A General Framework for Updating Belief Distributions (1306.6430v2)

Abstract: We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our proposed framework uses loss-functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.

• The paper proposes a loss-based updating mechanism for belief distributions that extends traditional Bayesian inference by integrating loss functions.
• It utilizes cumulative loss functions to calibrate prior beliefs, ensuring coherence in sequential or simultaneous data assimilation.
• The framework enhances robustness in complex, high-dimensional settings and offers practical utility for non-stochastic or partially informative data.

A General Framework for Updating Belief Distributions

The paper by Bissiri, Holmes, and Walker proposes a novel framework for Bayesian inference that extends the traditional reliance on likelihood functions by incorporating loss functions into the update mechanism of belief distributions. This research addresses challenges that arise in modern applications, where modeling the true data-generating process is complex or the parameter of interest is not directly linked to a family of density functions. The authors introduce a decision-theoretic approach using cumulative loss functions to update prior beliefs, which is coherent with Bayesian updating when a true likelihood is known but extends to more general settings.

Summary of Core Contributions

The authors suggest that a parameter of interest that minimizes expected loss can serve as the basis for updating belief distributions. The framework uses a loss function $l(\theta,x)$ to link data observations to parameters and adjust the prior belief $\pi(\theta)$ to a posterior $\pi(\theta|x)$ using: $$\pi(\theta|x) \propto \exp\{-l(\theta,x)\}\,\pi(\theta).$$

This formulation, akin to a Bayesian update, is built on coherence properties ensuring that the update process is consistent whether data is assimilated simultaneously or sequentially. A noteworthy feature is the realization that classical inference aligns with this framework by treating the negative log likelihood as a loss function, which provides a robust interpretation when the true model is known but also extends to the estimation of parameters without explicit model assumptions.

Implications and Applications

The implications of this framework are twofold:

• Practical Applications: This approach enhances the flexibility of Bayesian inference in fields where traditional modeling assumptions are untenable, allowing practitioners to employ robust estimation techniques prevalent in classical statistics without the need for complete model specification.
• Theoretical Insights: It emphasizes the importance of loss functions and coherence in inference, expanding the applicability of Bayesian analysis to scenarios with non-stochastic information or partial data. This is particularly relevant in high-dimensional problems or those involving parameters indirectly connected to observables.

Methodological Insights

The authors delve into several considerations and methodologies:

1. Type of Loss Functions: They differentiate scenarios based on the availability and type of data-generating models (M-closed, M-open), recommending specific loss functions ranging from self-information to robust alternatives like M-estimators.
2. Calibration of Loss Functions: Addressing potential arbitrary scaling of loss functions, they explore methods like unit information loss, hierarchical approaches, and subjective calibration. The latter connects Bayesian coherence with classical ANOVA-based methods to determine the influence level of data and prior in belief updates.
3. Application to Non-Stochastic and Partial Information: They extend this inference framework to non-stochastic and partially informative data, significant in fields like survival analysis using proportional hazards models or non-traditional applications such as clustering without probabilistic models.

Speculations and Future Directions

This research opens several avenues for future exploration, particularly in machine learning and AI, where traditional probabilistic models may be restrictive. Extensions of this framework could inform the development of novel algorithms capable of integrating various information types, enhancing the robustness and interpretability of AI systems. Furthermore, the interplay between decision-theoretic perspectives and Bayesian inference offers fertile ground for extending PAC Bayes approaches, providing robust guarantees for empirical performance.

Conclusion

Bissiri, Holmes, and Walker present a compelling argument for broadening the scope of Bayesian inference through a loss-based framework, offering a flexible, coherent approach to updating beliefs under challenging data scenarios. This framework not only expands the theoretical landscape of Bayesian statistics but also offers practical utilities in diverse, complex domains.