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.

翻译：本文提出了一种离散相型分布（DPH）的扩展，通过引入随机奖励机制。该机制允许对系统中访问特定状态时不产生确定性奖励的情形进行建模，转而采用伯努利分布或几何分布来刻画奖励。利用这种增强的灵活性，我们进一步通过引入惯性-升级模型（IEM）勾勒出一个可能的随机奖励应用场景，该模型通过两个参数（惯性ν和升级η）描述具有潜在严重性等级的过程。我们还讨论了此类模型的参数推断问题。为验证和探索随机奖励及IEM，我们进行了大量模拟，并将该模型应用于两个数据集：历史战争数据与电信客户流失数据集。