Estimating latent epidemic states and model parameters from partially observed, noisy data remains a major challenge in infectious disease modeling. State-space formulations provide a coherent probabilistic framework for such inference, yet fully Bayesian estimation is often computationally prohibitive because evaluating the observed-data likelihood requires integration over a latent trajectory. The Sequential Monte Carlo squared (SMC2) algorithm offers a principled approach for joint state and parameter inference, combining an outer SMC sampler over parameters with an inner particle filter that estimates the likelihood up to the current time point. Despite its theoretical appeal, this nested particle filter imposes substantial computational cost, limiting routine use in near-real-time outbreak response. We propose Ensemble SMC2 (eSMC2), a computationally efficient variant that replaces the inner particle filter with an Ensemble Kalman Filter (EnKF) to approximate the incremental likelihood at each observation time. While this substitution introduces bias via a Gaussian approximation, we mitigate finite-sample effects using an unbiased Gaussian density estimator and adapt the EnKF for epidemic data through state-dependent observation variance. This makes our approach particularly suitable for overdispersed incidence data commonly encountered in infectious disease surveillance. Simulation experiments with known ground truth and an application to 2022 United States (U.S.) monkeypox incidence data demonstrate that eSMC2 achieves substantial computational gains while producing posterior estimates comparable to SMC2. The method accurately recovers latent epidemic trajectories and key epidemiological parameters, providing an efficient framework for sequential Bayesian inference from imperfect surveillance data.

An efficient variational method is presented for estimating the diffusion coefficients and free-energy profiles along selected collective variables from projected molecular dynamics trajectories under both equilibrium and non-equilibrium conditions. This method is based on the assumption that the short-time transition probability density of the coordinate moves can be approximated by a Gaussian form. Defining a loss function as the sum of Kullback-Leibler divergences between the analytical short-time propagators of an overdamped Langevin model and those estimated directly from the projected trajectories maximizes the agreement between the two and allows for its analytic evaluation. For cases where the Gaussian approximation is insufficient, we present a robust alternative. To efficiently minimize this loss function by varying diffusion and free-energy profiles along collective variables, we use an adaptive Monte Carlo scheme. The method is applied to two model systems exhibiting diffusive dynamics, as well as to water diffusion across the interface of a biomolecular condensate, demonstrating its robustness and accuracy.

Quantum hypergraph states form a generalisation of the graph state formalism that goes beyond the pairwise (dyadic) interactions imposed by remaining inside the Gaussian approximation. Networks of such states are able to achieve universality for continuous variable measurement based quantum computation with only Gaussian measurements. For normalised states, the simplest hypergraph states are formed from k -adic interactions among a collection of k harmonic oscillator ground states. However such powerful resources have not yet been observed in experiments and their robustness and scalability have not been tested. Here we develop and analyse necessary criteria for hypergraph nonclassicality based on simultaneous nonlinear squeezing in the nullifiers of hypergraph states. We put forward an essential analysis of their robustness to realistic scenarios involving thermalisation or loss and suggest several basic proof-of-principle options for experiments to observe nonclassicality in hypergraph states.

In human-robot interaction, external force measurement is fundamental to achieving robot compliance control. Parameter identification based on robot dynamics enables external force detection without expensive sensors. However, the unmodeled dynamic errors inherent in real robots pose a challenge to force estimation accuracy. In addition, existing force estimation methods often suffer from high computational dimensionality and an excessive number of tuning parameters, which limits their generalizability and migration to other platforms. In this letter, we employ a Variational Approximate Gaussian Process Regression (VAGP) model to learn the robot's dynamic errors, capturing both the mean and covariance of the error. Then we introduced the indirect measurement form and proposed a dimension-reduced Kalman filter (DRKF) to simplify the state space equation. Finally, we propose a VAGP-based adaptive Kalman filter (VAGAKF) that utilizes the least squares method to reduce the number of tuning parameters. VAGAKF effectively separates external forces from dynamics model uncertainty, reducing reliance on highly accurate robot and external force models. VAGAKF reduces average RMSE and time delay by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${23.06\%}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${66.7\%}$</tex-math></inline-formula> respectively, relative to existing methods.

This article is motivated by challenges in conducting Bayesian inferences on unknown discrete distributions, with a particular focus on count data. To avoid the computational disadvantages of traditional mixture models, we develop a novel Bayesian predictive approach. In particular, our Metropolis-adjusted Dirichlet (mad) sequence model characterizes the predictive measure as a mixture of a base measure and Metropolis-Hastings kernels centered on previous data points. The resulting mad sequence is asymptotically exchangeable and the posterior on the data generator takes the form of a martingale posterior. This structure leads to straightforward algorithms for inference on count distributions, with easy extensions to multivariate, regression, and binary data cases. We obtain a useful asymptotic Gaussian approximation and illustrate the methodology on a variety of applications.

It is well known that the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian can be identified with a planar random walk taking steps of unequal length. In this paper we push this identification further and propose to study the chaotic content of a Hamiltonian H via its associated random walk seen as a fractal, using the tools of fractal geometry. In particular, we conjecture that for chaotic Hamiltonians the Hausdorff dimension of the frontier of the corresponding random walk approaches the universal value d_{F}=4/3-the same value obtained when the random walk describes a Wiener process. Our numerical simulations for nonintegrable models confirm this expectation while for quasifree integrable models we obtain a value d_{F}=1. Additionally, we numerically show that "Bethe-Ansatz walkers" fall into a category similar to the nonintegrable walkers. To motivate this conjecture we consider many-body Hamiltonians with degenerate but rationally independent eigenvalues. We prove that if the degeneracies satisfy certain Lyapunov conditions, the random walk becomes a Wiener process, d_{F}=4/3, and the distribution of the SFF becomes Gaussian. This is the familiar Gaussian approximation for the SFF which we show to be violated at very low temperature. We also compute the moments of the SFF exactly under milder hypotheses thus solving the classical problem of determining the moments of a random walker taking steps of unequal lengths. Finally, we consider quasifree Fermionic models with possibly degenerate but rationally independent one-particle spectra. We show that in this case the distribution of the SFF becomes lognormal and also give the exact form of the moments under milder hypotheses.

Additive fitness landscapes—also called Mount Fuji landscapes—are the simplest and most widely used models of sequence-function relationships. As such, they play essential roles across multiple areas of biology, including evolutionary theory, quantitative genetics, gene regulation, and protein science. One of the most basic properties of any fitness landscape is its genotypic density—the number of sequences near a given fitness value. Understanding this density is especially important near fitness peaks, as it quantifies the supply of high-fitness genotypes. Here I study the genotypic density of additive landscapes near fitness peaks. Although this density is well known to be approximately Gaussian near the middle of the fitness range, its behavior near maximal fitness has not been reported. I begin by deriving a saddle-point approximation that accurately describes the genotypic density of additive landscapes over virtually the entire fitness range. I then show that the log density follows a power law near maximal fitness, with an exponent determined by how much the best allele at each position outperforms its nearest competitor. This power-law behavior holds over a substantial fraction of fitness values, besting the Gaussian approximation on both simulated and empirical landscapes across roughly a quarter to a third of the fitness range. Under certain conditions this behavior also extends to globally epistatic landscapes (defined as nonlinear functions over one or more additive traits), though with a reduced range of validity. These findings advance our understanding of one of the most fundamental models of sequence-function relationships. In particular, they reveal that the uppermost reaches of Mount Fuji landscapes, rather than being sharply peaked, are actually quite stubby.

ABSTRACT While generalized additive models are widely used to estimate smooth nonlinear relationships between responses and covariates, their application to temporal data analysis is limited, as the smooth functions may fail to accurately capture temporal evolution in the data and may yield unstable out-of-sample predictions. To address this limitation, dynamic generalized additive models have been proposed, which comprise two components: a generalized additive component and a component of random effects that evolve according to latent stochastic processes. The model falls within the scope of non-Gaussian state space models. For posterior inference in a Bayesian perspective of the model, Markov chain Monte Carlo algorithms require many iterations to converge, particularly in cases involving high-dimensional non-Gaussian time series observations. Therefore, we employ a variational inference scheme to obtain reasonable results efficiently. Specifically for the coefficients of the spline bases and the random effects, a Gaussian variational approximation is assumed. The optimization of the evidence lower bound is performed using a coordinate ascent variational inference algorithm. The proposed variational approach is more efficient than several competing methods in the dynamic generalized additive model framework. We apply the method to study death counts as a function of observed predictors and temporally dependent multivariate random effects that incorporate dependence structures among geographical locations and among causes of death, using an Italian mortality dataset from January 2015 to December 2020.

The formation of extended sulfur vacancies in MoS2 monolayers is closely associated with catalytic activity and may also be the basis for its memristive behavior. Nanosecond-scale molecular dynamics simulations using machine learning interatomic potentials (MLIPs) reveal key mechanisms of cooperative vacancy transport, including incorporation of vacancies into clusters of arbitrary size. The simulations provide a coherent atomistic explanation for irradiation-induced vacancy patterns observed experimentally, especially the formation of line defects spanning tens of nanometers. Results and performance are compared of two MLIP frameworks: (i) on-the-fly learning with Gaussian approximation potential, and (ii) fine-tuning of an equivariant foundation model.

Bimetallic Pt–Cu nanoparticles are promising catalysts for oxidation and hydrogenation reactions due to their tunable electronic and geometric properties. However, first-principles simulations of realistic nanoparticle sizes remain computationally prohibitive. In this study, Gaussian Approximation Potential (GAP) models were developed for Pt–Cu nanoparticles functionalized with a single O2 or CO molecule, achieving near-DFT accuracy in energies and forces while drastically reducing computational cost. The training dataset, derived from ab initio molecular dynamics (AIMD) trajectories at 300–1000 K, spans various morphologies (pure, core–shell, Janus, and ordered alloys) and particle sizes (38–260 atoms), capturing both thermal and structural fluctuations representative of realistic catalytic conditions. The resulting GAP models successfully reproduce DFT-level energetics and atomic forces with root-mean-square errors below 0.4 meV atom-1 for energies and 70 meV Å-1 for forces, without overfitting to any specific morphology. AIMD simulations reveal that alloying Pt with Cu enhances thermal and mechanical stability, with core–shell and Janus configurations maintaining ordered atomic coordination up to 1000 K. Radial distribution function (RDF) analysis confirms that short-range order persists at elevated temperatures, ensuring structural integrity under reactive conditions. These results demonstrate that machine-learning-based interatomic potentials provide a robust and transferable framework for exploring adsorption-driven restructuring, morphology evolution, and catalytic stability of Pt–Cu nanoparticles beyond the accessible limits of conventional DFT.

Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. To address this issue, this paper reconstructs the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein's lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step's objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we derive its natural gradient that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new iterative filter for nonlinear systems called Natural Gradient Gaussian Approximation filter, or NANOfilter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. Furthermore, the estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like property across consecutive time steps. Real-world experiments demonstrate that, compared to popular Gaussian filters such as EKF, UKF, iterated EKF, and posterior linearization filter, NANO filter reduces the average root mean square error by approximately 45% while maintaining a comparable computational burden.

Abstract We study the impact of introducing fractional dynamics into a tight-binding model on a simple cubic lattice by replacing the standard Laplacian with an approximate fractional counterpart. We derive the dispersion relation in closed form and observe a progressive flattening of the energy bands as the degree of fractional order increases. Using the Gaussian approximation, we compute the density of states (DOS) and find that it becomes narrower and shifts toward the upper band edge with increasing fractionality, indicating a trend toward degeneracy. For an initially localized excitation, the participation ratio increases as the system becomes more fractional, a clear sign of a tendency towards delocalization. The root mean square (RMS) displacement remains ballistic across the parameter range, with a finite propagation speed for the fractional exponents examined. In the impurity problem, and for fractionality near s = 1, a minimum impurity strength is always required to induce a single localized impurity mode, and the increase of fractionality shifts the bound state energy further away from the conduction band when the impurity strength is negative. For positive impurity there is no appreciable shift.

Atrial fibrillation (AF) is a common cardiac arrhythmia characterised by disordered electrical activity in the atria. The standard treatment is catheter ablation, which is invasive and irreversible. Recent advances in computational electrophysiology offer the potential for patient-specific models that can be used to guide clinical decisions. To be of practical value, we must be able to rapidly calibrate physics-based models using routine clinical measurements. We pose this calibration task as a static inverse problem, where the goal is to infer spatially homogenous tissue-level electrophysiological parameters from the available observations. To make this tractable, we replace the expensive forward model with Gaussian process emulators (GPEs), and propose a novel adaptation of the ensemble Kalman filter (EnKF) for static non-linear inverse problems. The approach yields parameter samples that can be interpreted as coming from the best Gaussian approximation of the posterior distribution. We compare our results with those obtained using Markov chain Monte Carlo (MCMC) sampling and demonstrate the potential of the approach to enable near-real-time patient-specific calibration, a key step towards predicting outcomes of AF treatment within clinical timescales. The approach is readily applicable to a wide range of static inverse problems in science and engineering.

Problem definition: During pandemics, policymakers must make critical decisions about public health interventions and allocations of scarce resources in response to rapidly evolving diseases under high levels of uncertainty. Epidemiological models, such as the Susceptible-Exposed-Infectious-Recovered-type (SEIR-type) compartmental model, are indispensable tools for predicting how a pandemic may spread over time and how different public health interventions could affect the outcome. Based on such predictions, deterministic compartmental optimization models can be adopted to attain effective public health intervention decisions. However, deterministic models often neglect parameter uncertainty and the risks inherent in the stochastic compartment dynamics, leading to less robust solutions. Methodology/results: To address these issues, we develop an epidemiological analytics framework based on the ambiguity tolerance measure and stochastic compartmental models. We introduce a robust epidemiological optimization model that lexicographically minimizes the ambiguity tolerances associated with violating healthcare resource constraints. Leveraging the asymptotic Gaussian property, we employ Gaussian approximation to enhance the efficiency of evaluating robust epidemiological constraints. To streamline and automate its application for practitioners and policymakers, we develop a Python-based robust epidemiological analytics modeling (REALM) toolkit. Managerial implications: Employing real-world data from Singapore, we investigate various resource management scenarios, including testing, bed, and vaccine capacity allocations. Our numerical results showcase that our robust epidemiological analytics models outperform deterministic counterpart benchmarks, particularly in the number of hospitalized cases and deaths, given healthcare resource capacity constraints. The results demonstrate the benefits of accounting for risk and ambiguity in disease propagation when addressing epidemiological optimization models. Funding: The research of C. Fu was supported by the National Natural Science Foundation of China [Grants 72401229, 72310107003, and 72271201]. The research of M. Zhou was supported by the National Natural Science Foundation of China [Grants 72301075 and 72293564/72293560]. The research of J. Xie was supported by the Deutsche Forschungsgemeinschaft [Grant 543063591]. The research of M. Sim was supported by the Ministry of Education, Singapore under its 2019 Academic Research Fund Tier 3 [Grant MOE-2019-T3-1-010]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0984 .

Laplace approximation provides a Gaussian approximation of a posterior distribution via a second-order Taylor expansion. Although the Bernstein–von Mises theorem guarantees asymptotic normality as the sample size approaches infinity, the Gaussian approximation may be unreliable when the sample size is finite. This is particularly true when the posterior distribution is skewed, which is a common occurrence in the insurance ratemaking process, where the use of a Gaussian distribution may not yield an accurate approximation. In this study, by utilizing the generalized version of Taylor expansion [Widder (1928). A generalization of Taylor's series. Transactions of the American Mathematical Society, 30(1), 126–154], we introduce a generalized version of Laplace approximation where the posterior distribution is approximated by various parametric distributions in the exponential family. We apply this method to random effects models, connecting it to credibility premium, in the insurance context. While credibility premium provides an affine posterior mean approximation, it lacks further distributional information. Our method introduces the ability to approximate the posterior distribution, while still providing the same point approximation as credibility premium. Numerical analysis confirms the effectiveness of the proposed approach.

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of probability measures open up new avenues for algorithm developments. This paper makes three contributions to this approach to sampling, by scrutinizing the design components of such gradient flows. Any instantiation of a gradient flow for sampling needs an energy functional and a metric to determine the flow, as well as numerical approximations of the flow to derive algorithms. Our first contribution is to show that the Kullback-Leibler (KL) divergence, as an energy functional, has the unique property (among all f f -divergences) that gradient flows resulting from it do not depend on the normalization constant of the target distribution; this justifies the widespread use of the KL divergence in sampling. Our second contribution is to study the choice of metric from the perspective of invariance. The Fisher-Rao metric is known as the unique choice (up to scaling) that is diffeomorphism invariant. As a computationally tractable alternative, we introduce a relaxed, affine invariance property for the metrics and gradient flows. In particular, we construct various affine invariant Wasserstein and Stein gradient flows. Affine invariant gradient flows are shown to behave more favorably than their non-affine-invariant counterparts when sampling highly anisotropic distributions, in theory and by using particle methods. Our third contribution is to study, and develop efficient algorithms based on Gaussian approximations of the gradient flows; this leads to an alternative to particle methods. We establish connections between various Gaussian approximate gradient flows, discuss their relation to gradient methods arising from parametric variational inference, and study their convergence properties. Our theory and numerical experiments demonstrate the strengths and potential limitations of the Gaussian approximate Fisher-Rao gradient flow, which is affine invariant, by considering a wide range of target distributions.

This paper presents the design and hardware implementation of a Gaussian filter using approximate computing techniques to achieve efficient and resource-optimized image processing. Conventional Gaussian filters rely on exact arithmetic units, which increase hardware complexity and power consumption. To address this, the proposed architecture employs approximate adders and multipliers, reducing computational overhead while maintaining acceptable image quality. The design was implemented on the FPGA platform and evaluated across different noisy image datasets, including Gaussian noise, salt-and-pepper noise, and high-frequency images. Experimental results demonstrate significant reductions in hardware resource utilization, with notable improvements in delay. Furthermore, quantitative analysis of image quality metrics such as PSNR, MSSIM, MAE, and MSE confirmed that the approximate Gaussian filter preserved structural details and, in several cases, enhanced noise suppression compared to the exact filter. The results highlight the suitability of approximate arithmetic for embedded and real-time image processing applications, making this work a promising contribution toward energy-efficient and high-performance image processing systems.

Recent advances in transient microscopy have enabled high-resolution imaging of charge carrier dynamics. However, reliance on Gaussian fits to quantify population broadening can lead to misinterpretation when multiple species coexist. Transient scattering microscopy (TScM) provides a powerful alternative, yet its sensitivity to diverse species accentuates the limitations of traditional Gaussian fits. Here, we use TScM to visualize exciton transport in bulk transition metal dichalcogenides (TMDCs) and reveal that exciton populations exhibit non-Gaussian profiles by analyzing their excess kurtosis. Simulations incorporating anomalous diffusion reproduce these experimental observations and find that the signature of the kurtosis is distinct for coexisting populations and trap-dominated regimes. Additionally, we implement a discrete variable calculation to extract the variances which yields robust, consistent diffusivity values where Gaussian fits fail to do so. Our results establish kurtosis as a vital diagnostic parameter for identifying anomalous diffusion and demonstrate the necessity of moving beyond Gaussian approximations for analysis of TScM data.

Gaussian variational approximations are widely used for summarizing posterior distributions in Bayesian models, especially in high-dimensional settings. However, a drawback of such approximations is the inability to capture skewness or more complex features of the posterior. Recent work suggests applying skewness corrections to existing Gaussian or other symmetric approximations to address this limitation. We propose to incorporate the skewness correction into the definition of an approximating variational family. We consider approximating the posterior for hierarchical models, in which there are “global” and “local” parameters. A baseline variational approximation is defined as the product of a Gaussian marginal posterior for global parameters and a Gaussian conditional posterior for local parameters given the global ones. Skewness corrections are then considered. The adjustment of the conditional posterior term for local variables is adaptive to the global parameter value. Optimization of baseline variational parameters is performed jointly with the skewness correction. Our approach allows the location, scale, and skewness to be captured separately, without using additional parameters for skewness adjustments. The proposed method substantially improves accuracy for only a modest increase in computational cost compared to state-of-the-art Gaussian approximations. Good performance is demonstrated in generalized linear mixed models and multinomial logit discrete choice models.

We measure three-dimensional bispectra of halo intrinsic alignments (IA) and dark matter overdensities in real space from N-body simulations for halos of mass 10 12 − 10 12.5 M ⊙ / h . We show that their multipoles with respect to the line of sight can be accurately described by a tree-level perturbation theory model on large scales ( k ≲ 0.11 h /Mpc) at z = 0 . For these scales and in a simulation volume of 1 (Gpc/ h ) 3 , we detect the bispectrum monopole B δ δ E 00 at . We also report similar for the lowest order multipoles of B δ E E and B E E E , although these are largely driven by stochastic contributions. We show that the first and second order EFT parameters are consistent with those obtained from fitting the IA power spectrum analysis at next-to-leading order, without requiring any priors to break degeneracies for the quadratic bias parameters. Moreover, the inclusion of higher multipole moments of B δ δ E greatly reduces the errors on second order bias parameters, by factors of 5 or more. The IA bispectrum thus provides an effective means of determining higher order shape bias parameters, thereby characterizing the scale dependence of the IA signal. We also detect parity-odd bispectra such as B δ δ B and B δ E B at ∼ 10 σ significance or more for k &lt; 0.15 h /Mpc and they are consistent with the parity-even sector. Furthermore, we check that the Gaussian covariance approximation works reasonably well on the scales we consider here. These results lay the groundwork for using the bispectrum of IA in cosmological analyses.