change of variables

This paper addresses the limitations of conventional generalized additive models (GAMs), where covariate transformations typically rely on ad hoc preprocessing and hinder joint estimation and uncertainty quantification. We propose an end-to-end modeling framework that directly embeds differentiable parametric transformations into the GAM structure. Our method jointly optimizes transformation parameters, smooth function coefficients, and hyperparameters—enabling, for the first time, differentiable transformation and unified GAM fitting for high-dimensional, complex covariates. Within an empirical Bayes framework, we integrate maximum a posteriori estimation, Laplace approximation, and implicit differentiation to enable efficient inference and joint uncertainty quantification. Empirical evaluation on UK net electricity demand forecasting and London housing price modeling demonstrates superior flexibility and predictive performance. The proposed methodology has been implemented in the open-source R package *gamFactory*.

In Bayesian modeling, expert priors are often specified directly on observable or derived quantities (e.g., survival rates, R²), yet translating such domain knowledge into informative priors for latent model parameters remains a fundamental challenge. Method: We propose a hyperparameter optimization framework grounded in prior predictive distribution matching. It parameterizes a prior family and employs multi-stage Bayesian global optimization to minimize the discrepancy between the induced prior predictive distribution and the target expert-specified distribution—supporting mixed-type and nonstandard targets, as well as censored data, nonlinear structures, and complex derived statistics. Contribution/Results: Across three case studies—cure-rate survival models, R²-driven modeling, and nonlinear regression—the resulting informative priors substantially improve posterior stability and interpretability. Our approach provides the first systematic, computationally tractable, and empirically verifiable methodology for transforming marginal expert beliefs about observables into joint priors over latent parameters.

This paper addresses foundational issues in conditioning within Bayesian statistics, identifying a fundamental flaw in the conventional identification of conditional distributions with restrictions of density functions to observed subsets—particularly problematic on smooth manifolds, where disintegration of measures may lack mathematical justification. Method: The authors rigorously distinguish “restricted densities” from properly defined “disintegration densities”, construct explicit counterexamples demonstrating their substantial divergence, and integrate differential geometry with measure theory to develop a computable framework for disintegration densities on manifolds, including an explicit algorithm. Contribution/Results: They prove that the widely used “conditional mode” corresponds to the mode of the restricted measure—not the disintegration measure—leading to theoretical inconsistency in approximate Bayesian inference and Bayesian inverse problems. The work clarifies essential prerequisites for valid conditional modeling and establishes a rigorous mathematical foundation for Bayesian methods on high-dimensional and manifold-structured parameter spaces.

This work addresses the longstanding limitation in conditional density estimation—namely, the absence of closed-form solutions for multivariate conditional densities under non-Gaussian assumptions. We propose a generative conditional density estimation framework grounded in copula modeling and analytic conditionalization in latent space. Methodologically, we first establish the inheritability of “conditional stability” under mixture and transformation operations, thereby extending analytically tractable conditional families to non-Gaussian, nonlinear, and cross-dimensional settings. The core components include a Gaussian Mixture Copula Model (GMCM), an explicit latent-space conditionalization mechanism, and joint copula modeling. Experiments on synthetic and real-world datasets demonstrate substantial improvements in conditional density estimation accuracy and robustness to missing data imputation. Crucially, our approach enables efficient, differentiable, and sampling-free deterministic conditional inference.

To address insufficient Bayesian prior information in few-shot personalized inference, this work proposes a novel paradigm for constructing transferable, informative priors from population data. The core method introduces data-consistent stochastic inversion (DCI) to learn pullback probability measures from population-level observations, thereby deriving structured Gaussian priors tailored to individual inverse problems. Theoretically, we prove that the resulting prior strictly improves information gain—measured by both determinant and trace of the posterior precision matrix—and reduces the posterior KL divergence in linear Gaussian inverse problems. Numerical experiments demonstrate substantial improvements in inference accuracy and uncertainty calibration across digital twin and biomedical modeling tasks. This work establishes a provably sound, transferable, and physics-agnostic framework for data-driven prior construction.

In Bayesian analysis, translating domain expertise into computationally tractable prior distributions remains challenging due to expressive limitations and cognitive gaps. This paper introduces an interactive visual prior elicitation method that reframes prior specification as a “hypothetical data construction” process: users iteratively generate synthetic datasets aligned with their beliefs via drag-and-drop operations, constraint imposition, and forward simulation; the system then automatically infers the corresponding prior distribution and provides real-time feedback via prior predictive checks. Integrating visual reasoning, probabilistic modeling, and predictive calibration, the approach enhances the intuitiveness, controllability, and credibility of prior encoding. A user study demonstrates that, compared to conventional parametric prior specification, 92% of participants formulated priors more faithfully reflecting their domain beliefs; moreover, prior clarity and debugging efficiency improved significantly.

This work proposes TIED, a method for solving inverse problems involving unknown data transformations on general Lie groups. By modeling the transformation posterior as a Boltzmann distribution defined via an energy function and constructing a diffusion process in the Lie algebra to preserve manifold structure, TIED enables efficient posterior sampling. Its key innovation is the introduction of a trivialized objective score identity, which— for the first time—enables efficient score-based posterior inference on Lie groups. The approach supports test-time equivariance, substantially enhancing model robustness. In tasks involving image homography and PDE symmetries, TIED successfully recovers transformed inputs back to the original training distribution, outperforming existing normalization and sampling baselines.

This study addresses the stability of the solution operator with respect to perturbations in the input parameter distribution within the framework of nonparametric Bayesian computer model calibration. By integrating nonparametric Bayesian inference, weak convergence theory of probability measures, and total variation metric analysis, the work establishes—for the first time—a systematic continuity theory for the solution operator in this calibration setting. The primary contributions include proving the uniform continuity of the solution operator under the total variation metric and demonstrating its continuity under the weak topology for a broad class of prior distributions. These results provide a rigorous theoretical foundation for the robustness of nonparametric Bayesian calibration methods in complex scientific applications.

In Bayesian sequential inference, the marginal likelihood is often treated as a static constant, overlooking its role in modulating the pace of belief updates. This work reveals that the marginal likelihood not only governs the magnitude of individual updates but also encodes frequency patterns embedded in historical data, which the authors reformulate as a dynamic regularizer. By introducing three diagnostic metrics to control online estimation gain and integrating prior and posterior distributions into a hybrid probabilistic mechanism, the proposed approach adaptively adjusts to distributional drift. The resulting framework unifies Bayesian updating with frequentist characteristics within a two-layer probabilistic architecture, substantially enhancing the robustness of sequential estimation and offering a novel paradigm for online risk quantification.

This study addresses the vulnerability of high-dimensional linear regression to model selection uncertainty, which can lead to spurious manipulation of covariate coefficient signs and thereby undermine the reliability of empirical conclusions. We systematically demonstrate, for the first time, that auxiliary variables—termed SHAVE (Sign-Heuristic Auxiliary Variables with Expansive effects)—occupying a set of positive Lebesgue measure can induce sign reversals in target coefficients while simultaneously inflating associated test statistics. Leveraging tools from high-dimensional regression theory and measure-theoretic analysis, we provide a rigorous mathematical characterization of this sign-manipulation phenomenon and propose detection strategies based on either extended or independent datasets. Extensive simulations and empirical applications reveal the pervasiveness of this issue and confirm that our proposed methods effectively identify such manipulations, thereby enhancing inferential robustness.