A Counterfactual Analysis of the Dishonest Casino

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.