Fast algorithm solves BVPs in linear time with probabilistic uncertainty.
problem Solving boundary value problems efficiently and accurately.
method Gauss--Markov prior tailored to BVPs, linear-time computation.
result Probabilistic solution with linear time complexity and comparable quality.
Solves optimal stopping for Gauss-Markov bridges using time-space transformation.
problem Optimal stopping problem of a Gauss-Markov bridge.
method Time-space transformation approach, Picard iteration algorithm.
result Lipschitz continuity of the optimal stopping boundary and its characterization.
Extended Gauss-Markov theorem for linear estimation with bounded bias.
problem Linear estimation with bounded bias operator.
method Derive optimal estimator formulas for Nuclear and Spectral norms, analyze generalization error.
result Cross-validated Nuclear and Spectral regressors outperform Ridge regression in simulations.
The paper shows how to answer future and past questions from high-dimensional time series data.
problem Challenges in answering probabilistic inference questions from high-dimensional time series data.
method Temporal contrastive learning to learn Gaussian representations that enable compact closed-form solutions.
result Representations learned via contrastive learning follow a Gauss-Markov chain, enabling efficient inference and planning.
Optimal sensor placement minimizes information loss from simulations.
problem Designing efficient sensor networks for spatiotemporal processes.
method Model-based sensor placement criterion with sparse variational inference and Gauss-Markov priors.
result Our method identifies sensor networks that minimize information loss from simulated data.
Bayesian approach improves ODE solution accuracy.
problem Improving numerical solutions of ordinary differential equations.
method Bayesian inference with Gaussian filtering and smoothing.
result Maximum a posteriori estimate converges to true solution at polynomial rate.
Fenrir uses probabilistic numerics to simplify solving initial value problems.
problem Solving initial value problems in ordinary differential equations.
method Probabilistic numerics and Gauss--Markov regression.
result The method simplifies parameter estimation in ODEs, making it easier and more robust.
New method improves Kalman filtering and smoothing for large state spaces.
problem High computational cost and uncertainty in large-scale Kalman filtering.
method Probabilistic numerical method leveraging GPU acceleration and tunable trade-off.
result Mitigates scaling issues and provides more accurate uncertainty estimates.
New Hida-Matérn kernels enable flexible process priors and efficient GP inference.
problem Flexible modeling of stationary processes with oscillatory components.
method Introducing a new class of covariance functions (Hida-Matérn kernels) and their state space representations.
result Efficient Gaussian Process inference and improved numerical stability.
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability d…
A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives \emph{a priori} as a Gauss--Markov process X, which is…
Enhances machine learning interpretability using category theory.
problem Improving machine learning interpretability and social implementation.
method Develops a categorical framework for structured understanding of supervised learning.
result Introduces the Gauss-Markov Adjunction for clarifying residuals and parameters.
Fermat-Torricelli points help assess investment risks by smoothing series data.
problem Analyzing investment risks in series with large variance, nonlinear trends, or non-normal distributions.
method Construct Fermat-Torricelli points to reduce random component influence.
result Smoothing series by Fermat-Torricelli points reduces risk assessment errors.
Improves deep learning performance on noisy datasets using inverse-variance weighting.
problem Heteroscedastic regression with varying noise levels.
method Batch Inverse-Variance (BIV) loss function for neural networks.
result Significantly improves network performance on noisy datasets compared to other methods.
The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, cor…
We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, …
Paper proposes a bias-constrained deep learning approach to non-linear estimation.
problem Designing unbiased estimators for non-linear models.
method Bias Constrained Estimator (BCE) using deep learning with bias constraints.
result Asymptotic MVUEs with Cramer Rao bound performance.
Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds
problem Convergent approximation of Gaussian Whittle-Matern fields on Riemannian manifolds
method Finite Element approximation of SPDEs
result Universal approximation of precision and covariance matrices
Gaussian Graphical Models (GGMs) or Gauss Markov random fields are widely used in many applications, and the trade-off between the modeling capacity and the efficiency of learning and inference has been an important research problem. In this paper, we study the family of GGMs with small feedback vertex sets (FVSs), whe…
A new Weyl prior is proposed for Bayesian statistics, offering a more canonical choice for parameter α.
problem Choosing a prior distribution for Bayesian inference.
method Proposed a new Weyl prior based on the Weyl structure on a statistical manifold.
result The Weyl prior is a special case of the α-parallel prior with α = -n, where n is the dimension of the statistical manifold.
Improves AI-prior reliability for Bayesian inference.
problem Error propagation from predictive models into posterior inference.
method Rectified AI-informed prior elicitation framework.
result Significant reduction in bias and improvement in predictive performance.
Informative Bayesian priors are often difficult to elicit, and when this is the case, modelers usually turn to noninformative or objective priors. However, objective priors such as the Jeffreys and reference priors are not tractable to derive for many models of interest. We address this issue by proposing techniques fo…
While Bayesian methods are praised for their ability to incorporate useful prior knowledge, in practice, convenient priors that allow for computationally cheap or tractable inference are commonly used. In this paper, we investigate the following question: for a given model, is it possible to compute an inference result…
Researchers derive exact priors for finite Bayesian neural networks.
problem Understanding non-Gaussian priors in finite Bayesian neural networks.
method Analytical derivation of function space priors for finite fully-connected feedforward networks.
result Exact solutions for priors of finite networks, including Meijer G-function for linear networks and mixtures for ReLU networks.
PRCD-MAP learns to trust imperfect priors in causal discovery, improving accuracy and robustness.
problem Tackles the brittle trade-off between blind trust and rejection of external priors in causal discovery.
method Proposes PRCD-MAP, a soft prior-consumption layer that assigns per-edge trust to imperfect priors and modulates regularization in a MAP objective.
result Enjoys a population-level safety guarantee and outperforms existing methods on real-world causal discovery tasks.
Bayesian metalearning improves performance in linear bandits with misspecified priors.
problem Improper priors lead to suboptimal performance in sequential decision-making.
method Proves performance bounds for metalearning priors in stochastic linear bandits and develops a metalearning algorithm.
result Metalearning can improve performance by learning the prior from multiple tasks.
The paper extends and applies a new shrinkage prior in Bayesian factor analysis.
problem Estimating the number of factors in sparse Bayesian factor analysis.
method Introduces and extends a generalized cumulative shrinkage process (CUSP) prior.
result Exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index increases.
Bayesian method corrects for model selection multiplicity in regression.
problem Model selection multiplicity in regression analysis.
method Developed a Bayesian prior distribution based on Holm procedure analogy.
result Adequate multiplicity correction requires sparsity not provided by recommended priors.
Review of priors in Bayesian deep learning models.
problem The importance of prior choices in Bayesian deep learning models.
method Overview of different priors and methods of learning priors from data.
result Motivate practitioners to think carefully about prior specification.
Study characterizes training and test risks for MAP regression with Gaussian priors.
problem Understanding high-dimensional behavior of regularized linear regression with informative priors.
method Maximum a posteriori (MAP) regression with Gaussian priors, using random matrix theory.
result Closed-form risk formulas reveal the bias-variance-prior tradeoff and explain double descent.
This work tackles the challenge of Bayesian deep learning by proposing a new framework for matching Gaussian process priors with neural network parameters.
problem The challenge of specifying priors over neural network parameters, which affects the induced functional prior and is uncontrolled.
method The approach involves defining functional priors using Gaussian processes and matching these priors with the functional prior of neural networks through the minimization of Wasserstein distance.
result The proposed framework offers systematic performance improvements over alternative priors and approximate Bayesian deep learning approaches.
This paper uses reference priors to improve deep learning models with unlabeled and labeled data.
problem Improving deep learning models with limited labeled data and unlabeled data from the same or related tasks.
method Develops and applies generalizations of reference priors for deep networks to exploit unlabeled and labeled data.
result Demonstrates new semi-supervised learning and pretraining methods for transfer learning.
Proposes a new prior for complex models to improve prediction accuracy.
problem Difficulty in specifying priors for complex models like neural networks.
method Predictive complexity priors defined by comparing model predictions to a reference model, transferred to parameters via change of variables.
result Improves model predictions by reducing unintuitive effects of traditional priors.
We construct geometric shrinkage priors for Kählerian signal filters. Based on the characteristics of Kähler manifolds, an efficient and robust algorithm for finding superharmonic priors which outperform the Jeffreys prior is introduced. Several ansätze for the Bayesian predictive priors are also suggested. In particul…
Proposes NUV priors for half-space and box constraints.
problem Adding constraints to linear Gaussian models without computational cost.
method Introduces NUV representations for half-space and box constraints.
result Adds constraints to linear Gaussian models without affecting computational tractability.
Statsformer validates and adapts LLM-derived semantic priors for improved supervised learning.
problem Unreliable semantic priors from LLMs can degrade supervised learning performance.
method Adapts LLM-derived feature scores into a family of learner-specific prior-injection mechanisms, calibrating their influence using out-of-fold validation.
result Improves prediction performance by adaptively downweighting unreliable LLM priors, ensuring a guardrailed statistical learning system.
New priors can update posteriors without re-estimating likelihoods.
problem Degradation of classification approaches when class priors change.
method Recompute posteriors using recovered likelihoods from original posteriors and new priors.
result Dynamic update of original posteriors is possible without re-estimating likelihoods.
SAHMM-VAE separates sources adaptively using hidden Markov priors.
problem Unsupervised blind source separation.
method Source-wise adaptive Hidden Markov prior variational autoencoder.
result Different latent dimensions align with different source-specific temporal organizations.
BNNpriors library improves Bayesian neural network inference with various prior distributions.
problem Challenges in choosing good prior distributions for Bayesian neural networks.
method State-of-the-art Markov Chain Monte Carlo inference with a wide range of predefined priors.
result Facilitates foundational discoveries on the nature of the cold posterior effect.
GOAT improves attention mechanisms by learning better priors.
problem Standard attention mechanisms use a naive uniform prior, limiting flexibility and generalization.
method GOAT introduces a trainable, continuous prior that replaces the uniform assumption, maintaining compatibility with optimized kernels.
result GOAT avoids representational trade-offs and learns an extrapolatable prior that combines positional flexibility with length generalization.
We study the problem of learning shared structure \emph{across} a sequence of dynamic pricing experiments for related products. We consider a practical formulation where the unknown demand parameters for each product come from an unknown distribution (prior) that is shared across products. We then propose a meta dynami…
Weak diffusion priors can still perform well in inverse problems.
problem Using mismatched or low-fidelity diffusion priors in inverse problems.
method Extensive experiments and theoretical analysis combining Bayesian-consistency theory and local-correlation analysis.
result Weak priors succeed when measurements are highly informative, and they fail in other regimes.
New method samples Jeffreys prior for objective Bayesian inference.
problem Sampling from Jeffreys prior is challenging.
method Metropolis-Adjusted Langevin Algorithm
result Samples can be directly used in Bayesian methods.
Optimality of TS with noninformative priors proven for Pareto model.
problem Optimality of Thompson Sampling with noninformative priors for Pareto bandits.
method Proved optimality of TS with certain probability matching priors, showed suboptimality with others, and found effectiveness of truncation procedures.
result TS with certain probability matching priors achieves optimal regret bound for Pareto model.
We study the robustness of active learning (AL) algorithms against prior misspecification: whether an algorithm achieves similar performance using a perturbed prior as compared to using the true prior. In both the average and worst cases of the maximum coverage setting, we prove that all α-approximate algorithms are …
New meta-reinforcement learning method improves performance in finite-horizon MDPs.
problem Improving meta-reinforcement learning in finite-horizon MDPs with shared optimal action-value functions.
method Proposes MTSRL and MTSRL+ algorithms with learned priors and covariance, coupled with prior-alignment technique for meta-regret guarantees.
result Achieves meta-regret guarantees with learned priors and covariance, outperforming prior-independent RL and bandit-only meta-baselines.
The paper discusses the impact of prior densities on Bayesian model selection.
problem The sensitivity of marginal likelihood to prior choice in Bayesian model selection.
method Analyzes the role of prior densities in model selection, discusses improper priors, and proposes solutions.
result Marginal likelihood can be sensitive to prior choice, but improper priors can still be used with caution.
Paper introduces a method to generate physically feasible dynamics with physical priors.
problem Challenges in generating physically feasible dynamics under physical priors.
method Seamlessly incorporates physical priors into diffusion-based generative models.
result Efficient generation of physically realistic dynamics across various physical phenomena.