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 $(\set{1, 2}, \set{(1,2)}), (Z, x)$.1
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.
- Future editions will include a visualization. ↩︎