probabilisons

Probabilisons are a mathematical construct used to represent uncertainty by aggregating multiple probabilistic assessments about a domain. A probabilison consists of a collection of probability measures on a shared measurable space, together with a weighting mechanism that assigns relative importance to each component. The resulting predictive distribution is a mixture of the component distributions, enabling flexible modeling of both epistemic and aleatory uncertainty.

Formally, let (Ω, F) be a measurable space and {P_i: i ∈ I} a family of probability measures on (Ω, F). Let w_i ≥ 0 be weights with ∑_i w_i = 1. The probabilison defines the mixture distribution P by P(A) = ∑_i w_i P_i(A) for all A in F. In hierarchical variants, the weights themselves may be random variables, yielding a hierarchical probabilison. Observed data can update the weights via Bayes’ rule, producing posterior weights w_i|D ∝ w_i P_i(D).

Properties of probabilisons include linearity of the mixture: P(A) = ∑ w_i P_i(A) for measurable events A, and corresponding linearity for expectations: E_P[X] = ∑ w_i E_{P_i}[X]. They generalize single distributions and connect to mixture models and Bayesian model averaging.

Examples and applications include ensemble forecasting where multiple models contribute to a single forecast, sensor fusion that combines signals from different sources, and risk assessment that blends competing probability assessments. In machine learning, probabilisons underpin ensemble methods and uncertainty quantification. Limitations involve potential identifiability issues for component contributions, selection of weights, and increased computational cost.

See also: mixture models, Bayesian model averaging, ensemble learning, epistemic uncertainty.