State Definition Arbitrariness
A Markov Chain Trading Model does not operate on continuous returns — it operates on a finite set of states, which the modeller must construct by discretising the return distribution into buckets. Two things must be chosen: how many buckets and where to place the thresholds. The failure mode is that there is no objective way to make either choice, and the model’s reported performance depends heavily on both. This is the discrete-Markov-specific instance of the broader State-Space Design problem, which is the same failure mode for the state vector of a Markov decision process.
Wilinski 2019 frames the dilemma exactly. The return range Ymax - Ymin must be split into intervals, and the two failure directions are sharp: too few intervals and the state almost never changes — the chain predicts “stay put” and forecasts nothing useful — while too many intervals and each state contains too few observations to estimate a reliable transition probability. There is no setting that escapes both; the modeller is choosing a point on a trade-off, not finding a correct answer. Wiliński’s response is revealing: rather than fix the discretisation, he makes the number of intervals a tunable hyper-parameter optimised by machine learning alongside window length and window count. That is an honest engineering choice, but it converts the arbitrariness directly into a search over the backtest.
State Definition Arbitrariness [part-of] State-Space Design Wilinski 2019 [relates] State Definition Arbitrariness
Aronsson Folkesson 2023 make the same trade-off visible from the data side. They tested a six-state scheme (D3-U3, split at ±0.5% and ±1%) against an aggregated two-state up/down scheme. The two definitions gave different and incommensurable results: the six-state ensemble scored 17.1% accuracy against a 16.7% random benchmark, while the two-state first-order model scored exactly 50.0% (= benchmark) and the two-state second-order model scored 42.7% (below benchmark). The bucket thresholds ±0.5% / ±1% are themselves unmotivated round numbers — nudge them and the transition counts, and therefore the estimated probabilities, all shift. Their honest workaround was to ensemble over ten chains with random window sizes, smearing out one degree of freedom; the discretisation itself was still hand-chosen.
Aronsson Folkesson 2023 [supports] State Definition Arbitrariness
Why this is a genuine risk and not a cosmetic complaint: state count and thresholds are free parameters, and if they are tuned to maximise the performance figure that is then reported, they inflate the effective number of strategies tried. With enough discretisations to choose from, some state grid will look profitable on any finite history — the textbook mechanism of Data-Snooping Bias and Overfitting in Quantitative Trading. The arbitrariness also interacts with the First-Order Memory Assumption: higher-order chains multiply the state space (n^k rows for order k), so adding memory makes the data-sparsity horn of the dilemma far sharper. The verdict for this vault is confirmed: state definition arbitrariness is a real degree of freedom in every discrete-Markov backtest, and unless the discretisation was fixed by genuine out-of-sample procedure (cross-validation or pre-registration) the reported edge should be discounted accordingly.
State Definition Arbitrariness [causes] Overfitting in Quantitative Trading State Definition Arbitrariness [causes] Data-Snooping Bias State Definition Arbitrariness [relates] First-Order Memory Assumption
Connections
- State-Space Design — part-of, discrete-Markov instance of the sibling MDP failure mode, source: https://arxiv.org/html/2410.14504v2
- Markov Chain Trading Model — suffers_overfitting_risk, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
- Wilinski 2019 — relates, frames the too-few-vs-too-many dilemma and tunes interval count by ML, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
- Aronsson Folkesson 2023 — supports, six-state vs two-state schemes give incommensurable results, source: https://kth.diva-portal.org/smash/get/diva2:1823899/FULLTEXT01.pdf
- Overfitting in Quantitative Trading — suffers_overfitting_risk, state grid inflates the effective parameter count, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
- Data-Snooping Bias — suffers_overfitting_risk, tuning thresholds on the test set is data snooping, source: https://www.sciencedirect.com/science/article/abs/pii/S0957417419303033
- First-Order Memory Assumption — relates, higher-order chains multiply the state space and sharpen sparsity, source: https://kth.diva-portal.org/smash/get/diva2:1823899/FULLTEXT01.pdf