Adversarial robustness of machine learning models is critical to ensuring reliable performance under data perturbations. Recent progress has been on point estimators, and this paper considers distributional predictors. First, using the link between exponential families and Bregman divergences, we formulate an adversarial Bregman divergence loss as an adversarial negative log-likelihood. Using the geometric properties of Bregman divergences, we compute the adversarial perturbation for such models in closed-form. Second, under such losses, we introduce \emph{adversarially robust posteriors}, by exploiting the optimization-centric view of generalized Bayesian inference. Third, we derive the \emph{first} rigorous generalization certificates in the context of an adversarial extension of Bayesian linear regression by leveraging the PAC-Bayesian framework. Finally, experiments on real and synthetic datasets demonstrate the superior robustness of the derived adversarially robust posterior over Bayes posterior, and also validate our theoretical guarantees.
A near-field motion parameter estimation method is proposed. In contract to far-field sensing systems, the near-field sensing system leverages spherical-wave characteristics to enable full-vector location and velocity estimation. Despite promising advantages, the near-field sensing system faces a significant challenge, where location and velocity parameters are intricately coupled within the signal. To address this challenge, a novel subarray-based variational message passing (VMP) method is proposed for near-field joint location and velocity estimation. First, a factor graph representation is introduced, employing subarray-level directional and Doppler parameters as intermediate variables to decouple the complex location-velocity dependencies. Based on this, the variational Bayesian inference is employed to obtain closed-form posterior distributions of subarray-level parameters. Subsequently, the message passing technique is employed, enabling tractable computation of location and velocity marginal distributions. Two implementation strategies are proposed: 1) System-level fusion that aggregates all subarray posteriors for centralized estimation, or 2) Subarray-level fusion where locally processed estimates from subarrays are fused through Guassian product rule. Cram\'er-Rao bounds for location and velocity estimation are derived, providing theoretical performance limits. Numerical results demonstrate that the proposed VMP method outperforms existing approaches while achieving a magnitude lower complexity. Specifically, the proposed VMP method achieves centimeter-level location accuracy and sub-m/s velocity accuracy. It also demonstrates robust performance for high-mobility targets, making the proposed VMP method suitable for real-time near-field sensing and communication applications.
A central statistical problem in population genetics is to infer evolutionary and biological parameters such as the strength of natural selection and allele age from DNA samples extracted from a contemporary population. That all samples come only from the present-day has long been known to limit statistical inference; there is potentially more information available if one also has access to ancient DNA so that inference is based on a time-series of historical changes in allele frequencies. We introduce a Markov Chain Monte Carlo (MCMC) method for Bayesian inference from allele frequency time-series data based on an underlying Wright--Fisher diffusion model of evolution, through which one can infer the parameters of essentially any selection model including those with frequency-dependent effects. The chief novelty is that we show this method to be exact in the sense that it is possible to augment the state space explored by MCMC with the unobserved diffusion trajectory, even though the transition function of this diffusion is intractable. Through careful design of a proposal distribution, we describe an efficient method in which updates to the trajectory and accept/reject decisions are calculated without error. We illustrate the method on data capturing changes in coat colour over the past 20,000 years, and find evidence to support previous findings that the mutant alleles ASIP and MC1R responsible for changes in coat color have experienced very strong, possibly overdominant, selection and further provide estimates for the ages of these genes.
Inverse problems arise almost everywhere in science and engineering where we need to infer on a quantity from indirect observation. The cases of medical, biomedical, and industrial imaging systems are the typical examples. A very high overview of classification of the inverse problems method can be: i) Analytical, ii) Regularization, and iii) Bayesian inference methods. Even if there are straight links between them, we can say that the Bayesian inference based methods are the most powerful, as they give the possibility of accounting for prior knowledge and can account for errors and uncertainties in general. One of the main limitations stay in computational costs in particular for high dimensional imaging systems. Neural Networks (NN), and in particular Deep NNs (DNN), have been considered as a way to push farther this limit. Physics Informed Neural Networks (PINN) concept integrates physical laws with deep learning techniques to enhance the speed, accuracy and efficiency of the above mentioned problems. In this work, a new Bayesian framework for the concept of PINN (BPINN) is presented and discussed which includes the deterministic one if we use the Maximum A Posteriori (MAP) estimation framework. We consider two cases of supervised and unsupervised for training step, obtain the expressions of the posterior probability of the unknown variables, and deduce the posterior laws of the NN's parameters. We also discuss about the challenges of implementation of these methods in real applications.
In Bayesian inference, computing the posterior distribution from the data is typically a non-trivial problem, which usually requires approximations such as mean-field approaches or numerical methods, like the Monte Carlo Markov Chain. Being a high-dimensional distribution over a set of correlated variables, the posterior distribution can undergo the notorious replica symmetry breaking transition. When it happens, several mean-field methods and virtually every Monte Carlo scheme can not provide a reasonable approximation to the posterior and its marginals. Replica symmetry is believed to be guaranteed whenever the data is generated with known prior and likelihood distributions, namely under the so-called Nishimori conditions. In this paper, we break this belief, by providing a counter-example showing that, under the Nishimori conditions, replica symmetry breaking arises. Introducing a simple, geometrical model that can be thought of as a patient zero retrieval problem in a highly infectious regime of the epidemic Susceptible-Infectious model, we show that under the Nishimori conditions, there is evidence of replica symmetry breaking. We achieve this result by computing the instability of the replica symmetric cavity method toward the one step replica symmetry broken phase. The origin of this phenomenon -- replica symmetry breaking under the Nishimori conditions -- is likely due to the correlated disorder appearing in the epidemic models.
Bayesian and frequentist inference are two fundamental paradigms in statistical estimation. Bayesian methods treat hypotheses as random variables, incorporating priors and updating beliefs via Bayes' theorem, whereas frequentist methods assume fixed but unknown hypotheses, relying on estimators like maximum likelihood. While extensive research has compared these approaches, the frequentist paradigm of obtaining point estimates has become predominant in deep learning, as Bayesian inference is challenging due to the computational complexity and the approximation gap of posterior estimation methods. However, a good understanding of trade-offs between the two approaches is lacking in the regime of amortized estimators, where in-context learners are trained to estimate either point values via maximum likelihood or maximum a posteriori estimation, or full posteriors using normalizing flows, score-based diffusion samplers, or diagonal Gaussian approximations, conditioned on observations. To help resolve this, we conduct a rigorous comparative analysis spanning diverse problem settings, from linear models to shallow neural networks, with a robust evaluation framework assessing both in-distribution and out-of-distribution generalization on tractable tasks. Our experiments indicate that amortized point estimators generally outperform posterior inference, though the latter remain competitive in some low-dimensional problems, and we further discuss why this might be the case.
Tuberculosis (TB) remains a formidable global health challenge, driven by complex spatiotemporal transmission dynamics and influenced by factors such as population mobility and behavioral changes. We propose an Epidemic-Guided Deep Learning (EGDL) approach that fuses mechanistic epidemiological principles with advanced deep learning techniques to enhance early warning systems and intervention strategies for TB outbreaks. Our framework is built upon a modified networked Susceptible-Infectious-Recovered (MN-SIR) model augmented with a saturated incidence rate and graph Laplacian diffusion, capturing both long-term transmission dynamics and region-specific population mobility patterns. Compartmental model parameters are rigorously estimated using Bayesian inference via the Markov Chain Monte Carlo approach. Theoretical analysis leveraging the comparison principle and Green's formula establishes global stability properties of the disease-free and endemic equilibria. Building on these epidemiological insights, we design two forecasting architectures, EGDL-Parallel and EGDL-Series, that integrate the mechanistic outputs of the MN-SIR model within deep neural networks. This integration mitigates the overfitting risks commonly encountered in data-driven methods and filters out noise inherent in surveillance data, resulting in reliable forecasts of real-world epidemic trends. Experiments conducted on TB incidence data from 47 prefectures in Japan and 31 provinces in mainland China demonstrate that our approach delivers robust and accurate predictions across multiple time horizons (short to medium-term forecasts), supporting its generalizability across regions with different population dynamics.
Across many domains of science, stochastic models are an essential tool to understand the mechanisms underlying empirically observed data. Models can be of different levels of detail and accuracy, with models of high-fidelity (i.e., high accuracy) to the phenomena under study being often preferable. However, inferring parameters of high-fidelity models via simulation-based inference is challenging, especially when the simulator is computationally expensive. We introduce MF-NPE, a multifidelity approach to neural posterior estimation that leverages inexpensive low-fidelity simulations to infer parameters of high-fidelity simulators within a limited simulation budget. MF-NPE performs neural posterior estimation with limited high-fidelity resources by virtue of transfer learning, with the ability to prioritize individual observations using active learning. On one statistical task with analytical ground-truth and two real-world tasks, MF-NPE shows comparable performance to current approaches while requiring up to two orders of magnitude fewer high-fidelity simulations. Overall, MF-NPE opens new opportunities to perform efficient Bayesian inference on computationally expensive simulators.
Reverberant speech, denoting the speech signal degraded by the process of reverberation, contains crucial knowledge of both anechoic source speech and room impulse response (RIR). This work proposes a variational Bayesian inference (VBI) framework with neural speech prior (VINP) for joint speech dereverberation and blind RIR identification. In VINP, a probabilistic signal model is constructed in the time-frequency (T-F) domain based on convolution transfer function (CTF) approximation. For the first time, we propose using an arbitrary discriminative dereverberation deep neural network (DNN) to predict the prior distribution of anechoic speech within a probabilistic model. By integrating both reverberant speech and the anechoic speech prior, VINP yields the maximum a posteriori (MAP) and maximum likelihood (ML) estimations of the anechoic speech spectrum and CTF filter, respectively. After simple transformations, the waveforms of anechoic speech and RIR are estimated. Moreover, VINP is effective for automatic speech recognition (ASR) systems, which sets it apart from most deep learning (DL)-based single-channel dereverberation approaches. Experiments on single-channel speech dereverberation demonstrate that VINP reaches an advanced level in most metrics related to human perception and displays unquestionable state-of-the-art (SOTA) performance in ASR-related metrics. For blind RIR identification, experiments indicate that VINP attains the SOTA level in blind estimation of reverberation time at 60 dB (RT60) and direct-to-reverberation ratio (DRR). Codes and audio samples are available online.
The horseshoe prior, a widely used handy alternative to the spike-and-slab prior, has proven to be an exceptional default global-local shrinkage prior in Bayesian inference and machine learning. However, designing tests with frequentist false discovery rate (FDR) control using the horseshoe prior or the general class of global-local shrinkage priors remains an open problem. In this paper, we propose a frequentist-assisted horseshoe procedure that not only resolves this long-standing FDR control issue for the high dimensional normal means testing problem but also exhibits satisfactory finite-sample FDR control under any desired nominal level for both large-scale multiple independent and correlated tests. We carry out the frequentist-assisted horseshoe procedure in an easy and intuitive way by using the minimax estimator of the global parameter of the horseshoe prior while maintaining the remaining full Bayes vanilla horseshoe structure. The results of both intensive simulations under different sparsity levels, and real-world data demonstrate that the frequentist-assisted horseshoe procedure consistently achieves robust finite-sample FDR control. Existing frequentist or Bayesian FDR control procedures can lose finite-sample FDR control in a variety of common sparse cases. Based on the intimate relationship between the minimax estimation and the level of FDR control discovered in this work, we point out potential generalizations to achieve FDR control for both more complicated models and the general global-local shrinkage prior family.
Neural networks are fragile when confronted with data that significantly deviates from their training distribution. This is true in particular for simulation-based inference methods, such as neural amortized Bayesian inference (ABI), where models trained on simulated data are deployed on noisy real-world observations. Recent robust approaches employ unsupervised domain adaptation (UDA) to match the embedding spaces of simulated and observed data. However, the lack of comprehensive evaluations across different domain mismatches raises concerns about the reliability in high-stakes applications. We address this gap by systematically testing UDA approaches across a wide range of misspecification scenarios in silico and practice. We demonstrate that aligning summary spaces between domains effectively mitigates the impact of unmodeled phenomena or noise. However, the same alignment mechanism can lead to failures under prior misspecifications - a critical finding with practical consequences. Our results underscore the need for careful consideration of misspecification types when using UDA to increase the robustness of ABI.
As whole genomes become widely available, maximum likelihood and Bayesian phylogenetic methods are demonstrating their limits in meeting the escalating computational demands. Conversely, distance-based phylogenetic methods are efficient, but are rarely favoured due to their inferior performance. Here, we extend distance-based phylogenetics using an entropy-based likelihood of the evolution among pairs of taxa, allowing for fast Bayesian inference in genome-scale datasets. We provide evidence of a close link between the inference criteria used in distance methods and Felsenstein's likelihood, such that the methods are expected to have comparable performance in practice. Using the entropic likelihood, we perform Bayesian inference on three phylogenetic benchmark datasets and find that estimates closely correspond with previous inferences. We also apply this rapid inference approach to a 60-million-site alignment from 363 avian taxa, covering most avian families. The method has outstanding performance and reveals substantial uncertainty in the avian diversification events immediately after the K-Pg transition event. The entropic likelihood allows for efficient Bayesian phylogenetic inference, accommodating the analysis demands of the genomic era.
Neural ordinary differential equations (ODEs) provide expressive representations of invertible transport maps that can be used to approximate complex probability distributions, e.g., for generative modeling, density estimation, and Bayesian inference. We show that for a large class of transport maps $T$, there exists a time-dependent ODE velocity field realizing a straight-line interpolation $(1-t)x + tT(x)$, $t \in [0,1]$, of the displacement induced by the map. Moreover, we show that such velocity fields are minimizers of a training objective containing a specific minimum-energy regularization. We then derive explicit upper bounds for the $C^k$ norm of the velocity field that are polynomial in the $C^k$ norm of the corresponding transport map $T$; in the case of triangular (Knothe--Rosenblatt) maps, we also show that these bounds are polynomial in the $C^k$ norms of the associated source and target densities. Combining these results with stability arguments for distribution approximation via ODEs, we show that Wasserstein or Kullback--Leibler approximation of the target distribution to any desired accuracy $\epsilon > 0$ can be achieved by a deep neural network representation of the velocity field whose size is bounded explicitly in terms of $\epsilon$, the dimension, and the smoothness of the source and target densities. The same neural network ansatz yields guarantees on the value of the regularized training objective.
Simulation-based calibration checking (SBC) refers to the validation of an inference algorithm and model implementation through repeated inference on data simulated from a generative model. In the original and commonly used approach, the generative model uses parameters drawn from the prior, and thus the approach is testing whether the inference works for simulated data generated with parameter values plausible under that prior. This approach is natural and desirable when we want to test whether the inference works for a wide range of datasets we might observe. However, after observing data, we are interested in answering whether the inference works conditional on that particular data. In this paper, we propose posterior SBC and demonstrate how it can be used to validate the inference conditionally on observed data. We illustrate the utility of posterior SBC in three case studies: (1) A simple multilevel model; (2) a model that is governed by differential equations; and (3) a joint integrative neuroscience model which is approximated via amortized Bayesian inference with neural networks.
Subspace inference for neural networks assumes that a subspace of their parameter space suffices to produce a reliable uncertainty quantification. In this work, we mathematically derive the optimal subspace model to a Bayesian inference scenario based on the Laplace approximation. We demonstrate empirically that, in the optimal case, often a fraction of parameters less than 1% is sufficient to obtain a reliable estimate of the full Laplace approximation. Since the optimal solution is derived, we can evaluate all other subspace models against a baseline. In addition, we give an approximation of our method that is applicable to larger problem settings, in which the optimal solution is not computable, and compare it to existing subspace models from the literature. In general, our approximation scheme outperforms previous work. Furthermore, we present a metric to qualitatively compare different subspace models even if the exact Laplace approximation is unknown.
What computational structures emerge in transformers trained on next-token prediction? In this work, we provide evidence that transformers implement constrained Bayesian belief updating -- a parallelized version of partial Bayesian inference shaped by architectural constraints. To do this, we integrate the model-agnostic theory of optimal prediction with mechanistic interpretability to analyze transformers trained on a tractable family of hidden Markov models that generate rich geometric patterns in neural activations. We find that attention heads carry out an algorithm with a natural interpretation in the probability simplex, and create representations with distinctive geometric structure. We show how both the algorithmic behavior and the underlying geometry of these representations can be theoretically predicted in detail -- including the attention pattern, OV-vectors, and embedding vectors -- by modifying the equations for optimal future token predictions to account for the architectural constraints of attention. Our approach provides a principled lens on how gradient descent resolves the tension between optimal prediction and architectural design.
While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.
This paper considers a joint scattering environment sensing and data recovery problem in an uplink integrated sensing and communication (ISAC) system. To facilitate joint scatterers localization and multi-user (MU) channel estimation, we introduce a three-dimensional (3D) location-domain sparse channel model to capture the joint sparsity of the MU channel (i.e., different user channels share partially overlapped scatterers). Then the joint problem is formulated as a bilinear structured sparse recovery problem with a dynamic position grid and imperfect parameters (such as time offset and user position errors). We propose an expectation maximization based turbo bilinear subspace variational Bayesian inference (EM-Turbo-BiSVBI) algorithm to solve the problem effectively, where the E-step performs Bayesian estimation of the the location-domain sparse MU channel by exploiting the joint sparsity, and the M-step refines the dynamic position grid and learns the imperfect factors via gradient update. Two methods are introduced to greatly reduce the complexity with almost no sacrifice on the performance and convergence speed: 1) a subspace constrained bilinear variational Bayesian inference (VBI) method is proposed to avoid any high-dimensional matrix inverse; 2) the multiple signal classification (MUSIC) and subspace constrained VBI methods are combined to obtain a coarse estimation result to reduce the search range. Simulations verify the advantages of the proposed scheme over baseline schemes.
Big data and the rapid development of artificial intelligence (AI) provide unprecedented opportunities to enhance our understanding of the global carbon cycle and other biogeochemical processes. However, retrieving mechanistic knowledge from big data remains a challenge. Here, we develop a Biogeochemistry-Informed Neural Network (BINN) that seamlessly integrates a vectorized process-based soil carbon cycle model (i.e., Community Land Model version 5, CLM5) into a neural network (NN) structure to examine mechanisms governing soil organic carbon (SOC) storage from big data. BINN demonstrates high accuracy in retrieving biogeochemical parameter values from synthetic data in a parameter recovery experiment. We use BINN to predict six major processes regulating the soil carbon cycle (or components in process-based models) from 25,925 observed SOC profiles across the conterminous US and compared them with the same processes previously retrieved by a Bayesian inference-based PROcess-guided deep learning and DAta-driven modeling (PRODA) approach (Tao et al. 2020; 2023). The high agreement between the spatial patterns of the retrieved processes using the two approaches with an average correlation coefficient of 0.81 confirms BINN's ability in retrieving mechanistic knowledge from big data. Additionally, the integration of neural networks and process-based models in BINN improves computational efficiency by more than 50 times over PRODA. We conclude that BINN is a transformative tool that harnesses the power of both AI and process-based modeling, facilitating new scientific discoveries while improving interpretability and accuracy of Earth system models.
Graph representation learning has emerged as a cornerstone for tasks like node classification and link prediction, yet prevailing self-supervised learning (SSL) methods face challenges such as computational inefficiency, reliance on contrastive objectives, and representation collapse. Existing approaches often depend on feature reconstruction, negative sampling, or complex decoders, which introduce training overhead and hinder generalization. Further, current techniques which address such limitations fail to account for the contribution of node embeddings to a certain prediction in the absence of labeled nodes. To address these limitations, we propose a novel joint embedding predictive framework for graph SSL that eliminates contrastive objectives and negative sampling while preserving semantic and structural information. Additionally, we introduce a semantic-aware objective term that incorporates pseudo-labels derived from Gaussian Mixture Models (GMMs), enhancing node discriminability by evaluating latent feature contributions. Extensive experiments demonstrate that our framework outperforms state-of-the-art graph SSL methods across benchmarks, achieving superior performance without contrastive loss or complex decoders. Key innovations include (1) a non-contrastive, view-invariant joint embedding predictive architecture, (2) Leveraging single context and multiple targets relationship between subgraphs, and (3) GMM-based pseudo-label scoring to capture semantic contributions. This work advances graph SSL by offering a computationally efficient, collapse-resistant paradigm that bridges spatial and semantic graph features for downstream tasks. The code for our paper can be found at https://github.com/Deceptrax123/JPEB-GSSL