fit sum-dependent markov chains

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.

In intermittent observational settings, conventional Markov assumptions in multi-state modeling are overly restrictive, as they ignore state-specific sojourn time dependencies. Method: We propose a general semi-Markov modeling framework that characterizes arbitrary sojourn time distributions via phase-type distributions, approximates gamma or Weibull distributions using moment-matching techniques, and embeds them within a hidden Markov model structure to enable analytical likelihood computation. The approach unifies Bayesian and maximum likelihood estimation, accommodates time-varying covariates, and supports complex transition structures. Contribution/Results: We develop and release msmbayes—an open-source R package—filling a critical software gap for general semi-Markov modeling. Simulation studies and empirical analysis of cognitive decline demonstrate substantial improvements in estimation accuracy, stability, and practical applicability, thereby advancing the use of semi-Markov models for real-world intermittently observed longitudinal data.

This paper addresses the challenge of efficiently and analytically modeling process execution time statistics from event logs. Methodologically, it introduces the first end-to-end analytical performance analysis framework based on semi-Markov processes: it directly infers execution time means and probability density functions (PDFs) from logs—bypassing simulation entirely. For discrete-time execution times, it employs exact convolution; for continuous-time cases, it approximates PDFs using Gaussian mixture models (GMMs), balancing accuracy, model compactness, and interpretability. Experiments show that the discrete-time approach achieves up to one order of magnitude speedup over simulation under small support sets, while GMM-based representation drastically reduces model size, enabling rapid what-if analysis. The core contribution is the first fully analytical, log-driven inference of semi-Markov performance models—eliminating reliance on traditional simulation-based approaches and establishing a new paradigm for scalable, interpretable process performance analysis.

This work addresses the zero-shot inference challenge for continuous-time, discrete-state Markov jump processes (MJPs) under high noise and sparse observations. We propose the first general-purpose foundational inference model, built upon a generalized prior-driven synthetic data generation framework coupled with neural supervised learning. The model jointly infers the rate matrix and initial state distribution, and natively accommodates diverse observation noise models and irregular sampling patterns. Its key innovation is cross-dimensional and cross-physical-scenario zero-shot generalization: a single pre-trained model achieves accurate inference across MJPs with varying numbers of states and distinct dynamical mechanisms—without fine-tuning. Evaluated on four real-world tasks—discrete flashing ratchet systems, molecular conformational dynamics, ion channel recordings, and simplified protein folding—the model matches state-of-the-art fine-tuned methods in performance, significantly advancing the universality and practical applicability of MJP modeling.

This paper addresses the problem of sequential anomaly detection in a multi-process dynamic system: normal processes remain perpetually in a zero (quiescent) state, whereas anomalous processes evolve their latent states over time according to a Markov chain; observations are obtained only by sequentially probing a subset of processes, and each probe’s outcome depends stochastically on the probed process’s current latent state. Departing from conventional i.i.d. observation assumptions, we introduce, for the first time, a hidden Markov model (HMM) into this sequential search framework. We propose ADHM—an adaptive probing algorithm that jointly models latent-state evolution and observation uncertainty via Bayesian belief updating and statistical evidence accumulation. We establish its asymptotic optimality and derive a fundamental oracle lower bound on detection delay. Simulation results demonstrate that, under strict false-alarm probability constraints, ADHM reduces the average detection time by 32% compared to state-of-the-art methods.

This paper addresses the challenge of parameter estimation for stable continuous-state branching processes (CSBPs) under partial observation. We propose a novel inference framework grounded in the subordinator representation of CSBPs. Our core innovation lies in fully mapping the stochastic dynamics of CSBPs into the subordinator domain, thereby circumventing reliance on closed-form transition densities. Specifically, we reconstruct the likelihood function assumption-free via Laplace transforms and their numerical inversion, while simultaneously developing a differentiable discrete-time trajectory simulator. The method achieves statistical consistency and computational feasibility for stable CSBPs, markedly improving both estimation accuracy and efficiency. To our knowledge, this is the first approach enabling end-to-end parameter inference and simulation within the subordinator domain.

This work addresses the high-fidelity discretization of continuous-time, continuous-state stochastic processes—such as heat diffusion and geometric Brownian motion—whose first- and second-order moments evolve linearly in time. We propose a novel construction of discrete-time Markov chains on non-uniform spatial grids: transition probabilities are designed via moment recurrence relations, and the grid is adaptively refined to ensure exact matching of the target process’s mean and variance at any user-specified time points. Unlike conventional methods relying on uniform grids or asymptotic moment matching, our approach achieves strict moment preservation for arbitrary finite horizons, thereby significantly improving long-term statistical fidelity. Numerical experiments demonstrate stable and small Wasserstein-1 distance over extended simulation periods, accurately reproducing the prescribed moment dynamics. The method provides an efficient, analytically tractable discretization framework for linear-moment-driven stochastic systems.

This study addresses the challenge of uncovering latent synergistic interactions among components in multivariate stochastic systems when only aggregate observations are available. The authors propose a sum-dependent Markov chain model and introduce a synergy index capable of distinguishing positive synergy, negative synergy, and independence. They develop a consistent estimator for this index and design a stepwise hypothesis testing procedure with asymptotic control of Type I error and guaranteed statistical power. The theoretical framework leverages continuous-time multivariate Markov processes, hidden Markov modeling, and asymptotic statistical inference to establish consistency and asymptotic normality of the parameter estimators. Numerical experiments on both simulated data and real electrophysiological recordings demonstrate the method’s effectiveness and reliability in detecting synergy and accurately recovering underlying parameters.

Traditional epidemiological models struggle to capture the multi-wave, non-stationary transmission dynamics induced by interventions and emerging variants. This work proposes a semi-Markov state-space model that represents the time-varying transmission rate as a sequence of stable states with random durations. It uniquely integrates semi-Markov processes, particle filtering, and gradient-based optimization to enable both batch and sequential Bayesian inference, accommodating complex observation mechanisms. Experiments on UK COVID-19 data demonstrate that jointly leveraging case and death counts substantially improves the accuracy and stability of parameter estimation.