Murphy’s Decomposition

It decomposes a proper scoring rule (like the Brier score) into three components:

1. Uncertainty
   • Measures the inherent difficulty of the prediction task (variance of the true outcomes).
   • High uncertainty = hard to predict (e.g., if events happen about 50% of the time).
2. Resolution
   • Measures how well the forecast separates different outcome situations.
   • High resolution = model gives different probabilities for different situations (not always the same probability).
3. Reliability (Calibration)
   • Measures how close predicted probabilities are to the true observed frequencies.
   • A perfectly calibrated model has high reliability.

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Formula (for the Brier Score)

The Brier Score (BS) can be decomposed as:

$BS = \text{Uncertainty} – \text{Resolution} + \text{Reliability}$

• Uncertainty = variance of actual outcomes
• Resolution = how much forecasts differ across subsets of data
• Reliability = calibration error

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Example in ML

Suppose you train a binary classifier:

• Model predicts 0.7 probability of “rain” on 100 days → it should rain ~70 days (calibration).
• If model predicts always 0.5 → resolution is low.
• If actual rain frequency is 50%, uncertainty is high (hard task).

Murphy’s decomposition lets you understand whether poor performance comes from:

• the task being uncertain,
• the model not distinguishing cases (low resolution),
• or the model being miscalibrated (low reliability).

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So in short:
Murphy’s decomposition = a way to break down forecast error into calibration (reliability), sharpness (resolution), and task difficulty (uncertainty).