bayesian Predictive Synthesis
Masuda, R.; Irie, K.. 2023. Working paper; arXiv:2308.15910.
Theory and computation
Equivariant online predictions of non-stationary time series
Takanashi, K.; McAlinn, K. 2023. Working paper; arXiv:1911.08662 (first posted 2019; revised 2023).
On the Proofs of the Predictive Synthesis Formula
Masuda, R.; Irie, K.. 2024. Working paper; arXiv:2409.09660.
Predictive Concordance for Parameter Optimisation and Mixture Synthesis
Adrian, T.; Giannone, D.; Luciani, M.; West, M.. 2026. Working paper; arXiv:2606.14382.
Causal inference
Bayesian Causal Synthesis for Meta-Inference on Heterogeneous Treatment Effects
Sugasawa, S.; Takanashi, K.; McAlinn, K.; Airoldi, E. M. 2026. Working paper.
Ensemble Doubly Robust Bayesian Inference via Regression Synthesis
Babasaki, K.; Sugasawa, S.; Takanashi, K.; McAlinn, K. Observational Studies (accepted for publication).
From Trial Averages to Target Effects: Bayesian CATE Synthesis with Randomized and Observational Evidence
Ogawa, H.; Sugasawa, S.; Takanashi, K.; McAlinn, K. 2026. Working paper.
Decision synthesis
Bayesian predictive decision synthesis
Tallman, E.; West, M.. 2024. Journal of the Royal Statistical Society Series B: Statistical Methodology, 86(2), 340–363.
General Bayesian Predictive Synthesis
Kato, M.. 2024. Working paper; arXiv:2406.09254.
Decision synthesis in monetary policy
Chernis, T.; Koop, G.; Tallman, E.; West, M.. 2025. Working paper; Bank of Canada Staff Working Paper 2024-30; arXiv:2406.03321, revised 2025.
Predictive Decision Synthesis for Portfolios: Betting on Better Models
Tallman, E.; West, M.. 2025. Recent Developments in Bayesian Econometrics and Their Applications, 223–249.
Extensions and applications
Economics
Adaptive Variable Selection for Sequential Prediction in Multivariate Dynamic Models
Lavine, I.; Lindon, M.; West, M.. 2021. Bayesian Analysis, 16, 1059–1083.
Quantifying Time-Varying Forecast Uncertainty and Risk for the Real Price of Oil
Aastveit, K. A.; Cross, J. L.; van Dijk, H. K.. 2023. Journal of Business & Economic Statistics, 41(2), 523–537.
Predictive Density Combination Using Bayesian Machine Learning
Chernis, T.; Hauzenberger, N.; Huber, F.; Koop, G.; Mitchell, J.. 2025. International Economic Review, 66(3), 1287–1315.
PaperarXivData 1Data and code 2Google Scholar
Earlier title: Predictive Density Combination Using a Tree-Based Synthesis Function.
Scenario Synthesis and Macroeconomic Risk
Adrian, T.; Giannone, D.; Luciani, M.; West, M.. 2025. Working paper; Finance and Economics Discussion Series 2025-036; IMF Working Paper 2025/105; CEPR Discussion Paper20219.
Dynamic Bayesian regression quantile synthesis for forecasting outlook-at-risk
Kobayashi, G.; Sugasawa, S.; Yamauchi, Y.; Han, D.. 2026. Working paper; arXiv:2603.11474.
Predictive Synthesis under Sporadic Participation: Evidence from Inflation Density Surveys
Johnson, M. C.; Luciani, M.; Zhang, M.; McAlinn, K. 2026. Working paper.
Risks and Uncertainty in Monetary Policy
Adrian, T.; Giannone, D.; Luciani, M.; West, M.. 2026. Working paper; Finance and Economics Discussion Series 2026-061.
Finance
Divide and Conquer: Financial Ratios and Industry Returns Predictability
Bianchi, D.; McAlinn, K. 2021. Working paper.
SSRNEarlier arXiv versionGoogle Scholar
Earlier title: Large-Scale Dynamic Predictive Regressions.
Bayesian Portfolio Optimization by Predictive Synthesis
Kato, M.; Baba, K.; Kaibuchi, H.; Inokuchi, R.. 2025. Working paper; arXiv:2510.07180.
Bayesian Predictive Synthesis with Outcome-Dependent Pools
Johnson, M. C.; West, M.. 2025. Statistical Science, 40(1), 109–127.
Health
Predicting COVID-19 hospitalisation using a mixture of Bayesian predictive syntheses
Kobayashi, G.; Sugasawa, S.; Kawakubo, Y.; Han, D.; Choi, T.. 2024. The Annals of Applied Statistics, 18(4), 3383–3404.
Real estate and ecology
Bayesian Spatial Predictive Synthesis
Cabel, D.; Sugasawa, S.; Kato, M.; Takanashi, K.; McAlinn, K. Bayesian Analysis (accepted for publication).
Others
Combining Large Numbers of Density Predictions with Bayesian Predictive Synthesis
Chernis, T.. 2024. Studies in Nonlinear Dynamics & Econometrics, 28(2), 293–317.
Bayesian Predictive Synthesis for Dynamic Networks: Forecasting and Identifying Structural Mechanisms
Papamichalis, M.; Ruane, R.; Papamichalis, T.. 2026. Working paper; arXiv:2606.26136.
Graphon-Level Bayesian Predictive Synthesis for Random Network
Papamichalis, M.; Ruane, R.. 2026. Working paper; arXiv:2512.18587; first posted 2025, revised 2026.
Review and related work
Bayesian Predictive Synthesis: Forecast Calibration and Combination
Johnson, M. C.. 2017. Doctoral thesis, Duke University.
Dynamic Modeling and Bayesian Predictive Synthesis
McAlinn, K. 2017. Doctoral thesis, Duke University.
Discussion of “Using Stacking to Average Bayesian Predictive Distributions,” by Yao et al.
McAlinn, K.; Aastveit, K. A.; West, M. 2018. Bayesian Analysis, 13(3), 971–973.
Bayesian forecasting of multivariate time series: scalability, structure uncertainty and decisions
West, M.. 2020. Annals of the Institute of Statistical Mathematics, 72, 1–31.
Bayesian Predictive Decision Synthesis: Methodology and Applications
Tallman, E. E.. 2024. Doctoral thesis, Duke University.
Bayesian forecasting in economics and finance: A modern review
Martin, G. M.; Frazier, D. T.; Maneesoonthorn, W.; Loaiza-Maya, R.; Huber, F.; Koop, G.; Maheu, J.; Nibbering, D.; Panagiotelis, A.. 2024. International Journal of Forecasting, 40(2), 811–839.
Ensembling under Statistical Pluralism: A Taxonomy of Formulations, Diagnostics, and Reporting
McAlinn, K. 2026. Working paper.
Code and data
Code links are given where available. A paper without a code link may have additional materials available from its authors.
BPS (MATLAB)
Multivariate Bayesian Predictive Synthesis in Macroeconomic Forecasting; Dynamic Bayesian predictive synthesis in time series forecasting
Multivariate-Bayesian-Predictive-Synthesis-in-Macroeconomic-Forecasting (MATLAB)
Multivariate Bayesian Predictive Synthesis in Macroeconomic Forecasting
BSPS (R)
BCS (R)
Bayesian Causal Synthesis for Meta-Inference on Heterogeneous Treatment Effects
Predictive Concordance for Parameter Optimisation and Mixture Synthesis
dynbps — independent implementation (Python)
Dynamic Bayesian predictive synthesis in time series forecasting
chhkm-ier-bps (Data)
Predictive Density Combination Using Bayesian Machine Learning
Data and code package for Predictive Density Combination Using Bayesian Machine Learning (Data and code)
Predictive Density Combination Using Bayesian Machine Learning
BPDS-portfolio (Python)
Bayesian predictive decision synthesis; Predictive Decision Synthesis for Portfolios: Betting on Better Models
Adaptive-Variable-Selection (R)
Adaptive Variable Selection for Sequential Prediction in Multivariate Dynamic Models
Updated: October 2026.