Full Distribution Forecast

1. Definition

• A full distribution describes the complete probability law of a random variable rather than only a point summary, such as its mean or median, or a selected set of quantiles.
• Knowing the full distribution allows probabilities, quantiles, prediction intervals, and expectations to be calculated when the relevant quantities exist.

Formally:

• The distribution of any real-valued random variable $X$ can be characterized by its cumulative distribution function (CDF) $F(x) = P(X \leq x)$.
• For an absolutely continuous random variable, the distribution can also be represented by a probability density function (PDF) $f(x)$.
• For a discrete random variable, it can be represented by a probability mass function (PMF) $p(x) = P(X=x)$.

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2. Why Full Distribution Forecasts Matter

• A point forecast provides one predicted value, such as a mean or median.
• A quantile forecast commonly provides selected percentiles of the possible outcome.
• A full distribution forecast provides an entire predictive probability distribution for a future outcome.

With the full predictive distribution, you can derive:

• Mean forecasts, when the expected value exists.
• Medians and other quantiles.
• Prediction intervals.
• Event probabilities, such as $P(Y > 100)$.
• Risk measures, such as Value-at-Risk and Expected Shortfall.

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3. Examples

Weather

• Point forecast: “Tomorrow’s temperature will be 25°C.”
• Quantile forecast: The 10th, 50th, and 90th percentiles are approximately 22.4°C, 25°C, and 27.6°C.
• Full distribution forecast: “Tomorrow’s temperature follows a Normal distribution with $\mu=25$ and $\sigma=2$, allowing probabilities, quantiles, and intervals to be derived.”

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Retail Demand

• Full distribution forecast: “Next week’s demand follows a LogNormal distribution with $\mu=6.2$ and $\sigma=0.3$ on the log scale.”
  • Mean ≈ 516 units.
  • 90th percentile ≈ 724 units.
  • $P(\text{demand} > 700) \approx 0.12$.

If demand must be represented as whole units, a discrete count distribution or another suitable demand model may be more appropriate than a continuous LogNormal distribution.

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Finance

• Stock return forecast: “Returns follow a location-scale Student’s t-distribution with $\nu=5$, location $\mu=0.01$, and scale $\sigma=0.02$.”
  • The predictive distribution can be used to estimate Value-at-Risk, Expected Shortfall, and other tail-risk measures.

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4. Methods to Estimate Full Distributions

• Classical models:
  • ARIMA models with Gaussian innovations can produce Normal predictive distributions under their assumptions.
  • GARCH models combine forecasts of conditional volatility with an assumed innovation distribution to produce return distributions.
• Bayesian models: Combine uncertainty about parameters with outcome uncertainty to form posterior predictive distributions.
• Ensemble and simulation methods: Generate many plausible future outcomes or paths to approximate an empirical predictive distribution.
• Deep learning:
  • DeepAR models the parameters of a chosen predictive likelihood.
  • Mixture Density Networks predict the parameters of a probability mixture.
  • Normalizing flows can learn flexible predictive densities.
  • Temporal Fusion Transformers commonly produce quantile forecasts and may produce full predictive distributions when paired with an appropriate distributional output layer.

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5. Evaluation

Because the forecast represents a probability distribution rather than only a single value, it should be evaluated for both calibration and sharpness.

• CRPS (Continuous Ranked Probability Score): Measures the distance between a predictive CDF and the observed outcome while rewarding calibrated, concentrated forecasts.
• Logarithmic score: Uses the log predictive density assigned to the observed outcome; higher values are better.
• Negative Log-Likelihood: The negative of the log predictive density; lower values are better.
• Calibration checks: Examine whether predicted probabilities and intervals match observed frequencies over many forecasts.

CRPS and the logarithmic score are proper scoring rules, meaning that their expected values encourage forecasters to report their true predictive distributions.

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6. Key Contrast

Type of Forecast	What It Gives	Limitation
Point Forecast	One value, such as a mean or median	Does not express the forecast’s uncertainty by itself
Selected Quantile Forecast	A set of cut points, such as the 10th, 50th, and 90th percentiles	Describes only part of the distribution
Full Distribution Forecast	An entire predictive probability distribution	Requires stronger modeling choices and can be harder to estimate and evaluate

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Summary:
A full distribution forecast provides a predictive probability distribution over possible future outcomes. It supports the calculation of point forecasts, quantiles, prediction intervals, event probabilities, and risk measures. Its quality can be assessed using calibration diagnostics and proper scoring rules such as CRPS and the logarithmic score.