The assumptions of stationarity and isotropy often stated over spatial processes have not aged well during the last two decades, partly explained by the combination of computational developments and the increasing availability of high-resolution spatial data. While a plethora of approaches have been developed to relax these assumptions, it is often a costly tradeoff between flexibility and a diversity of computational challenges. In this paper, we present a class of covariance functions that relies on fixed, observable spatial information that provides a convenient tradeoff while offering an extra layer of numerical and visual representation of the flexible spatial dependencies. This model allows for separate parametric structures for different sources of nonstationarity, such as marginal standard deviation, geometric anisotropy, and smoothness. It simplifies to a Mat\'ern covariance function in its basic form and is adaptable for large datasets, enhancing flexibility and computational efficiency. We analyze the capabilities of the presented model through simulation studies and an application to Swiss precipitation data.

One of the main challenges of federated learning (FL) is handling non-independent and identically distributed (non-IID) client data, which may occur in practice due to unbalanced datasets and use of different data sources across clients. Knowledge sharing and model personalization are key strategies for addressing this issue. Clustered federated learning is a class of FL methods that groups clients that observe similarly distributed data into clusters, such that every client is typically associated with one data distribution and participates in training a model for that distribution along their cluster peers. In this paper, we present a unified Bayesian framework for clustered FL which associates clients to clusters. Then we propose several practical algorithms to handle the, otherwise growing, data associations in a way that trades off performance and computational complexity. This work provides insights on client-cluster associations and enables client knowledge sharing in new ways. The proposed framework circumvents the need for unique client-cluster associations, which is seen to increase the performance of the resulting models in a variety of experiments.

Conventional medical artificial intelligence (AI) models face barriers in clinical application and ethical issues owing to their inability to handle the privacy-sensitive characteristics of medical data. We present a novel personalized federated learning (pFL) method for medical visual question answering (VQA) models, addressing privacy reliability challenges in the medical domain. Our method introduces learnable prompts into a Transformer architecture to efficiently train it on diverse medical datasets without massive computational costs. Then we introduce a reliable client VQA model that incorporates Dempster-Shafer evidence theory to quantify uncertainty in predictions, enhancing the model's reliability. Furthermore, we propose a novel inter-client communication mechanism that uses maximum likelihood estimation to balance accuracy and uncertainty, fostering efficient integration of insights across clients.

This paper presents a novel Integer Programming (IP) approach for discovering the Markov Equivalent Class (MEC) of Bayesian Networks (BNs) through observational data. The MEC-IP algorithm utilizes a unique clique-focusing strategy and Extended Maximal Spanning Graphs (EMSG) to streamline the search for MEC, thus overcoming the computational limitations inherent in other existing algorithms. Our numerical results show that not only a remarkable reduction in computational time is achieved by our algorithm but also an improvement in causal discovery accuracy is seen across diverse datasets. These findings underscore this new algorithm's potential as a powerful tool for researchers and practitioners in causal discovery and BNSL, offering a significant leap forward toward the efficient and accurate analysis of complex data structures.

Dimensionality reduction (DR) offers a useful representation of complex high-dimensional data. Recent DR methods focus on hyperbolic geometry to derive a faithful low-dimensional representation of hierarchical data. However, existing methods are based on neighbor embedding, frequently ruining the continual relation of the hierarchies. This paper presents hyperboloid Gaussian process (GP) latent variable models (hGP-LVMs) to embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation. We adopt generative modeling using the GP, which brings effective hierarchical embedding and executes ill-posed hyperparameter tuning. This paper presents three variants that employ original point, sparse point, and Bayesian estimations. We establish their learning algorithms by incorporating the Riemannian optimization and active approximation scheme of GP-LVM. For Bayesian inference, we further introduce the reparameterization trick to realize Bayesian latent variable learning. In the last part of this paper, we apply hGP-LVMs to several datasets and show their ability to represent high-dimensional hierarchies in low-dimensional spaces.

Bayesian inference offers a robust framework for updating prior beliefs based on new data using Bayes' theorem, but exact inference is often computationally infeasible, necessitating approximate methods. Though widely used, these methods struggle to estimate marginal likelihoods accurately, particularly due to the rigid functional structures of deterministic models like Gaussian processes and the limitations of small sample sizes in stochastic models like the ensemble Kalman method. In this work, we introduce BI-EqNO, an equivariant neural operator framework for generalized approximate Bayesian inference, designed to enhance both deterministic and stochastic approaches. BI-EqNO transforms priors into posteriors conditioned on observation data through data-driven training. The framework is flexible, supporting diverse prior and posterior representations with arbitrary discretizations and varying numbers of observations. Crucially, BI-EqNO's architecture ensures (1) permutation equivariance between prior and posterior representations, and (2) permutation invariance with respect to observational data. We demonstrate BI-EqNO's utility through two examples: (1) as a generalized Gaussian process (gGP) for regression, and (2) as an ensemble neural filter (EnNF) for sequential data assimilation. Results show that gGP outperforms traditional Gaussian processes by offering a more flexible representation of covariance functions. Additionally, EnNF not only outperforms the ensemble Kalman filter in small-ensemble settings but also has the potential to function as a "super" ensemble filter, capable of representing and integrating multiple ensemble filters for enhanced assimilation performance. This study highlights BI-EqNO's versatility and effectiveness, improving Bayesian inference through data-driven training while reducing computational costs across various applications.

Reconstructing cosmological initial conditions (ICs) from late-time observations is a difficult task, which relies on the use of computationally expensive simulators alongside sophisticated statistical methods to navigate multi-million dimensional parameter spaces. We present a simple method for Bayesian field reconstruction based on modeling the posterior distribution of the initial matter density field to be diagonal Gaussian in Fourier space, with its covariance and the mean estimator being the trainable parts of the algorithm. Training and sampling are extremely fast (training: $\sim 1 \, \mathrm{h}$ on a GPU, sampling: $\lesssim 3 \, \mathrm{s}$ for 1000 samples at resolution $128^3$), and our method supports industry-standard (non-differentiable) $N$-body simulators. We verify the fidelity of the obtained IC samples in terms of summary statistics.

Stein variational inference (SVI) is a sample-based approximate Bayesian inference technique that generates a sample set by jointly optimizing the samples' locations to minimize an information-theoretic measure of discrepancy with the target probability distribution. SVI thus provides a fast and significantly more sample-efficient approach to Bayesian inference than traditional (random-sampling-based) alternatives. However, the optimization techniques employed in existing SVI methods struggle to address problems in which the target distribution is high-dimensional, poorly-conditioned, or non-convex, which severely limits the range of their practical applicability. In this paper, we propose a novel trust-region optimization approach for SVI that successfully addresses each of these challenges. Our method builds upon prior work in SVI by leveraging conditional independences in the target distribution (to achieve high-dimensional scaling) and second-order information (to address poor conditioning), while additionally providing an effective adaptive step control procedure, which is essential for ensuring convergence on challenging non-convex optimization problems. Experimental results show our method achieves superior numerical performance, both in convergence rate and sample accuracy, and scales better in high-dimensional distributions, than previous SVI techniques.

We show that for a given DAG $G$, among all observational distributions of Bayesian networks over $G$ with arbitrary outcome spaces, the faithful distributions are `typical': they constitute a dense, open set with respect to the total variation metric. As a consequence, the set of faithful distributions is non-empty, and the unfaithful distributions are nowhere dense. We extend this result to the space of Bayesian networks, where the properties hold for Bayesian networks instead of distributions of Bayesian networks. As special cases, we show that these results also hold for the faithful parameters of the subclasses of linear Gaussian -- and discrete Bayesian networks, giving a topological analogue of the measure-zero results of Spirtes et al. (1993) and Meek (1995). Finally, we extend our topological results and the measure-zero results of Spirtes et al. and Meek to Bayesian networks with latent variables.

Generative Flow Networks (GFlowNets) are amortized inference models designed to sample from unnormalized distributions over composable objects, with applications in generative modeling for tasks in fields such as causal discovery, NLP, and drug discovery. Traditionally, the training procedure for GFlowNets seeks to minimize the expected log-squared difference between a proposal (forward policy) and a target (backward policy) distribution, which enforces certain flow-matching conditions. While this training procedure is closely related to variational inference (VI), directly attempting standard Kullback-Leibler (KL) divergence minimization can lead to proven biased and potentially high-variance estimators. Therefore, we first review four divergence measures, namely, Renyi-$\alpha$'s, Tsallis-$\alpha$'s, reverse and forward KL's, and design statistically efficient estimators for their stochastic gradients in the context of training GFlowNets. Then, we verify that properly minimizing these divergences yields a provably correct and empirically effective training scheme, often leading to significantly faster convergence than previously proposed optimization. To achieve this, we design control variates based on the REINFORCE leave-one-out and score-matching estimators to reduce the variance of the learning objectives' gradients. Our work contributes by narrowing the gap between GFlowNets training and generalized variational approximations, paving the way for algorithmic ideas informed by the divergence minimization viewpoint.