Parameterized Distributions

Definition

A conditional distribution (density) $q: Y \times X \to \R $ is functionally parametrizable if there exists a function $f: X \to \Theta$ and distribution (density) family $\set{p^{(\theta )}: Y \to \R }_{\theta \in \Theta}$ satisfying $q(y, x) \equiv p^{(f(x))}(\gamma )$ for all $x \in X$ and $y \in Y$.

In this case we call $f$ the parameterizer and we call $\set{p^{(\theta )}}_{\theta \in \Theta}$ the parameterized family. A parameterized conditional distribution is an ordered pair whose first coordinate is a function from $X$ to $\Theta$ and whose second coordinate is a family of distributions on $X$ with parameter set $\Theta$. For a particular choice of parameterizer and family, it induces a conditional distribution.

Since all conditional distributions are functionally parametrizable (consider $\set{q(\cdot , \xi )}_{\xi \in X}$ with parameters $X$ and identity parameterizer), we are interested in parameterizers and parameterized families that are simple. Said differently, we are interested in approximating a conditional distribution by selecting an appropriate parameterizer and parameterized family.

If $\set{f_{\phi }: X \to \Theta}_{\phi \in \Phi}$ is a family of functions and $\set{q^{(\theta )}}$ is a family of distributions, then $\set{p^{(\phi )}: X \times Z \to \R }_{\phi }$ defined by $p^{(\phi )}(\cdot , \zeta ) \equiv q^{f_\phi (\zeta )}$ is a conditional distribution family called a functionally parameterized conditional distribution family. In other words, by selecting some parameters $\phi $, we induce a conditional distribution $p^{(\phi )}: X \times Z \to \R $

We similarly define parameterized conditional densities and functionally parameterized conditional density families.

Basic example

Let $Z = \set{1, 2}$ and $X = \R $. Let $f: \set{1, 2} \to \R \times \R _+$ be defined by $f(1) = (\mu _1, \sigma _1)$ and $f(2) = (\mu _2, \sigma _2)$. Let $\set{g^{(\theta )}}_{\theta }$ be the normal family. Then $(f, \set{g^{(\theta )}})$ is a functionally parameterized conditional density.¹

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