Conditioning a mixture - Normix theory

Fix the model first¶

Let $Z$ be standard normal and let $W\geq0$ be independent of $Z$ . For constants $\mu,\beta\in\mathbb R$ and $\sigma>0$ , define

X=\mu+\beta W+\sigma\sqrt W\,Z.

(1)

Conditional on $W=w>0$ , this is normal with mean $\mu+\beta w$ and variance $\sigma^2w$ . At $w=0$ it is a point mass at $\mu$ . This conditional description does not require choosing a particular distribution for $W$ .

One exact calculation¶

Assume in addition that $\mathbb E[W^2]<\infty$ . Conditional expectation gives

\mathbb E[X]=\mu+\beta\mathbb E[W].

(2)

The law of total variance separates variation inside each conditional normal from variation of its mean:

\operatorname{Var}(X) =\mathbb E[\operatorname{Var}(X\mid W)] +\operatorname{Var}(\mathbb E[X\mid W]) =\sigma^2\mathbb E[W]+\beta^2\operatorname{Var}(W).

(3)

For the limiting case $W=1$ almost surely, this reduces to a normal variable with mean $\mu+\beta$ and variance $\sigma^2$ . The second-moment assumption is a sufficient condition for the displayed calculation, not a definition of every possible normal mixture.

The normal-mixtures note extends this calculation to vectors. The GH note uses the literature symbols $Y=W$ , $\gamma=\beta$ , and $\Sigma=\sigma^2$ in one dimension, while keeping $Z$ standard normal and independent. Choosing a GIG law for the mixing variable leads to the posterior calculations in EM for GH.

This original derivation introduces a mathematical model, not an assertion about a specific normix constructor or parameterization. Consult the upstream package when writing executable package examples. Compare counting exceedances for a quantity that describes observations without specifying their model.