1. Definition
- A full distribution describes the complete probability structure of a random variable — not just a point (mean, median) or a few quantiles, but the entire PDF/CDF.
- Knowing the full distribution means you can compute any probability, quantile, or expectation.
Formally:
- For continuous random variable $X$: the full distribution is given by its PDF $f(x)$ or CDF $F(x)$.
- For discrete random variable: by its PMF $p(x)$.
2. Why Full Distribution Forecasts Matter
- A point forecast (mean, median) = one number.
- A quantile forecast = selected percentiles.
- A full distribution forecast = the entire probability distribution for future outcomes.
With the full distribution, you can derive:
- Mean forecast (expected value).
- Median, quantiles.
- Prediction intervals.
- Event probabilities (e.g., $P(Y > 100)$).
3. Examples
Weather
- Point forecast: “Tomorrow’s temp = 25°C.”
- Quantile forecast: 10th = 23, 50th = 25, 90th = 27.
- Full distribution forecast: “Temp tomorrow follows Normal(μ=25, σ=2), so any probability or interval can be derived.”
Retail Demand
- Full distribution forecast: “Next week’s demand ~ LogNormal(μ=6.2, σ=0.3).”
- Mean = 500 units.
- 90th percentile = 650 units.
- $P(\text{demand > 700}) = 0.08$.
Finance
- Stock return forecast: “Returns follow Student-t(ν=5, μ=0.01, σ=0.02).”
- Allows computing Value-at-Risk (VaR), Expected Shortfall, tail risks.
4. Methods to Estimate Full Distributions
- Classical models:
- ARIMA with Gaussian residuals → Normal distribution forecasts.
- GARCH for return volatility → distributional outputs.
- Bayesian models: posterior distributions over parameters & outcomes.
- Ensemble / Bootstrapping: simulate many future paths → empirical distribution.
- Deep learning:
- DeepAR (RNN forecasting likelihoods).
- Mixture Density Networks.
- Temporal Fusion Transformer (outputs full predictive distributions).
5. Evaluation
Because we’re evaluating distributions (not single values), special metrics apply:
- CRPS (Continuous Ranked Probability Score).
- Logarithmic Score (Negative Log-Likelihood).
- Calibration checks (do forecast probabilities match observed frequencies?).
6. Key Contrast
| Type of Forecast | What It Gives | Limitation |
|---|---|---|
| Point Forecast | One number (e.g., mean) | Ignores uncertainty |
| Quantile Forecast | A few cut-points (e.g., 10th, 50th, 90th) | Partial info only |
| Full Distribution Forecast | Entire probability distribution | Most informative, hardest to model |
Summary:
A full distribution forecast provides the complete probability distribution of possible outcomes. It allows deriving point forecasts, intervals, quantiles, and event probabilities from a single forecast. It’s the richest form of forecasting and is evaluated with proper scoring rules like CRPS and log score.