Generalized Bayesian Inference for Neural Networks

• Generalized Bayesian inference is a framework that replaces traditional likelihood with arbitrary loss functions and divergences to update beliefs about neural network parameters.
• It employs strategies like low-dimensional subspace inference, hybrid covariance filtering, and Q-posterior calibration to achieve robust and scalable computation.
• These methods offer practical uncertainty quantification and risk guarantees, enabling flexible applications in sequential decision making and function-space analysis.

Generalized Bayesian inference of neural network parameters refers to a suite of methodologies that extend traditional likelihood-based Bayesian inference by allowing for arbitrary loss functions, explicit regularization, structured geometry, and robust handling of high-dimensional or misspecified models. This perspective unifies and systematizes many contemporary advances in Bayesian neural networks (BNNs), notably those addressing scalability, robustness, constrained domains, and sequential or online learning. Rather than centering solely on the likelihood, these approaches deploy divergences, scoring rules, surrogate losses, and geometric projections to update beliefs about neural network parameters—promoting better uncertainty calibration, flexibility, and computational tractability.

1. Foundations and Framework of Generalized Bayesian Inference

Generalized Bayesian inference (GBI) systematically replaces the likelihood in Bayes' theorem with a user-specified loss or divergence. The posterior for parameter vector $\theta$ given data $x=(x_1, \ldots, x_n)$ and prior $\pi(\theta)$ becomes: $$\pi_{\eta}(\theta|x) \propto \exp(-\eta L(\theta; x)) \, \pi(\theta)$$ where $L(\theta; x)$ is a cumulative loss (e.g., negative log-likelihood, scoring rule, or robust divergence), and $\eta > 0$ is a learning rate (also called temperature). This formulation encompasses standard Bayesian inference as the special case when $L$ is the negative log-likelihood and $\eta=1$. Modern developments emphasize:

• Loss-based and Loss-free Approaches: The loss $L$ may be the negative log-likelihood (for well-specified likelihoods), a scoring rule (for likelihood-free or simulator-based models), or a surrogate divergence (for robustness).
• Learning Rate Estimation: $\eta$ controls the degree of trust in the data versus the prior. Direct estimation of $\eta$ by placing a prior and using held-out data yields hyperparameter posteriors that concentrate near optimal predictive performance, as established by asymptotic theory (2506.12532).
• Calibration via Q-posterior: The Q-posterior replaces the Gibbs posterior with one derived from the sandwich variance of the loss gradients, leading to empirically calibrated uncertainty quantification even in the presence of model misspecification (2311.15485).

The GBI framework can further accommodate multi-modular settings, combining separate data sources or inference modules with their own learning rates and loss hyperparameters for robust and flexible aggregation.

2. Algorithmic Strategies for High-Dimensional and Structured Neural Networks

Scaling Bayesian inference to complex neural network models presents challenges in both computation and uncertainty quantification. Recent approaches leverage:

• Subspace Inference: By constructing low-dimensional subspaces—typically via principal components of SGD trajectories—Bayesian inference (via MCMC or variational methods) can be performed efficiently in a manifold that contains most of the posterior mass and predictive diversity (1907.07504).
• Hybrid Covariance and Online Filtering: Scalable online GBI in sequential decision settings uses block-diagonally structured parameter error covariances: low-rank approximations for hidden layers and full-rank (or low-rank) for output layers, admitting fast Kalman-style updates and well-defined predictive distributions (2506.11898).
• Mixture Distributions with Logconcave Components: For neural networks with $\ell_1$-controlled weights, a mixture decomposition of the posterior enables log-concave MCMC sampling when $K d \gtrsim (\beta N)^2 \log (\beta N)$, leading to theoretically justified, rapid computation (2411.17667).
• Amortized Cost Estimation: In simulation-based inference for neural-network-driven models, generalized Bayesian posteriors are constructed via amortized neural cost estimators (ACE), which learn to predict distances between simulated and observed data, yielding robust GBI posteriors without expensive repeated simulation (2305.15208).

3. Posterior Structure, Risk Guarantees, and Robustness

The intricate structure of neural network posteriors—often highly multimodal and heavy-tailed—demands strategies attuned to both computational tractability and generalization risk:

• Heavier Tails in Finite-Width BNNs: The distribution of hidden units in finite-depth and finite-width BNNs develops increasingly heavy (generalized Weibull) tails with each succeeding layer, enhancing the expressivity but also complicating inference theory (2110.02885). The tail parameter at layer $\ell$ is determined by

$$\frac{1}{\beta^{(\ell)}} = \sum_{k=1}^\ell \frac{1}{\beta_w^{(k)}}$$

where $\beta_w^{(k)}$ is the base-layer tail parameter.

• Risk Bounds and Temperature Selection: For discretely supported priors under bounded parameterizations, generalized Bayesian learning with temperature $\beta \sim (\log d / N)^{1/4}$ achieves predictive regret of order $O((\log d / N)^{1/4})$, improving to order $O((\log d/N)^{1/3})$ in KL divergence for Gaussian noise with matched inverse temperature (2411.17667).
• Uncertainty Calibration: Standard metrics—log-likelihood, RMSE, interval coverage—often conflate model error and inference error, failing to discern true posterior quality. The Q-posterior cures this by matching credible set coverage to nominal levels even for misspecified or loss-based posteriors (2311.15485).
• Learning Rate and Hyperparameter Posteriors: GBI enables data-driven calibration of inference hyperparameters (learning rate $\eta$, loss choices) by cross-validated Bayesian updating:

$$\rho(\eta | y, x) \propto \rho(\eta) \prod_{j=1}^J p_\eta(y_{(j)}|x)$$

with theoretical guarantees for posterior contraction and optimal risk performance (2506.12532).

4. Methodologies for Diverse Settings: Likelihood-Free, Constraint-Aware, and Function-Space Inference

The generality of GBI accommodates a wide range of practical modeling scenarios:

• Likelihood-Free Inference with Scoring Rules: Proper scoring rules (e.g., energy, kernel/MMD) provide surrogate losses for GBI when likelihoods are intractable but model simulation is possible. Gradient-based MCMC (SG-MCMC) affords scalability in high dimensions and direct inference on NN weights in dynamical systems or scientific simulators (2104.03889).
• Function-Space Variational Inference: Rather than approximating parameter-space posteriors, explicit variational inference in function-space leads to controllable, interpretable priors on function behavior, reliable predictive uncertainty, and improved robustness under distribution shift (2312.17199). The tractable objective maximizes data likelihood while enforcing function-space KL matching on context sets of interest.
• Constraint-Aware and Geometric Priors: Generalized Bayes linear inference recasts parameter or function estimation as projection onto constrained cones (e.g., non-negativity, monotonicity) under geometry informed by the prior (covariance-weighted norm), supporting fast and principled uncertainty quantification even in partial or nonprobabilistic belief systems (2405.14145).

5. Practical Applications, Case Studies, and Empirical Evaluation

Generalized Bayesian neural inference is realized in varied applied domains:

• Sequential Decision Making and Bandits: Online GBI with block-diagonal and low-rank covariance structures supports rapid adaptation and expressive uncertainty quantification for contextual bandits and Bayesian optimization, showing strong empirical speed/accuracy tradeoffs and robust uncertainty calibration for exploration (2506.11898).
• Engineering Inverse Problems: In engineering simulation and optimization, NNK-based variational inference captures multimodal and irregular parameter posteriors, outperforming MAP, standard BNNs, and MCMC on complex PDE-governed systems (2205.03681).
• Text Clustering and Modular Models: GBI hyperparameter posteriors enable calibrated modular inference in multi-domain settings, improving performance on real-world text analysis by tuning the learning rate to optimize predictive accuracy rather than adhering to a fixed Bayes-optimality (2506.12532).
• Safety-Critical Uncertainty Calibration: Function-space variational techniques deliver reliable uncertainty quantification for medical imaging diagnosis, especially under distribution shift, outperforming standard MC-dropout, mean-field VI, and deep ensembles on both in-domain and OOD cases (2312.17199).

6. Comparative Summary of Methodologies

Approach	Posterior Model	Scalability/Computation	Risk/Calibration
HMC/Fully Bayesian (TensorBNN)	Full parameter posterior	Accurate but slow for large nets	Best for small/medium models
Subspace Inference	Low-dim affine manifold	Fast, expressive if subspace well chosen	Strong predictive UQ, tractable
Generalized Bayes + Scoring Rule	Loss-based surrogate posterior	Rejection-free SG-MCMC; scalable	Robustness, consistency under misspecification
Q-posterior	Score-corrected posterior	Moderate; extra covariance computations	Calibrated uncertainty even under misspecification
Function-space VI	Posterior over function outputs	Scalable; local linearization	Better prior control and OOD UQ
Bayesian linear/projection	Moment or constrained update	Very fast; approximate	Useful for rapid approximate UQ, with domain constraints
Mixture/log-concave inference	Mixture over log-concave	Fast MCMC; polynomial-time if dim high	Suboptimal rates, but path to guarantees

7. Open Directions and Theoretical Challenges

Several avenues remain for further research:

• Bridging Sampling and Risk Guarantees: Ongoing work aims to develop prior structures that permit both rapid log-concave MCMC sampling and explicit statistical generalization control, closing the gap between continuous and discrete support priors (2411.17667).
• Scalability to Deep and Wide Models: Although online low-rank filtering and function-space methods are scalable, extending theoretical calibration guarantees (e.g., for Q-posteriors) to very high-dimensional NNs remains an open problem (2311.15485).
• Heavy-Tailed Priors and Robustness: Systematic paper of heavy-tailed and structured priors in finite BNNs is needed to harness expressivity without undermining calibration and tractability (2110.02885).
• Optimal Hyperparameter Learning: The theory and practice of learning the loss structure and temperature from data or hold-out sets is an active area; empirical results show practical advantage in predictive risk, but further theory is desirable (2506.12532).
• Constraint-Aware Learning in Deep Nets: Integrating geometric constraints in deep learning architectures, particularly via Bayes linear projections or constrained variational objectives, is an emerging direction for uncertainty-aware, domain-faithful inference (2405.14145).

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Generalized Bayesian inference of neural network parameters encompasses loss-based, constraint-informed, and robustness-enhancing perspectives that extend standard Bayesian inference to the realities of modern, large-scale, and often misspecified models. These frameworks support principled uncertainty quantification, computational scalability, flexible prior specification, and accommodate complex domain knowledge, establishing a foundation for future advances in statistical learning with neural networks.