expectation-maximization

The latent Markov modeling community suffers from fragmented formalisms, inconsistent terminology, and disparate inference methods and software tools—severely hindering practical adoption. To address this, we propose a unified latent-variable Markovian framework for time-series and sequential data, systematically integrating paradigms including hidden Markov models, state-space models, and Markov-modulated Poisson processes under a recursive structural perspective. We introduce a modular (Lego-style) modeling language and develop the efficient R package *LaMa*, whose core implements numerically stable maximum-likelihood estimation in C++ with dynamic programming, forward–backward algorithms, and optimizations for state-dependent structures. This framework substantially lowers the modeling barrier, enabling rapid, robust, and reproducible parameter estimation across all supported models. Moreover, it provides a data-driven, practical roadmap for model selection—bridging theoretical flexibility with empirical usability.

This study addresses the challenge of efficiently modeling high-dimensional time series data exhibiting both spatial dependencies and temporal trends, which traditional hidden Markov models (HMMs) struggle to handle. The authors propose integrating Gaussian random fields into the HMM framework to capture structured latent variables and develop an enhanced forward algorithm that explicitly exploits sparsity in the Hessian matrix of the likelihood function. By leveraging stochastic partial differential equations (SPDEs), Laplace approximation, automatic differentiation, and sparse optimization techniques—implemented within the Template Model Builder (TMB) framework—the method achieves, for the first time, scalable and efficient joint inference of Gaussian fields and HMMs in high dimensions. Empirical evaluations on stellar flare detection and lion movement trajectory modeling demonstrate substantial improvements in computational speed and scalability, confirming the approach’s effectiveness and practical utility.

This work addresses the challenge of posterior inference in high-dimensional continuous-time, discrete-state coupled hidden Markov models under noisy discrete observations—where exact inference is analytically intractable due to Doob’s *h*-transform. We propose an approximate inference framework based on an interacting particle system over latent variables. Our key contribution is a learnable forward-looking function parameterization that explicitly incorporates future observation information; combined with a twisted potential function and sequential Monte Carlo sampling, it yields efficient, low-variance posterior approximations. The method is validated on two complex systems: a graph-structured latent-variable SIRS epidemic model and a real-data-driven neural dynamical model of wildfire propagation. Results demonstrate substantial improvements in both inference accuracy and computational efficiency for high-dimensional continuous-time Markov chains under noisy observations.

Standard Expectation-Maximization (EM) algorithms are ill-suited for statistical models with discrete latent variables due to their reliance on differentiability and continuous latent spaces. Method: We propose the first unified framework integrating Mirror Descent (MD) with Sequential Monte Carlo (SMC) for joint parameter estimation and posterior inference. Our approach jointly minimizes a variational functional over both the parameter space and the space of probability measures, enabling maximum likelihood estimation (MLE) without requiring latent variable continuity. Contribution/Results: This work breaks EM’s dependence on latent variable continuity, marking the first application of MD to MLE in discrete latent variable models, with rigorous convergence guarantees established. Experiments demonstrate significant improvements over standard EM across multiple discrete latent variable tasks; on real-valued latent variable benchmarks, our method matches state-of-the-art performance, validating both theoretical soundness and empirical robustness.

This work addresses the tractability of exact inference and learning in exponential-family latent variable models (LVMs), seeking to characterize the precise boundary of models admitting closed-form analytical solutions without approximation. Method: We derive necessary and sufficient conditions for prior–posterior conjugacy in exponential-family LVMs, providing the first systematic characterization of exact solvability. We further propose a composable graphical model construction framework that preserves structural flexibility while guaranteeing analytic tractability throughout. A general-purpose exact Bayesian inference and parameter learning algorithm is developed, accompanied by an open-source implementation supporting empirical validation across diverse models. Contribution/Results: Our results substantially broaden the class of LVMs amenable to exact inference—bypassing variational approximations or Monte Carlo sampling—and establish a rigorous theoretical foundation and practical toolkit for interpretable, high-precision latent-variable modeling.

This work addresses the challenge of inefficient posterior exploration in hierarchical discrete models with latent variables, where conventional MCMC methods struggle due to the need to integrate out latent variables. The authors propose a similarity-driven MCMC approach that constructs a proposal mechanism based on a data-driven measure of discrepancy between observations and model predictions, thereby guiding transitions toward regions of higher posterior support without explicitly integrating latent variables. This method represents the first application of similarity-driven proposals to discrete-space MCMC and is naturally suited to complex hierarchical discrete models. Experiments on both synthetic and real-world data demonstrate substantial improvements in sampling efficiency and posterior exploration, confirming its effectiveness in models such as Dirichlet–Multinomial regression.

This work addresses the challenge of parameter estimation in dynamic models where the likelihood function is intractable, and existing likelihood-free methods either rely on handcrafted summary statistics or computationally expensive neural networks. To overcome these limitations, the authors propose a simulation-based inference approach leveraging random features. Their key innovation lies in introducing embedding theory from nonlinear dynamical systems into simulation-based inference, enabling identification of a p-dimensional parameter model by matching only a small number (2p+1) of random features between observed and simulated data. The method applies to both stationary and non-stationary processes and, under mild regularity conditions, yields consistent estimators. This framework establishes a new paradigm for dynamic system parameter estimation that is efficient, broadly applicable, and theoretically grounded.

This work addresses the scalability bottleneck in Bayesian latent position models, whose likelihood computation incurs a prohibitive $O(|V|^2)$ complexity. To overcome this limitation, the authors propose two efficient MCMC sampling algorithms that leverage a novel auxiliary data structure termed a “cheap sketch” to enable dynamic updates. The “fast” algorithm reduces computational complexity to nearly $O(|E|)$ while achieving higher accuracy than existing approaches; the “faster” variant further approaches $O(|V|)$ time complexity and yields modest accuracy gains. By integrating stochastic graph modeling, MCMC inference, and dynamic data structure optimization, this study delivers a scalable solution for Bayesian latent space inference in large-scale networks.

This work proposes a Bayesian dynamic latent space model tailored for weighted temporal networks exhibiting complex characteristics such as integer-valued edge weights, zero-inflation, dynamically evolving node latent positions, and time-varying sparsity. The model captures the temporal dependencies of node latent features through vector autoregression and, for the first time in latent space network modeling, incorporates both contemporaneous and lagged dependencies across nodes and latent dimensions. To enhance inference efficiency, a non-recursive block-updating multi-step sampler is developed, integrating auxiliary mixture sampling, Laplace approximation, and partially collapsed Gibbs sampling to substantially improve Markov chain mixing and computational scalability. The framework flexibly accommodates both integer and continuous edge weights and can be readily extended to static or dynamic settings, enabling accurate and efficient inference for complex temporal networks.

This study addresses the computational challenges in Bayesian inference for statistical models involving intractable normalizing functions. It provides a systematic review and comparison of mainstream inference algorithms, including Markov chain Monte Carlo (MCMC), approximate Bayesian computation, pseudo-likelihood, and general likelihood-free methods. The work innovatively constructs a diagnostic framework to assess the accuracy of approximate algorithms, thereby enhancing the reliability of algorithmic tuning. By elucidating the intrinsic connections and applicability boundaries among these approaches, the paper clarifies the trade-offs between theoretical properties and empirical performance. Furthermore, it offers practitioners a clear guideline for algorithm selection, significantly improving the feasibility and credibility of inference in complex models with intractable normalizing constants.