But, they are limited to unimodal, Gaussian approximations of state uncertainty.

Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles.

Latent space Energy-Based Models (EBMs), also known as energy-based priors, have drawn growing interests in the field of generative modeling due to its flexibility in the formulation and strong modeling power of the latent space. However, the common practice of learning latent space EBMs with non-convergent short-run MCMC for prior and posterior sampling is hindering the model from further progress; the degenerate MCMC sampling quality in practice often leads to degraded generation quality and instability in training, especially with highly multi-modal and/or high-dimensional target distributions. To remedy this sampling issue, in this paper we introduce a simple but effective diffusion-based amortization method for long-run MCMC sampling and develop a novel learning algorithm for the latent space EBM based on it. We provide theoretical evidence that the learned amortization of MCMC is a valid long-run MCMC sampler.

One of the fundamental cognitive abilities of humans is to quickly resolve uncertainty by generating hypotheses and testing them via active trials.

Our forward and backward processes will be defined through jump diffusions. A jump diffusion process has two components, the diffusion part and the jump part.

The causal revolution has stimulated interest in understanding complex relationships in various fields.

Clustered Federated Learning (CFL) improves performance under non-IID client heterogeneity by clustering clients and training one model per cluster, thereby balancing between a global model and fully personalized models. However, most CFL methods require the number of clusters K to be fixed a priori, which is impractical when the latent structure is unknown. We propose DPMM-CFL, a CFL algorithm that places a Dirichlet Process (DP) prior over the distribution of cluster parameters. This enables nonparametric Bayesian inference to jointly infer both the number of clusters and client assignments, while optimizing per-cluster federated objectives. This results in a method where, at each round, federated updates and cluster inferences are coupled, as presented in this paper. The algorithm is validated on benchmark datasets under Dirichlet and class-split non-IID partitions.

Classical joint modeling approaches often rely on competing risks or recurrent event formulations to account for complex real-world processes involving evolving longitudinal markers and discrete event occurrences. However, these frameworks typically capture only limited aspects of the underlying event dynamics. Multi-state joint models offer a more flexible alternative by representing full event histories through a network of possible transitions, including recurrent cycles and terminal absorptions, all potentially influenced by longitudinal covariates. In this paper, we propose a general framework that unifies longitudinal biomarker modeling with multi-state event processes defined on arbitrary directed graphs. Our approach accommodates both Markovian and semi-Markovian transition structures, and extends classical joint models by coupling nonlinear mixed-effects longitudinal submodels with multi-state survival processes via shared latent structures. We derive the full likelihood and develop scalable inference procedures based on stochastic gradient descent. Furthermore, we introduce a dynamic prediction framework, enabling individualized risk assessments along complex state-transition trajectories. To facilitate reproducibility and dissemination, we provide an open-source Python library \texttt{jmstate} implementing the proposed methodology, available on \href{https://pypi.org/project/jmstate/}{PyPI}. Simulation experiments and a biomedical case study demonstrate the flexibility and performance of the framework in representing complex longitudinal and multi-state event dynamics. The full Python notebooks used to reproduce the experiments as well as the source code of this paper are available on \href{https://gitlab.com/felixlaplante0/jmstate-paper/}{GitLab}.