The dishonest casino is a well-known hidden Markov model (HMM) used in educational settings to introduce HMMs and graphical models. Here, a sequence of die rolls is observed, with the casino switching between a fair and a loaded die. Typically, the goal is to use the observed rolls to infer the pattern of fair and loaded dice, leading to filtering, smoothing, and Viterbi algorithms. This paper, however, explores how much of the winnings is attributable to the casino's cheating, a counterfactual question beyond the scope of HMM primitives. To address this, we introduce a structural causal model (SCM) consistent with the HMM and show the expected winnings attributable to cheating (EWAC) (which is only partially identifiable) can be bounded using linear programs (LPs). Through numerical experiments, we compute these bounds and develop intuition using benchmark SCMs based on independence, comonotonic, and counter-monotonic copulas. We show that tighter bounds are obtained with a time-homogeneity condition on the SCM, while looser bounds allow for an almost explicit LP solution. Domain-specific knowledge such as pathwise monotonicity or counterfactual stability can be incorporated via linear constraints. We also show the time-average EWAC is fully identifiable in the limit as the number of time periods goes to infinity. Our work contributes to bounding counterfactuals in causal inference and is the first to develop LP bounds in a dynamic HMM setting, benefiting educational contexts where counterfactual inference is taught.

翻译：不诚实赌场是一个著名的隐马尔可夫模型，常用于教学场景以介绍HMM和概率图模型。在该模型中，我们观测到一系列骰子投掷结果，赌场在公平骰子和灌铅骰子之间切换。通常，目标是利用观测到的投掷结果推断公平与灌铅骰子的使用模式，从而引出滤波、平滑和维特比算法。然而，本文探讨赌场赢利中有多少可归因于作弊行为，这是一个超越HMM基本框架的反事实问题。为解决此问题，我们构建了一个与HMM相容的结构因果模型，并证明可归因于作弊的期望收益（部分可识别）可通过线性规划进行边界估计。通过数值实验，我们基于独立性、同单调及反单调连接函数构建的基准SCM计算这些边界并建立直观理解。研究表明，在SCM满足时间齐次性条件时可获得更紧的边界，而较宽松的边界则允许近乎显式的LP解。路径单调性或反事实稳定性等领域知识可通过线性约束融入模型。我们还证明当时段数趋于无穷时，时间平均EWAC具有完全可识别性。本研究为因果推断中的反事实边界估计作出贡献，首次在动态HMM框架中建立LP边界，对开展反事实推断教学的教育场景具有积极意义。