Brier score for event probabilities

Question

Implement the mean Brier score for one-dimensional binary outcomes and probability forecasts. Define handling of invalid probabilities and empty input.

Short answer

The binary Brier score is the mean squared difference between probability and outcome. Lower is better; reject invalid probabilities, shapes and empty input. This is a proper scoring rule for probabilities, not a direct trading-profit metric.

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Lessons behind this question

The Supervised Learning Problem and the ML Pipeline
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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