1. Definition

  • A probability forecast predicts the likelihood of different outcomes, instead of giving a single point value.
  • It’s a type of probabilistic forecast that outputs probabilities for discrete categories or ranges.

Formally:

$P(Y \in A \mid \text{information up to time } t)$

where $A$ is an event or range of outcomes.

2. Characteristics

  • Instead of saying “Sales will be 500”, a probability forecast says:
    “There’s a 20% chance sales < 480, 50% chance 480–520, 30% chance > 520.”
  • Useful when decisions depend on risk, uncertainty, or multiple scenarios.

3. Examples

Weather Forecasting

  • Deterministic: “It will rain tomorrow.”
  • Probability forecast: “There’s a 70% chance of rain tomorrow.”

Sports

  • Predicting match outcome:
    • Home win: 60%
    • Draw: 25%
    • Away win: 15%

Finance

  • Stock price next week:
    • 10% chance it falls below –2% return.
    • 60% chance between –2% and +2%.
    • 30% chance it rises above +2%.

4. How Probability Forecasts Are Produced

  • Logistic regression / classification models: output class probabilities.
  • Ensemble methods (random forests, gradient boosting): give probability distributions across outcomes.
  • Bayesian methods: naturally produce posterior probabilities.
  • Neural networks (with softmax or sigmoid outputs): give class probability forecasts.

5. Evaluation of Probability Forecasts

Special scoring rules are used, since accuracy alone doesn’t apply:

  • Brier Score: $\text{Brier} = \frac{1}{N}\sum (p_i – o_i)^2$ where $p_i$ = predicted probability, $o_i$ = observed outcome (0 or 1).
  • Logarithmic Score (Log Loss): penalizes overconfident wrong probabilities.
  • Calibration plots / Reliability curves: check whether probabilities match observed frequencies (e.g., when model says 70% rain, does it rain ~70% of the time?).

6. Benefits

  • Uncertainty quantification: better decision-making under risk.
  • Flexibility: supports interval forecasts, risk assessments.
  • Interpretability: probabilities are intuitive (70% chance of rain).

7. Contrast

  • Deterministic forecast: Single best guess.
  • Probabilistic forecast: Full distribution or intervals.
  • Probability forecast (subset of probabilistic): Focuses on categorical events or ranges with assigned probabilities.

Summary:
A probability forecast predicts the likelihood of outcomes (e.g., rain tomorrow = 70%) instead of a single value. They are produced by models like logistic regression, ensembles, and neural networks, and evaluated using scoring rules such as the Brier Score and Log Loss. They’re especially important in weather, finance, sports, and risk management.