Papers

Gaussian process (GP) modeling is widely used in computational science and engineering. However, fitting a GP to high-dimensional inputs remains challenging due to the curse of dimensionality. While various methods have been proposed to reduce input dimensionality, they typically follow a two-stage approach, performing dimension reduction and GP fitting separately. We introduce a Bayesian framework that seamlessly integrates dimensionality reduction with GP modeling and inference. Our approach, built on a hierarchical Bayesian model with priors on the Stiefel manifold, enforces orthonormality on the projection matrix and enables posterior inference via Hamiltonian Monte Carlo with geodesic flow. Additionally, we extend this framework by incorporating Deep Gaussian Processes (DGP) with built-in dimension reduction, providing a more flexible and powerful tool for complex datasets. Through extensive numerical studies, we demonstrate that while the proposed Bayesian method incurs higher computational costs, it improves predictive performance and uncertainty quantification, providing a principled and robust alternative to existing methods.

Interrupted time series analysis has been used to model the effect of policy and other interventions on public health by forecasting a counterfactual time series during the intervention period using data from prior to the intervention. However, due to typically relying on a single study unit, this approach risks not adjusting for other interruptions that precede and co-occur during the intervention period. The COVID-19 pandemic is a prominent example of this phenomenon of nuisance interruptions in contemporary public health research. To address this complication, we propose using Bayesian stacking over a range of functional forms for the impact of the nuisance interruption in order to make counterfactual forecasts for the intervention period. We used our proposed methods to estimate the impact of the 2021 Texas six-week abortion ban on documented pregnancies among women in Texas while adjusting for the impact of the COVID-19 pandemic.

Modeling galaxy-galaxy strong gravitational lenses to infer the brightness of the source galaxy and the mass distribution of the foreground galaxy is computationally challenging, particularly for high-resolution, high signal-to-noise ratio observations. In this regime, high-dimensional representations of both the source and the foreground mass distribution are necessary to model the data down to the noise level. This inference problem has been challenging for both traditional and machine learning-based methods because of its high dimensionality and its non-linearity in the foreground mass distribution. We present a method to generate joint posterior samples of the source galaxy and foreground mass distribution as pixelated images conditioned on observations. The method combines diffusion-based generative modeling and recurrent inference machines. It can model realistic gravitational lensing simulations with background and foreground galaxies drawn from cosmological hydrodynamical simulations down to the noise level.

We consider the problem of constructing a $K$-class classifier from the combination of $O(\log K)$ simple binary classifiers – this is a natural paradigm to construct a sophisticated classifier in a distributed manner with each agent performing a relatively straightforward task. We study the fundamental performance limits of such a classifier when the corresponding binary classifiers are hyperplanes. For a stylized Gaussian setting where the $K$ class centers are independent Gaussian points in $\mathbb R^d$ and the observations are corrupted by Gaussian noise, we derive explicit performance bounds across several decoding and dimensional regimes. Extensive simulation experiments provide strong empirical validation of the presented theoretical results.

Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

This paper introduces a class of observation-driven models whose systematic component includes a tempered fractional differencing term. This specification generalizes long-range dependent models based on the fractional differencing operator, enabling a more general and robust model specification while offering theoretical advantages. We propose a partial maximum likelihood approach for parameter estimation and address hypothesis testing, confidence intervals, goodness-of-fit assessment, and both in-sample and out-of-sample forecasting. A Monte Carlo simulation study evaluates the finite-sample performance of the proposed estimation method, and an empirical application illustrates the model's practical utility.

Tailings dam-breach analyses are essential for flood-hazard assessment, emergency planning and risk estimation, but their results are strongly affected by uncertainties in breach development, released volume and tailings rheology. This study proposes an efficient probabilistic methodology that integrates uncertainty quantification and global sensitivity analysis for tailings dam-breach studies. High-dimensional outputs (spatial maps) and the computational cost of deterministic simulations are addressed through dimensionality reduction and metamodeling. The methodology is demonstrated on a benchmark case with complex terrain using HEC-RAS v6.6 and considering uncertainties in breach parameters and rheological properties. The results quantify uncertainty in maximum flow depth and arrival time, characterize their statistical distributions and identify the spatial influence of the main input variables through sensitivity maps. Sensitivity indices reveal the dominance of breach parameters near the dam and yield stress farther downstream. The modular and non-intrusive framework can be coupled with other deterministic models and applied to different dam-breach scenarios, supporting more standardized and risk-informed assessments.

U-shaped relationships between prognostic biomarker levels and adverse event risk are commonly observed across diseases, where both low and high biomarker values are associated with elevated risk, with a well-defined minimum – the critical point – marking the biomarker value of the lowest risk. The U-shaped risk curve, especially the location of the critical point, informs the identification of high- and low-risk subgroups. However, existing methods are limited: U-shaped risk models rarely accommodate survival outcomes, and existing survival analysis methods do not enable estimation of or formal inference for the critical point. To fill this gap, we propose a semiparametric transformation model that explicitly parameterizes the critical point, a rank-based maximum C-index estimator for the parametric component, and a smoothed Kaplan-Meier estimation approach for the nonparametric component. The resulting framework estimates both subgroup-specific U-shaped risk curves and their critical points within a single survival model. We establish consistency and asymptotic normality of the proposed estimators and demonstrate their finite-sample performance through numerical studies. We apply the proposed methods to UK Biobank data to characterize subgroup-specific U-shaped associations between body mass index and all-cause mortality and to identify the corresponding critical points.

Heavy-tailed diffusion models replace Gaussian noise by a Gaussian variance mixture: denoising Levy probabilistic models (DLPM) take the mixing variables i.i.d. across coordinates, while Student-t EDM shares one mixing variable per sample. Neither has dynamics, yet temporal dependence of the noise amplitude - volatility clustering - is the defining stylized fact of financial returns. We introduce the Denoising Subordinated Probabilistic Model (DSPM), whose mixing vector is a stationary AR(1) chain driven by tempered-stable subordinator increments (the discrete Barndorff-Nielsen-Shephard volatility process) along the data axis. Conditionally on the chain the DDPM machinery survives verbatim; kurtosis and squared-noise autocorrelation are closed-form in the chain parameters, giving an exactly identified, analytically invertible calibration; DDPM, DLPM and Student-t noise are boundary cases of one memory parameter. We then prove a delimiting result: when the denoiser is conditioned on the mixing variables, their law is a nuisance - in the exact-denoiser limit the generated distribution is invariant to it and interventions on the chain do nothing. Experiments confirm both halves: conditioned models match the data's clustering whatever the mixing law, a designed x8 volatility shock moves the envelope by under 13%, while blind models transmit the mechanism exactly as calibrated. Finally, coupling the chain to the data by a variational volatility encoder - trained with the stochastic-volatility likelihood whose log-determinant the simplified denoising loss provably drops - restores control (shock response 3.07 vs. naive 2.83), recovers latent volatility (correlation 0.76), and learns the prior memory toward the true persistence.

Two misunderstandings, frequently arising in Bayesian predictive inference, are discussed. The first deals with the data generating mechanism, while the second consists in overestimating the role played by asymptotic exchangeability. Some consequences of such misunderstandings are highlighted through examples.

We study the numerical resolution of uniformly elliptic Dirichlet boundary value problems (BVPs) by minimization of the PDE residual over a hypothesis space of Neural Networks made to satisfy exactly the boundary conditions through multiplication by a user-defined distance-to-boundary approximation $\rho$. While an $L^2(\partial\Omega)$ boundary penalty cannot generally guarantee $H^2(\Omega)$ error control, we show that exact enforcement can, provided $\rho$ is chosen appropriately. More precisely, we identify a sharp sufficient condition on $\rho$ preserving the regularity needed for $H^2(\Omega)$ error control, provide practical constructions of such $\rho$ for general smooth domains $\Omega$, and refer to the resulting networks as boundary-adapted. For boundary-adapted Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we then derive explicit high-probability $H^2(\Omega)$ convergence rates by combining approximation-theoretic and statistical-learning arguments, making explicit the dependence on $\rho$. Numerical experiments support our theoretical results, showing that appropriate choices of $\rho$ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.

Probabilistic forecast models can be machine-learned from data using loss functions based on scoring rules such as the Continuous Ranked Probability Score (CRPS). This note summarises a preliminary study comparing versions of AIFS-CRPS, a global weather forecast model, trained with different univariate and multivariate scoring rules that aim to explicitly represent scale-awareness in the loss function. In the first part, we compare the (almost) fair CRPS, a fair global energy score, and a graph energy score based on node neighbourhoods. Across standard verification metrics, forecast skill is broadly similar. In the extratropics we find only small differences, while in the tropics the graph energy score setup performs somewhat better and the global energy score shows some degradation. These results suggest that multivariate scores are a viable alternative to CRPS-based training for global machine-learned weather forecasting. In the second part of the study, we analyse how different scoring rules and scale-aware loss constraints shape the spectra of forecast fields. It is apparent that any form of explicit scale-awareness improves realism. Here, the largest differences are likely associated with different effective weights per scale.

Considering that the family of bivariate Archimax copulas contains both the Archimedean and the extreme-value class, Archimax copulas constitute a flexible family allowing to model extreme and moderate levels of dependence. Despite their appeal, no fully nonparametric, consistent estimator that is itself an element of the Archimax family $\mathcal{C}_{am}$ has been established yet, mainly because Archimedean generators and Pickands dependence functions alone do not identify Archimax copulas. We resolve this identifiability issue by working with transformed generators and transformed Pickands dependence functions, and show that these functions do identify the Archimax copula uniquely. Building upon this result, we prove that uniform convergence of Archimax copulas is equivalent to uniform convergence of the corresponding transformed generators and Pickands dependence functions. Moreover, as for Archimedean and extreme-value copulas, uniform convergence in $\mathcal{C}_{am}$ is equivalent to weak convergence of almost all conditional distributions. Exploiting these equivalences, we construct two nonparametric estimators for Archimax copulas (a Pickands and a CFG type estimator, both elements of $\mathcal{C}_{am}$) and show that they are strongly consistent under mild regularity conditions. As a further consequence of the aforementioned weak conditional convergence, we obtain strongly consistent plug-in estimators for measures of directed dependence such as Chatterjee's $\xi$ and Trutschnig's $\zeta_1$. A large-scale simulation study shows that the proposed CFG type estimator outperforms both the standard empirical copula estimator and the Pickands type estimator; an application to precipitation data from Bregenz and Dornbirn (Austria) illustrates the practical use of our estimators on real data.

Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map $\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $\mu$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+t\delta_{x_0}$, we prove that the first-order limit $\mathbf{I}(x_0;\mathbf{Q}_P(z)) := \lim_{t\downarrow 0} [\mathbf{Q}_{(1-t)P+t\delta_{x_0}}(z)-\mathbf{Q}_P(z)]/t$ exists whenever $x_0\ne \mathbf{Q}_P(z)$ and characterize it uniquely. Specifically, $\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)$, where $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension $d\ge 2$, this influence function has a pole-type singularity. For fixed $z\in\operatorname{int}(\Omega_\mu)$, it remains bounded when $\mathbf{F}_P(x_0)$ stays away from $z$, where $\mathbf{F}_P=\mathbf{Q}_P^{-1}$ is the transport-based distribution function, but diverges as $x_0\to\mathbf{Q}_P(z)$, equivalently as $\mathbf{F}_P(x_0)\to z$. In fact, $\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d-1)}$. This contrasts with the bounded influence function of univariate quantiles and implies that $\mathbf{I}(X;\mathbf{Q}_P(z))$, for $X\sim P$, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.

A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.

Fast prediction of the response of adhesive soft viscoelastic contacts represents a current challenge in soft robotics and for gripping and manipulation tasks. Determining the complete time-resolved force trajectory requires full numerical simulations, whose computational cost is strongly parameter-dependent, making them impractical for real-time application or design-optimization loops. In this work, we overcome this limitation by training a scalar-conditioned, stateful, sequence-to-sequence deep learning model to predict the full force evolution from a prescribed displacement history for both short- and long-range adhesion regimes. The data set spans four orders of magnitude in loading and unloading rates and includes varied dwell times, with the Tabor parameter ranging from $0.2$ to $3.2$. To enable learning across these heterogeneous time scales, we introduce a fixed-measurement-step (FMS) representation that converts variable-length trajectories into fixed-length sequences while preserving their physical-time information. Different architectures were trained, including long short-term memory (LSTM) networks, temporal convolutional neural (TCN) networks, and time-distributed dense layers with three different Tabor-conditioning mechanisms. The models were compared using global waveform and error metrics. We found that the best-performing model has an LSTM architecture with concatenated conditioning, which achieves a held-out mean-squared error of $5.0\times10^{-4}$, a median pull-off-force error of $\approx2.2\%$, and a median hysteresis error of $\approx1.1\%$. For the held-out protocols, the model predicts a complete force trajectory with a median inference time of $0.16$ s. The model is tested across unseen parameter combinations and against analytical limiting cases, providing a rapid surrogate for repeated numerical evaluations with potential use in control-oriented applications.

Time-to-event data often include individuals who will never experience the failure event, and are therefore considered cured. In such settings, frequently encountered in clinical research, a substantial proportion of patients may remain event-free throughout the observation period, leading to the appearance of a survival plateau that is commonly interpreted as evidence of a cured fraction. Analysing such data requires inference on the cure fraction and on the survival function for the uncured subpopulation, tasks which are traditionally achieved with mixture cure models. However, assessing the causal effect of a treatment on these quantities is non-trivial. We consider principal stratification causal estimands, which have been proposed to evaluate effects on the cure fraction and on the survival for an always-uncured stratum. We additionally introduce a novel estimand, which considers the causal effect on the survival for a non-always-cured union of strata. We frame the problem from a Bayesian model-based perspective, which provides a flexible and unified estimation strategy while maintaining a direct link with classical mixture cure model quantities. The reliability of the proposed approach is validated through simulations, demonstrating competitive and robust performance relative to existing methods. Finally, we illustrate its practical usefulness through an application to a randomized trial comparing non-invasive ventilation with standard oxygen therapy in patients with hypoxemic respiratory failure following abdominal surgery.

In satellite design, it is common practice to add safety margins to the constraints to achieve conservative solutions that are robust to uncertainties. This robustness often comes at the expense of performance and it may be more appropriate to include uncertainties in the definition of the design problem. This work addresses such a challenge by applying techniques of multidisciplinary design optimization under uncertainty to an industrial use case. The latter is a pair of telecommunication satellites launched together in a stacked configuration. Each satellite is a strongly coupled multidisciplinary system while the two satellites are not coupled at all. This use case can be extended to an arbitrary number of satellites. By considering uncertainty quantification techniques such as sensitivity analysis and reliability-based design optimization, this study demonstrates that accounting for uncertainties in the design problem results in a 66% reduction in performance loss compared to the adding of a predefined safety margins, while guaranteeing the feasibility of constraints with high probability.

Spatial deformations offer a flexible route to nonstationary dependence by warping the coordinates of a stationary random field. While the exact induced covariance depends on the deformation map in its entirety, we show that its behavior in a neighborhood is approximated accurately by linearization. This produces a tangent covariance whose discrepancy from the true covariance we bound explicitly, and its Fourier transform yields a local spectrum in closed form. Building on this spectral description, we introduce a simulation scheme that generates a deformed Gaussian field in a neighborhood accounting for the local spectrum, so that the simulated field reproduces the finite dimensional tangent covariance by construction. For repeated sampling across many reference points, a truncated singular value decomposition compresses the space and frequency weights into a reusable form. We further apply the summaries based on the local Jacobian as an exploratory device for deformations estimated from images, using cardiac magnetic resonance data from the Automated Cardiac Diagnosis Challenge together with optical flow. The resulting local geometry exhibits differences across diagnostic groups through directional and anisotropic features of myocardial deformation that go beyond simple measures of local expansion or compression.

The Gasoil options market is illiquid, making it difficult to construct its implied volatility surface directly. However, it is closely linked to the highly liquid Brent options market. In this paper, we jointly model Brent and Gasoil futures prices through a correlated Bachelier local volatility model: the Brent factor is described by a normal mixture diffusion model, while the Gasoil-Brent spot volatility spread is estimated using a data-driven procedure that identifies clusters of historical crack-spread levels and Gasoil-Brent volatility spreads. The resulting bivariate model allows us to compute an implied volatility correction that maps Brent implied volatilities to Gasoil implied volatilities without using illiquid Gasoil option prices as inputs. Monte Carlo simulations demonstrate that the resulting implied volatilities closely match observed Gasoil implied volatilities when benchmarked against more direct approaches. These results suggest that the proposed framework is well suited for modeling refined products and pricing the corresponding financial derivatives.

We provide a simple and tight characterization of the types of inexact score oracle access that permit sampling with vanishing total variation bias, for a standard, well-behaved target family. Our main result shows that any weaker error than the sub-Gaussian assumption used by [YW26] rules out the tractability of unbiased sampling. This strengthens the conclusion of [CCSW26] to be algorithm-agnostic, and to hold for a wider range of error assumptions.

Specifying a prior over the space of correlation matrices is a persistent challenge in Bayesian analysis. The space is a curved manifold whose dimension grows quadratically with the number of variables, making substantive prior beliefs difficult to encode.\\ We propose a distance-based prior that assigns mass decaying exponentially in the Fisher arc-length distance from a user-specified reference correlation matrix, enabling shrinkage toward any target correlation structure rather than being confined to the identity matrix. Formally, this is constructed as a Penalised Complexity prior, but its interpretation shifts accordingly: unless the chosen target represents a structurally simpler state, the shrinkage penalises deviation rather than complexity in the usual sense. To accommodate conditional independence constraints, we introduce a parameterisation that constructs the correlation matrix via the Cholesky factor of the inverse correlation matrix with respect to a user-supplied graph, thereby reducing the number of free parameters from one per variable pair to one per graph edge. The prior is proper for every positive value of its rate parameter, accommodates correlations of either sign under any graph structure, and reduces to a fully unstructured prior when the graph is complete. A direct sampling algorithm is provided, enabling prior predictive checks and sensitivity analysis, implemented within the graphpcor package.

Building on the identity that expected regret equals the covariance between costs and decisions, this paper develops a derivative theory of the covariance regret functional. We derive the Gâteaux derivative, showing that the universal steepest-descent direction is the contrarian policy $-(c-\bar c)$, while ascent yields momentum. For linear policies $\hat\pi(c)=Ac+b$, the gradient is the cost covariance matrix $\Sigma_c$, with a zero Hessian implying boundary-optimal solutions such as the minimum-variance portfolio. We extend to constrained optimization, sign-gradient duality between regret minimization and alpha maximization, finite-sample convergence bounds paralleling Thompson Sampling, and gradient-descent algorithms requiring only input observations.

Sufficient dimension reduction seeks a low-dimensional covariate representation that preserves the conditional law of a response. We introduce SDR-COT, which represents that law by conditional optimal transport from an independent reference response. On separable Hilbert spaces, sufficiency forces the response component of the optimal triangular map to factor through the reduction. For quadratic cost, the induced interpolation has a Borel current-state velocity on every truncated time interval, without global injectivity of the terminal map, and this velocity has the same factorisation. These results motivate a conditional-flow-matching criterion. For linear reductions, we prove consistency using a suitably tuned relaxed empirical coupling. Euclidean responses are treated through slicewise Caffarelli bounds; Hilbert-valued responses are treated through Gaussian Sobolev regularity, interpolation compression and uniqueness of a Gaussian continuity equation. Numerical studies with Euclidean and functional data show competitive performance, especially when sufficient information is not solely contained in the conditional mean.

Determinantal point processes (DPPs) are widely used as probabilistic models for diverse random subsets, but their approximation error under model misspecification has not been fully characterized. We study the population-level approximation of a strictly positive target distribution p* by DPPs under the forward Kullback-Leibler divergence. Using information-geometric analysis and the standard quality-diversity decomposition of an L-ensemble kernel, in which the diagonal quality component Q encodes item-specific weights and the diversity component D controls repulsive interactions among items, we show that the quality component can be chosen uniquely to match all first-order inclusion probabilities of p*. The DPP approximation problem therefore reduces to the optimization of the diversity component. This reduction yields a global optimality result for attractively dependent targets: under conditions including weak positive association, the independent product distribution with the same first-order marginals, corresponding to D=I, is an optimal DPP approximation. For more general target distributions, positively correlated pairs yield lower bounds on the approximation error. On the repulsive side, a matching of disjoint negatively correlated pairs yields an upper bound on the approximation error, or equivalently a guaranteed improvement over the independent approximation. We further study local optimality around D=I and, more generally, around block-diagonal diversity matrices by analyzing perturbations between their blocks.

For hierarchical models, Pareto-smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) fails on the folds where a random-effect coordinate is data-driven and its group is small. We show that the Gelman-Pardoe pooling factor and structural leverage predict these folds from model structure and group sizes, without forming importance weights. In Gaussian linear mixed models the leverage reduces to group size, giving a design-time map that separates the failing ($\hat{k}>0.7$) folds with AUC 0.96; across replicated logistic GLMMs the post-fit, weight-free predictor reaches AUC 0.81. The cure is integrated importance sampling: marginalise the random-effect block and importance-sample only the base parameters. This is not new, but we contribute its observation-level specialisation for random-intercept GLMMs: an analytic Gaussian downdate and a 1-D quadrature for Bernoulli, binomial and Poisson responses, packaged as a drop-in rb_loo(fit). Against exact refits, this marginalised estimator (RB-LOO) is $3\times$ more accurate than moment matching on singleton-heavy logistic GLMMs. On overdispersed count data with 97 failing folds, moment matching leaves 37 uncorrected and is no more accurate than raw PSIS-LOO, while RB-LOO reproduces the 82-minute exact refit (elpd RMSE 0.04) at no cost. The error changes decisions: against a negative-binomial model, PSIS-LOO reports decisive evidence ($z=4.9$) and reloo reports significant evidence ($z=3.4$) for the more complex model, where an exact analysis, reproduced by RB-LOO, finds the two indistinguishable ($z=1.0$). A base-fiber Schur decomposition splits case-deletion influence into a vertical (pooling) term that governs where PSIS-LOO fails and a horizontal (variance-component) term that governs where RB-LOO is itself strained, giving a two-level triage that recovers the exact answer while refitting only the few folds that need it.

Algebraic statistics characterizes statistical models through polynomial constraints, but it has mainly been used for analytically specified model classes. This paper studies the inverse problem: identifying probabilistic structure from vanishing binomials observed in empirical probability tensors. We treat the vanishing binomials of a toric model as its algebraic signature, and turn the ideal-variety correspondence of algebraic statistics into an operational procedure for structural learning that identifies a model by signature matching without parameter estimation. By restricting attention to a computationally tractable class of configuration matrices, which we call the Kronecker-stack class, we make these signatures explicitly enumerable. Within this class we define minimum invariant constraint (MIC) as the atomic unit characterizing each signature and generalizing the notion of independence. We tested this approach employing MICs on synthetic data as well as on corpus-scale real language data. The results suggested the utility of the method, revealing that the identified rank-one structures correspond to interpretable sets of words. These results open up a new avenue for applying algebraic statistics to computational linguistics.

In the latent posterior model of transformer behavior, the next-token distribution arises from a posterior over latent predictive models conditioned on the context, mixed to generate continuations. We exploit this model in settings where it is exact, namely Bayes-filtered transformers (BFTs) meta-learned on sequences from a hierarchical prior, to introduce Posterior Prefix Tuning (PPT), a new method for eliciting behavior from a transformer: given a utility function on continuations, find a prompt under which the transformer generates continuations of high expected utility. For a BFT, the elicitation objective factors through the latent posterior, and the gradient of this objective can be estimated from samples of the prior alone. PPT optimizes the parameters of a distribution over hard prompts: it draws prior samples once from the BFT via predictive Monte Carlo (PMC), then estimates the gradient by importance sampling against them. The optimization performs no transformer forward passes and no backpropagation through the transformer, and the prior samples are utility-independent, so a single set of samples drives elicitation against any number of utilities at negligible marginal cost. We validate PPT on Beta–Bernoulli and reinforced urn BFTs across three utility families (reverse cross-entropy, frequency matching, Dyck validity).

Weighted distributions arise in situations where observations are selected with unequal probabilities or because of the non-observability of some events. Detecting such sampling bias is essential for ensuring valid statistical inference. In this paper, we propose a statistical test for detecting bias in a sample using the Renyi divergence measure. The proposed test statistic is formulated for both uncensored and type-I censored data and possesses several theoretical properties including asymptotic normality. Critical values are obtained through simulations under the Weibull distribution for a range of sample sizes and censoring proportions. A comprehensive power study compares the proposed test with the existing Kullback-Leibler divergence-based test and the likelihood ratio test. Two real datasets are analyzed to demonstrate the practical utility of the proposed test in detecting length bias.

Count data with overdispersion and high-dimensional predictors pose significant challenges in modern applications. While negative binomial regression offers a flexible modeling framework, existing Bayesian approaches rely on computationally expensive MCMC methods that become impractical in high-dimensional settings. This paper develops a variational Bayesian framework for sparse negative binomial regression using horseshoe and continuous spike-and-slab priors. Our proposed methods achieve estimation accuracy and variable selection performance comparable to MCMC benchmarks while offering substantial computational savings over MCMC. Extensive simulations demonstrate that the negative binomial specification is essential for overdispersed data, as Poisson-based approaches exhibit substantial performance degradation under overdispersion. Conversely, our methods remain robust when the data are Poisson, making them a safer default choice. Applications to real benchmark datasets further confirm the practical utility of our approach.