transient distribution computation

This work addresses the challenge of uniformly modeling multiple action delays—including stochastic, deterministic, and general delays—in concurrent systems under discrete time. The paper proposes the dtphPBC framework, which introduces discrete phase-type distributions into Petri Box Calculus for the first time. By leveraging the transition probability matrix of discrete-time Markov chains with an absorbing state, the approach precisely characterizes diverse delay behaviors. A structured operational semantics seamlessly integrates instantaneous and timed actions, yielding a step semantics grounded in labeled probabilistic transition systems. This model coherently accommodates both zero-delay and positive-delay actions, enabling accurate representation of complex timing characteristics. The framework’s semantic consistency and constructiveness are demonstrated through illustrative examples.

This paper addresses the challenge of parameter estimation for time-scaled inhomogeneous phase-type (IPH) distributions under discrete-time observations. We propose a maximum likelihood method that jointly estimates the baseline subintensity matrix Λ and the time-scaling parameter β. Our approach innovatively integrates Markov bridge reconstruction with a stochastic EM algorithm, leveraging time transformation and gradient-based updates to enable efficient latent-state path inference. This is the first work to systematically resolve both identifiability and computational feasibility of IPH distributions in modeling time-varying multistate processes from discrete data. Extensive experiments—including simulations based on matrix-Gompertz and matrix-Weibull models, as well as real-world data on coronary artery bypass graft disease progression—demonstrate high estimation accuracy and robust numerical performance.

Existing probabilistic programming languages lack native support for dynamic systems—particularly state-space models—hindering the broader adoption of Bayesian methods in this domain. This work introduces dynestyx, a library that provides first-class, unified, and user-friendly support for state-space models within a probabilistic programming framework. dynestyx enables flexible specification of priors, accommodates both discrete- and continuous-time dynamics, handles mixed-effects data, and facilitates joint Bayesian inference over latent states and model parameters with full uncertainty quantification. By doing so, this contribution substantially enhances the accessibility, flexibility, and practical utility of dynamic system modeling across statistics, signal processing, and machine learning.

This work proposes a parameter estimation method based on the Expectation–Maximization (EM) algorithm for Markovian Arrival Process (MAP)-driven Quasi-Birth–Death (QBD) queueing systems, tailored to realistic scenarios where only coarse-grained data such as system utilization are available. Within a maximum likelihood framework, the approach infers sufficient statistics—including sojourn times, phase transitions, and service dynamics—underlying the hidden states directly from utilization time series. To the best of our knowledge, this is the first method capable of fully estimating MAP-QBD model parameters using solely utilization data. The study further introduces an innovative use of the Akaike Information Criterion (AIC) to automatically select the number of MAP phases, thereby mitigating overfitting. Experimental results demonstrate that the method accurately recovers both arrival and service parameters, offering a practical performance modeling tool for real-world systems lacking fine-grained event logs.

This work addresses key challenges in particle-based Bayesian inference for continuous-discrete state-space models (CD-SSMs): (i) absence of closed-form transition densities for Itô diffusions; (ii) non-resamplability of deterministic ancestral paths over continuous trajectories; and (iii) posterior collapse of diffusion parameters to Dirac distributions. To overcome these, we introduce a novel path-space Feynman–Kac formulation that unifies guided path proposals, differentiable reparameterization, and particle filtering/smoothing—enabling the first differentiable and resamplable particle inference framework for CD-SSMs without requiring closed-form transition densities. The method supports hypoelliptic diffusions and integrates seamlessly with probabilistic programming frameworks, enabling both online and offline parameter learning while scaling to high-dimensional settings. It has been implemented in mainstream probabilistic programming libraries and empirically validated across multiple numerical experiments, demonstrating superior robustness and accuracy compared to state-of-the-art alternatives.

Existing causal meta-models are confined to Markovian systems and struggle to represent non-Markovian queueing systems with non-exponential service times. This work proposes the first causal meta-modeling framework tailored for non-Markovian systems: it approximates general distributions using phase-type distributions, extends modular dynamic Bayesian networks to the discrete-time domain, and introduces corresponding parameter learning and time discretization strategies. The approach is validated on systems such as G/M/1, achieving inference speeds several orders of magnitude faster than direct simulation while preserving high accuracy in both probabilistic and causal queries.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

Traditional discrete Phase-Type (PH) distributions struggle to capture the stochasticity of rewards associated with state visits, limiting their applicability in modeling latent severity dynamics. This work addresses this limitation by introducing, for the first time, a stochastic reward mechanism into the PH framework, proposing the Inertia–Escalation Model (IEM). The IEM allows state-dependent rewards to follow Bernoulli or geometric distributions and employs a two-parameter formulation to characterize the dynamic evolution of latent severity. Combining parameter inference with Monte Carlo simulation, the proposed approach is validated on historical warfare and telecommunications customer churn datasets, demonstrating its enhanced capability to accurately capture the underlying patterns of latent severity in complex sequential processes.

Traditional AR(p) models struggle to capture the time-varying dynamics and complex noise structures inherent in nonstationary time series. This work proposes a novel hybrid approach that integrates deep learning with time-varying autoregressive (TVAR) modeling, leveraging neural networks to flexibly estimate time-dependent coefficients. The method preserves model interpretability while effectively adapting to nonstationarity and enables probabilistic forecasting under both Gaussian and Laplacian noise assumptions. Empirical validation under a TVAR(1) framework demonstrates that the proposed approach accurately captures intricate temporal dynamics with a parsimonious structure and yields reliable prediction intervals, particularly excelling in scenarios involving heavy-tailed distributions or sharp volatility shifts.

This work addresses the challenge of causal inference with continuous-time marked point process data, for which existing methods lack a suitable identification framework. Building on martingale theory, the authors extend the core assumptions of discrete-time causal inference—consistency, exchangeability, and positivity—to the continuous-time setting. They formulate a dynamic treatment strategy and a potential outcomes model tailored to marked point processes and establish corresponding causal identification conditions. Leveraging this foundation, they derive a novel marginal g-formula that enables nonparametric identification of causal effects. The proposed framework subsumes existing results for discrete-time and counting process settings as special cases, demonstrating both theoretical compatibility and extensibility, thereby unifying survival analysis and causal inference within a coherent paradigm.