A great probability model is worthless without disciplined sizing and validation. This note maps the machinery that turns a calibrated probability into a sustainable edge: calibration, edge estimation, Kelly sizing, bankroll variance, backtesting, CLV measurement, and bet correlation.
Markets & strategy
Quantitative Betting Framework — Landscape Note (Area E)
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Summary
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Probability calibration
Models output p. Markets demand probabilities that are calibrated — when the model says 30%, the event happens 30% of the time over enough samples. Calibrated probabilities map to honest edges; uncalibrated ones don't.
Scoring rules
- Brier score: mean squared error against the realised one-hot outcome. Decomposes into reliability + resolution + uncertainty (Murphy decomposition). Rewards calibration and sharpness.
- Log-loss / log-likelihood:
−log p(realised). Heavier penalty for confident wrong predictions; the standard metric for 1X2 and binary markets. - Ranked probability score (RPS): ordinal-aware version of Brier, useful for 1X2 because outcomes have a natural ordering (home / draw / away).
For football 1X2, log-loss is standard; RPS is reported in academic comparisons.
Calibration plots and tests
Plot model probability vs empirical frequency, binned. Perfect calibration lies on y = x. Diagnostic for systematic over- or under-confidence at the tails. Quantitative tests: Hosmer-Lemeshow, expected calibration error (ECE).
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