# Latent Generation Pairs

# Definition

Let $Z$ and $X$ be sets, either of which may
or may not be finite.

A latent generation pair
from latents $Z$ to
observations $X$ is an
ordered pair $(p_z, p_{x \mid z})$ whose first
coordinate is a distribution (density) on $Z$
and whose second coordinate is a conditional
distribution (density) on $X$ from $Z$.

The joint function
$p_{zx}: Z \times X \to \R $ of the pair is
defined by $p_{zx}(\zeta , \xi ) =
p_{z}(\zeta )p_{x \mid z}(\xi , \zeta )$ for all
$\xi \in X$ and $\zeta \in Z$.
It is a distribution (density) if (not only
if) both $p_{z}$ and $p_{x \mid z}$ are
distributions (densities).
Regardless, we define the
marginal function $p_x: X
\to \R $ by $p_x(\xi ) = \int_Z p_{zx}(\xi ,
\cdot )$.
It too may be a distribution, density, or
neither.
In cases we construct, it is often a
distribution or a density, but it need not be
either.

## Interpretation as distribution graph

The latent generation pairs from $Z$ to $X$
are isomorphic to the graph distributions whose
typed graph $(\set{1, 2}, \set{(1,2)}), (Z, x)$.

# Parametrizations

By parameterizing either or both of the
coordinates of the pair, we have
latent generation family.

## Other terminology

Other terminology for latent generation pair
includes latent variable
model.
Some authorities refer to the marginal function
as the generative model,
still others use this term to refer to the
pair.