Probabilistic Forecasts

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.

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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.

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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).

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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).”

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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).

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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?).

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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).

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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.