  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 that the expected winnings attributable to
cheating (EWAC) 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 like pathwise monotonicity or counterfactual stability can be
incorporated via linear constraints. 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.
