Probabilistic forecasting provides a distribution of possible future values rather than a single point estimate, allowing for uncertainty quantification. Instead of predicting only the most likely outcome, it estimates a range of potential outcomes along with their probabilities by outputting a probability distribution.

A probabilistic forecast models the conditional probability distribution of a future value $(Y_t)$ given past observations $(\mathcal{H}_{t-1})$ . This can be expressed as:

$P(Y_t \mid \mathcal{H}_{t-1})$

where $(\mathcal{H}_{t-1})$ represents the historical data up to time $(t-1)$ . Instead of a single prediction $(\hat{Y}_t)$ , the model estimates a full probability distribution $(f(Y_t \mid \mathcal{H}{t-1}))$ , which can take different parametric forms, such as a Gaussian:

$Y_t \mid \mathcal{H}_{t-1} \sim \mathcal{N}(\mu_t, \sigma_t^2)$

where $(\mu_t)$ is the predicted mean and $(\sigma_t^2)$ represents the uncertainty in the forecast.

Probabilistic forecasting can be handled through various approaches, including variance forecasters, quantile forecasters, interval forecasters or distribution forecasters, each capturing uncertainty differently.

For example, you can try to forecast the target location by a gaussian density function (or a mixture), thus the model output follows the form:


  {

      "density": {

          "name": "normal",

          "params": {

              "loc": y_mean,

              "scale": y_var

          }

      },

      "weight": weight

  }

A mixture density, such as the gaussian mixture $\sum_{i=1}^{K} w_i \mathcal{N}(Y_t | \mu_i, \sigma_i^2)$ allows for capturing multi-modal distributions and approximate more complex distributions.

Mathematical Definition

The informal meaning of probabilistic forecast is a mixture of parametric univariate density functions where each is taken from a standard family (such as exponential, or gaussian).

A mixture of gaussian densities conspire to match a fat-tailed distribution.

Engineering Definition

The meaning of probabilistic forecast is made more precise by means of the Python density package which provides a dict specification of continuous univariate density function mixtures using the pydantic Python package. The function validatedensitydict will tell you whether or not your specification is valid.