Probabilistic Learners

Why

In general, a dataset may be both incomplete and inconsistent.

Why

Let $(Z, \mathcal{Z} )$ be a measurable space. Let $\mathcal{M} (Z, \mathcal{Z} )$ be the set of measures over $(Z, \mathcal{Z} )$. A probabilistic inductor for a dataset in $Z^n$ is a function mapping $Z^n$ to $\mathcal{M} (Z, \mathcal{Z} )$.

Suppose $(Z, \mathcal{Z} ) = (X \times Y, \mathcal{X} \times \mathcal{Y} )$ for measurable spaces $(X, \mathcal{X} )$ and $(Y, \mathcal{Y} )$. Then a probabilistic inductor yields a probabilistic functional inductor. The conditional measure on $Y$ given an observation $x \in X$ is exactly a family of measures indexed by $X$.