Bayesian Inference and Model Misspecification

Bayesian inference offers a principled mechanism for updating beliefs by combining prior distributions with observed data via Bayes’s theorem. Central to this approach is the specification of a statistical model that encapsulates the data-generating process through a likelihood function. When this model is misspecified—owing to incorrect functional forms, omitted variables or unrealistic assumptions—the posterior distribution may concentrate around parameter values that minimise Kullback–Leibler divergence rather than recover the true mechanism. Such misspecification can manifest as inconsistency, overconfidence or phenomena like hypercompression, where predictive intervals become misleadingly narrow even as predictive accuracy falters. To address these challenges, recent theoretical and methodological advances extend Bayesian updating beyond classical likelihoods, employing loss-based or tempered likelihood frameworks that introduce a learning rate to temper data influence, and using diagnostic measures to calibrate uncertainties post hoc. Modularisation strategies further isolate components suspected of misspecification, while novel algorithms ensure tractable inference in intractable or high-dimensional settings. Together, these developments bolster the resilience of Bayesian methods across diverse domains, from complex hierarchical models to modern machine-learning applications.

Research from Nature Portfolio

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Research from all publishers

Recent studies highlight diverse strategies to address model misspecification in Bayesian practice. Adaptive post-hoc calibration techniques extend temperature scaling to capture the link between predictive entropy and overconfidence, while entropy-based scaling offers a simple yet effective alternative under data scarcity. Robust generalised Bayesian methods employ discrepancies such as the Stein measure to update beliefs without explicit likelihoods, ensuring consistency and bias-robustness in intractable models. Modularisation approaches introduce cut distributions to isolate suspect components, and novel stochastic approximation algorithms facilitate sampling with convergence guarantees. Comparative evaluations of learning-rate selection in fractional-likelihood updates demonstrate that calibrated tempering can restore reliable coverage of credible intervals even when models diverge substantially from the true data-generating process.

Bayesian Inference and Model Misspecification publication trend

The graph below shows the total number of articles in bayesian inference and model misspecification across all publications each year (not limited to Nature Index journals).

Technical terms

Posterior distribution: The probability distribution representing updated beliefs about parameters after observing data.

Model misspecification: The situation in which the assumed statistical model differs from the true data-generating mechanism.

Temperature scaling: A post-hoc calibration technique that rescales predictive probabilities using a temperature parameter to adjust model confidence.

Predictive entropy: A measure of uncertainty in predictive distributions, quantifying the expected information content of predictions.

Stein discrepancy: A criterion derived from Stein’s method used as a loss function in generalised Bayesian updating when likelihoods are intractable.

Cut distribution: A modular Bayesian construct that blocks feedback from specific submodels to insulate inference from potential misspecification.

Learning rate (generalised Bayes): A fractional exponent on the likelihood term in Bayesian updating to temper the influence of observed data.

References

1. Adaptive temperature scaling for Robust calibration of deep neural networks. Neural Computing and Applications (2024).
2. A General Framework for Updating Belief Distributions. Journal of the Royal Statistical Society Series B Statistical Methodology (2016).
3. Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It. Bayesian Analysis (2017).
4. Robust Generalised Bayesian Inference for Intractable Likelihoods. Journal of the Royal Statistical Society Series B Statistical Methodology (2022).
5. On Posterior Concentration in Misspecified Models. Bayesian Analysis (2015).
6. Stochastic approximation cut algorithm for inference in modularized Bayesian models. Statistics and Computing (2021).
7. A Comparison of Learning Rate Selection Methods in Generalized Bayesian Inference. Bayesian Analysis (2022).