Generalized Bayesian Inference

Reliable uncertainty quantification (UQ) is essential for deploying neural networks in scientific and high-stakes applications, but full Bayesian inference over the network parameters is computationally infeasible. We propose a low-rank generalized Laplace approximation for neural-network UQ based on a small number of data-informed curvature directions. Starting from a generalized Bayesian posterior defined through an empirical loss, we construct a local Gaussian approximation around a pretrained set of weights in this active curvature subspace. The posterior variances in the retained subspace are available in closed form, and the prior variance is calibrated by an empirical Bayes procedure. The generalized Bayesian formulation allows us to compare two posterior scalings: the standard Bayesian scaling associated with the summed negative log likelihood, and a mean-loss scaling in which the empirical loss is normalized by the number of data. A central finding is that the standard scaling induces a data-size dependent contraction of the posterior variance in the leading active directions. In regression problems, this can force the low-rank framework to retain additional weak-curvature directions in order to achieve nominal coverage of calibration data. When posterior samples are propagated through the non-linear network, these additional directions can degrade the coherence of the predictive intervals and shift the posterior predictive mean away from the pretrained model. In contrast, the generalized mean-loss scaling yields a more stable, lower dimensional active subspace and produces calibrated, coherent predictive confidence intervals. These results indicate that generalized Laplace active subspaces provide a practical and scalable route to calibrated uncertainty quantification in neural networks.

Counterfactual explanations (CEs) enhance the interpretability of machine learning models by identifying the smallest change to an input required to obtain a desired output. Although CEs are conventionally formulated as a distance-minimization problem, the theoretical basis of this formulation has received limited attention. We show that a distance-minimization-based CE is mathematically equivalent to the maximum a posteriori (MAP) estimate of a Gibbs posterior within the generalized Bayes framework, specifically when a distance-based prior is used. We call this formulation the Distance-Prior Generalized Bayes CE (DP-GBCE). Building on this posterior perspective, we introduce two decision rules beyond MAP within a unified framework: a Bayes decision that minimizes expected decision loss and CVaR-CE, a risk-averse decision rule. We also propose an extension that uses Bayesian model weights to mix the posterior distributions of multiple models, thereby accounting for model multiplicity, where several models have comparable predictive performance. Finally, we define metrics for evaluating both individual CEs and the posterior distribution as a whole, and use experiments on simulated data and Google Trends data to quantify the trade-offs among the decision rules.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

Uncertainty quantification is central to many applications of causal machine learning, yet principled Bayesian inference for causal effects remains challenging. Standard Bayesian approaches typically require specifying a probabilistic model for the data-generating process, including high-dimensional nuisance components such as propensity scores and outcome regressions. Standard posteriors are thus vulnerable to strong modeling choices, including complex prior elicitation. In this paper, we propose a generalized Bayesian framework for causal inference. Our framework avoids explicit likelihood modeling; instead, we place priors directly on the causal estimands and update these using an identification-driven loss function, which yields generalized posteriors for causal effects. As a result, our framework turns existing loss-based causal estimators into estimators with full uncertainty quantification. Our framework is flexible and applicable to a broad range of causal estimands (e.g., ATE, CATE). Further, our framework can be applied on top of state-of-the-art causal machine learning pipelines (e.g., Neyman-orthogonal meta-learners). For Neyman-orthogonal losses, we show that the generalized posteriors converge to their oracle counterparts and remain robust to first-stage nuisance estimation error. With calibration, we thus obtain valid frequentist uncertainty even when nuisance estimators converge at slower-than-parametric rates. Empirically, we demonstrate that our proposed framework offers causal effect estimation with calibrated uncertainty across several causal inference settings. To the best of our knowledge, this is the first flexible framework for constructing generalized Bayesian posteriors for causal machine learning.

This study proposes a General Bayes framework for policy learning. We consider decision problems in which a decision-maker chooses an action from a given set to maximize expected welfare. Typical examples include treatment choice and portfolio optimization. In such problems, the statistical target is a decision rule, and predicting each potential outcome is not necessarily of primary interest. We formulate this policy-learning problem through loss-based Bayesian updating. Our main technical device is a squared-loss surrogate for welfare maximization. We show that maximizing empirical welfare over a policy class with a quadratic penalty controlled by a tuning parameter $ζ>0$ is equivalent to minimizing a scaled squared error in the outcome difference. The resulting General Bayes posterior over decision rules admits two equivalent characterizations: a Gaussian pseudo-likelihood representation and a decision-theoretic loss-based characterization. As one implementation, we introduce GBPLNet, a neural network implementation with a tanh-squashed output. Finally, we establish PAC-Bayes-type guarantees for the surrogate risk and the corresponding penalized welfare.

Bayesian optimal experimental design is a principled framework for conducting experiments that leverages Bayesian inference to quantify how much information one can expect to gain from selecting a certain design. However, accurate Bayesian inference relies on the assumption that one's statistical model of the data-generating process is correctly specified. If this assumption is violated, Bayesian methods can lead to poor inference and estimates of information gain. Generalised Bayesian (or Gibbs) inference is a more robust probabilistic inference framework that replaces the likelihood in the Bayesian update by a suitable loss function. In this work, we present Generalised Bayesian Optimal Experimental Design (GBOED), an extension of generalised Bayesian inference to the experimental design setting which achieves robustness in both design and inference. Using a generalisation of Bayesian decision theory, we derive a new acquisition function, the Gibbs expected information gain. Our empirical results demonstrate that GBOED can enhance robustness to outliers and incorrect assumptions about the outcome noise distribution.

Conjugate Bayes plays a key role in machine learning, ensuring computationally simple posterior updates. We generalize the variational principle behind this computation by using the Fenchel-Young (FY) inequality, which lets us define a generalized evidence lower bound beyond the KL geometry, and at equality, characterizes new generalized posteriors. A generalized notion of conjugacy follows, defined as closure of prior-potential families that are affine under translation by the loss. We show that such families not only preserve additive canonical updating, but also posterior inference with parameter-dependent supports and tuneably heavy tails. This combination is not feasible with standard conjugate Bayes. We derive a new FY generalization of expectation maximization (EM), which recovers classical and Hard EM, and amortized versions. We show examples, including generalized Bayesian linear regression with closed-form bounded support and heavy-tailed posteriors, and a Gaussian mixture with sparse cluster assignments.