Random Reward Phase-Type Distributions with Applications in Latent Severity Modeling

2026-04-21

AI Summary

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.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper proposes an extension to discrete Phase-Type distributions (DPH) by introducing random rewards. These allow for modeling a system in which a visit to a certain state does not emit a deterministic reward. Instead, the rewards follow either a Bernoulli or a geometric distribution. Utilizing this increased flexibility, we further sketch a possible use case for these random rewards by introducing the Inertia-Escalation model (IEM), a process with latent severity levels characterized through two parameters: Inertia ν and escalation η. We also discuss parameter inference for such models. To validate and explore random rewards and the IEM, we conducted extensive simulations and applied the model to two datasets: historical warfare and the Telco customer churn dataset.

Problem

• Phase-Type distributions
• random rewards
• latent severity
• Inertia-Escalation model
• stochastic modeling

Innovation

• Random Reward
• Discrete Phase-Type Distribution
• Latent Severity Modeling
• Inertia-Escalation Model
• Parameter Inference