1. Definition

  • A probabilistic forecast predicts a range or full probability distribution of possible future outcomes, rather than a single number.
  • Instead of just giving a “best guess,” it expresses uncertainty about the future.

Formally:

$\hat{F}_{t+h}(y) = P(Y_{t+h} \leq y \mid \text{information up to } t)$

where $\hat{F}_{t+h}(y)$ is the forecasted CDF of the future value.

2. Characteristics

  • Provides distribution (PDF/CDF) or intervals instead of just a point.
  • Allows decision-makers to assess risks and confidence.
  • More informative than deterministic forecasts, especially when outcomes are uncertain.

3. Examples

  • Weather:
    • Deterministic: “Tomorrow will be 25°C.”
    • Probabilistic: “There’s a 90% chance temperature will be between 23–27°C.”
  • Retail demand forecasting:
    • Deterministic: “Expected sales = 500 units.”
    • Probabilistic: “10% chance < 450, 50% chance ≈ 500, 90% chance < 560.”
  • Finance (Value-at-Risk):
    • Predicts lower quantiles of returns distribution (e.g., 5th percentile loss).

4. Types of Probabilistic Forecasts

Prediction Intervals

  • Specify bounds within which the future value is likely to fall.
  • Example: 95% prediction interval = [450, 560].

Quantile Forecasts

  • Predict specific quantiles (e.g., 10th, 50th, 90th percentiles).
  • Useful in supply chain (planning for worst- or best-case demand).

Full Distribution Forecasts

  • Provide an estimated PDF/CDF for the future outcome.
  • Example: “Next week’s demand follows approx. Normal(500, σ=30).”

5. Methods for Probabilistic Forecasting

  • Classical statistics: ARIMA with prediction intervals, GARCH (finance).
  • Quantile regression: estimates specific quantiles.
  • Bayesian methods: naturally produce posterior distributions.
  • Ensemble methods: bootstrapping, bagging produce distributions of forecasts.
  • Deep learning:
    • Probabilistic RNNs,
    • Bayesian neural networks,
    • Distributional forecasting (e.g., DeepAR, TFT, Informer).

6. Evaluation of Probabilistic Forecasts

Since accuracy isn’t just a single number, evaluation uses special metrics:

  • Pinball Loss (for quantiles).
  • Continuous Ranked Probability Score (CRPS) (for distributions).
  • Coverage & Width of prediction intervals (did actual values fall inside? were intervals too wide/narrow?).

7. Why Probabilistic Forecasts Matter

  • Better for risk management (finance, supply chain, energy).
  • More robust to uncertainty and shocks.
  • Increasingly required in competitions (e.g., M5 Competition focused on probabilistic forecasts).

Summary:
Probabilistic forecasts go beyond a single point prediction by providing intervals, quantiles, or full distributions of possible outcomes. They quantify uncertainty, allow better decision-making, and are evaluated with metrics like pinball loss and CRPS.