Isotonic Calibration
A non-parametric monotonic transformation that maps raw model scores to well-calibrated probabilities.
Definition
Most classifiers — gradient-boosted trees included — produce raw scores that are not true probabilities: a predicted '70%' might actually correspond to a 55% empirical hit rate. Isotonic regression fits a monotonic step function from raw scores to empirical frequencies, restoring calibration. Compared to Platt scaling (which assumes a logistic shape), isotonic is non-parametric and flexible but needs more data to avoid overfitting. Calibration matters because position sizing and expected-value calculations are useless on miscalibrated probabilities.
Formula
Given pairs (s_i, y_i):
Find non-decreasing f minimising sum (y_i - f(s_i))^2
Solved efficiently with Pool-Adjacent-Violators (PAV) algorithm:
Sort by s, repeatedly merge adjacent groups violating monotonicity
using weighted averages.Worked example
A LightGBM model assigns 0.78 to a trade signal. Historically signals in the 0.75-0.80 raw range hit 0.62. Isotonic calibration maps 0.78 -> 0.62. With this calibrated probability, expected value can be computed correctly: 0.62 * +R - 0.38 * -1 = positive only when reward/risk > 0.61.
How ARIA Analyst uses it
ARIA applies isotonic calibration to every LightGBM and XGBoost model in its ensemble before generating confidence scores for the Bull vs Bear debate.
Related terms
LightGBM
A fast gradient-boosted decision tree framework from Microsoft, dominant on tabular financial data.
XGBoost
Extreme Gradient Boosting — the original modern GBDT library, slightly slower but very robust.
Purged K-Fold Cross-Validation
A cross-validation scheme that removes overlapping training samples to prevent look-ahead leakage.
Walk-Forward Analysis
A backtesting procedure that retrains the model on a rolling window and tests on the next out-of-sample period.
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