A counterfactual analysis of the dishonest casino
File(s)
Published version
Author(s)
Haugh, Martin
SIngal, Raghav
Type
Journal Article
Abstract
The dishonest casino is a well-known hidden Markov model (HMM) often used in education to introduce HMMs and graphical models. A sequence of die rolls is observed with the casino switching between a fair and a loaded die. Instead of recovering the latent regime through filtering, smoothing, or the Viterbi algorithm, we ask a counterfactual question: how much of the gambler’s winnings are caused by the casino’s cheating? We introduce a class of structural causal models (SCMs) consistent with the HMM and define the expected winnings attributable to cheating (EWAC). Because EWAC is only partially identifiable, we bound it via linear programs (LPs). Numerical experiments help to develop intuition using benchmark SCMs based on independence, comonotonic, and countermonotonic copulas. Imposing a time homogeneity condition on the SCM yields tighter bounds, whereas relaxing it produces looser bounds that admit an explicit LP solution. Domain knowledge such as pathwise monotonicity or counterfactual stability can be incorporated through additional linear constraints. Finally, we show the time averaged EWAC becomes fully identifiable as the number of time periods tends to infinity. Our work is the first to develop LP bounds for counterfactuals in a HMM setting, benefiting educational contexts where counterfactual inference is taught.
Date Issued
2026-01-01
Date Acceptance
2026-01-03
Citation
Journal of Causal Inference, 2026, 14 (1)
ISSN
2193-3677
Publisher
De Gruyter
Journal / Book Title
Journal of Causal Inference
Volume
14
Issue
1
Copyright Statement
© 2026 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License.
License URL
Identifier
10.1515/jci-2024-0023
Subjects
causal inference
counterfactuals
hidden-Markov models
dishonest casino MSC 2020: 62M05
68T01
Publication Status
Published
Article Number
20240023
Date Publish Online
2026-04-14