Temporal prediction is inherently uncertain, but representing the ambiguity in natural image sequences is a challenging high-dimensional probabilistic inference problem. For natural scenes, the curse of dimensionality renders explicit density estimation statistically and computationally intractable. Here, we describe an implicit regression-based framework for learning and sampling the conditional density of the next frame in a video given previous observed frames. We show that sequence-to-image deep networks trained on a simple resilience-to-noise objective function extract adaptive representations for temporal prediction. Synthetic experiments demonstrate that this score-based framework can handle occlusion boundaries: unlike classical methods that average over bifurcating temporal trajectories, it chooses among likely trajectories, selecting more probable options with higher frequency. Furthermore, analysis of networks trained on natural image sequences reveals that the representation automatically weights predictive evidence by its reliability, which is a hallmark of statistical inference

Evidence-based deep learning represents a burgeoning paradigm for uncertainty estimation, offering reliable predictions with negligible extra computational overheads. Existing methods usually adopt Kullback-Leibler divergence to estimate the uncertainty of network predictions, ignoring domain gaps among various modalities. To tackle this issue, this paper introduces a novel algorithm based on H\"older Divergence (HD) to enhance the reliability of multi-view learning by addressing inherent uncertainty challenges from incomplete or noisy data. Generally, our method extracts the representations of multiple modalities through parallel network branches, and then employs HD to estimate the prediction uncertainties. Through the Dempster-Shafer theory, integration of uncertainty from different modalities, thereby generating a comprehensive result that considers all available representations. Mathematically, HD proves to better measure the ``distance'' between real data distribution and predictive distribution of the model and improve the performances of multi-class recognition tasks. Specifically, our method surpass the existing state-of-the-art counterparts on all evaluating benchmarks. We further conduct extensive experiments on different backbones to verify our superior robustness. It is demonstrated that our method successfully pushes the corresponding performance boundaries. Finally, we perform experiments on more challenging scenarios, \textit{i.e.}, learning with incomplete or noisy data, revealing that our method exhibits a high tolerance to such corrupted data.

Like many chronic diseases, human immunodeficiency virus (HIV) is managed over time at regular clinic visits. At each visit, patient features are assessed, treatments are prescribed, and a subsequent visit is scheduled. There is a need for data-driven methods for both predicting retention and recommending scheduling decisions that optimize retention. Prediction models can be useful for estimating retention rates across a range of scheduling options. However, training such models with electronic health records (EHR) involves several complexities. First, formal causal inference methods are needed to adjust for observed confounding when estimating retention rates under counterfactual scheduling decisions. Second, competing events such as death preclude retention, while censoring events render retention missing. Third, inconsistent monitoring of features such as viral load and CD4 count lead to covariate missingness. This paper presents an all-in-one approach for both predicting HIV retention and optimizing scheduling while accounting for these complexities. We formulate and identify causal retention estimands in terms of potential return-time under a hypothetical scheduling decision. Flexible Bayesian approaches are used to model the observed return-time distribution while accounting for competing and censoring events and form posterior point and uncertainty estimates for these estimands. We address the urgent need for data-driven decision support in HIV care by applying our method to EHR from the Academic Model Providing Access to Healthcare (AMPATH) - a consortium of clinics that treat HIV in Western Kenya.

Authorship attribution aims to identify the origin or author of a document. Traditional approaches have heavily relied on manual features and fail to capture long-range correlations, limiting their effectiveness. Recent advancements leverage text embeddings from pre-trained language models, which require significant fine-tuning on labeled data, posing challenges in data dependency and limited interpretability. Large Language Models (LLMs), with their deep reasoning capabilities and ability to maintain long-range textual associations, offer a promising alternative. This study explores the potential of pre-trained LLMs in one-shot authorship attribution, specifically utilizing Bayesian approaches and probability outputs of LLMs. Our methodology calculates the probability that a text entails previous writings of an author, reflecting a more nuanced understanding of authorship. By utilizing only pre-trained models such as Llama-3-70B, our results on the IMDb and blog datasets show an impressive 85\% accuracy in one-shot authorship classification across ten authors. Our findings set new baselines for one-shot authorship analysis using LLMs and expand the application scope of these models in forensic linguistics. This work also includes extensive ablation studies to validate our approach.

We study the behavior of Thompson sampling from the perspective of weak convergence. In the regime where the gaps between arm means scale as $1/\sqrt{n}$ with the time horizon $n$, we show that the dynamics of Thompson sampling evolve according to discrete versions of SDE's and stochastic ODE's. As $n \to \infty$, we show that the dynamics converge weakly to solutions of the corresponding SDE's and stochastic ODE's. Our weak convergence theory is developed from first principles using the Continuous Mapping Theorem, and can be easily adapted to analyze other sampling-based bandit algorithms. In this regime, we also show that the weak limits of the dynamics of many sampling-based algorithms -- including Thompson sampling designed for any exponential family of rewards, and algorithms involving bootstrap-based sampling -- coincide with those of Gaussian Thompson sampling. Moreover, in this regime, these algorithms are generally robust to model mis-specification.

Flow-based generative modeling is a powerful tool for solving inverse problems in physical sciences that can be used for sampling and likelihood evaluation with much lower inference times than traditional methods. We propose to refine flows with additional control signals based on a simulator. Control signals can include gradients and a problem-specific cost function if the simulator is differentiable, or they can be fully learned from the simulator output. In our proposed method, we pretrain the flow network and include feedback from the simulator exclusively for finetuning, therefore requiring only a small amount of additional parameters and compute. We motivate our design choices on several benchmark problems for simulation-based inference and evaluate flow matching with simulator feedback against classical MCMC methods for modeling strong gravitational lens systems, a challenging inverse problem in astronomy. We demonstrate that including feedback from the simulator improves the accuracy by $53\%$, making it competitive with traditional techniques while being up to $67$x faster for inference.

Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics. Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between efficiency and flexibility. In this paper, we introduce Point Set Diffusion, a diffusion-based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function. By directly learning to stochastically interpolate between noise and data point sets, our approach enables efficient, parallel sampling and flexible generation for complex conditional tasks defined on the metric space. Experiments on synthetic and real-world datasets demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

Multivariate distributions often carry latent structures that are difficult to identify and estimate, and which better reflect the data generating mechanism than extrinsic structures exhibited simply by the raw data. In this paper, we propose a model-free approach for estimating such latent structures whenever they are present, without assuming they exist a priori. Given an arbitrary density $p_0$, we construct a multiscale representation of the density and propose data-driven methods for selecting representative models that capture meaningful discrete structure. Our approach uses a nonparametric maximum likelihood estimator to estimate the latent structure at different scales and we further characterize their asymptotic limits. By carrying out such a multiscale analysis, we obtain coarseto-fine structures inherent in the original distribution, which are integrated via a model selection procedure to yield an interpretable discrete representation of it. As an application, we design a clustering algorithm based on the proposed procedure and demonstrate its effectiveness in capturing a wide range of latent structures.

Motivated by deep neural networks, the deep Gaussian process (DGP) generalizes the standard GP by stacking multiple layers of GPs. Despite the enhanced expressiveness, GP, as an $L_2$ regularization prior, tends to be over-smooth and sub-optimal for inhomogeneous subjects, such as images with edges. Recently, Q-exponential process (Q-EP) has been proposed as an $L_q$ relaxation to GP and demonstrated with more desirable regularization properties through a parameter $q>0$ with $q=2$ corresponding to GP. Sharing the similar tractability of posterior and predictive distributions with GP, Q-EP can also be stacked to improve its modeling flexibility. In this paper, we generalize Q-EP to deep Q-EP to enjoy both proper regularization and improved expressiveness. The generalization is realized by introducing shallow Q-EP as a latent variable model and then building a hierarchy of the shallow Q-EP layers. Sparse approximation by inducing points and scalable variational strategy are applied to facilitate the inference. We demonstrate the numerical advantages of the proposed deep Q-EP model by comparing with multiple state-of-the-art deep probabilistic models.

Patients who are undergoing physical rehabilitation, benefit from feedback that follows from reliable assessment of their cumulative performance attained at a given time. In this paper, we provide a method for the learning of the recovery trajectory of an individual patient, as they undertake exercises as part of their physical therapy towards recovery of their loss of movement ability, following a critical illness. The difference between the Movement Recovery Scores (MRSs) attained by a patient, when undertaking a given exercise routine on successive instances, is given by a statistical distance/divergence between the (posterior) probabilities of random graphs that are Bayesianly learnt using time series data on locations of 20 of the patient's joints, recorded on an e-platform as the patient exercises. This allows for the computation of the MRS on every occasion the patient undertakes this exercise, using which, the recovery trajectory is drawn. We learn each graph as a Random Geometric Graph drawn in a probabilistic metric space, and identify the closed-form marginal posterior of any edge of the graph, given the correlation structure of the multivariate time series data on joint locations. On the basis of our recovery learning, we offer recommendations on the optimal exercise routines for patients with given level of mobility impairment.