Shared notation - Mathematical notes

This page is the single shared notation canon for Incerto, Information Geometry, and Normix theory. New notes must reuse these symbols for the same concepts. Do not invent parallel symbols that mean the same thing in another track.

Track hubs and body notes may still define local symbols for one-off arguments. When a concept already appears here, prefer the shared form and link here rather than redefining it.

Probability and expectation¶

Symbol	Meaning
$P$ , $P_\theta$	A probability law; the model law at parameter $\theta$
$p_\theta$ , $p(x\mid\theta)$	Density of $P_\theta$ with respect to a stated base measure
$\mathbb E$ , $\mathbb E_\theta$	Expectation; expectation under $P_\theta$
$\operatorname{Cov}$ , $\operatorname{Var}$	Covariance and variance under the stated law
$X$ , $x$	Observable random element and a realized value
$Z$ , $z$	Latent or noise variable when that role is explicit (see Normix note below)

Logarithms are natural unless a note states otherwise. Densities are taken with respect to a common measure named in the note (Lebesgue or counting measure).

Information geometry¶

Symbol	Meaning
$\theta$	Natural / parameter coordinates of a regular model or exponential family
$\eta$	Expectation coordinates; for an exponential family $\eta=\nabla\psi(\theta)$
$\psi$	Log-partition function of an exponential family
$s_\theta(x)=\nabla_\theta\log p_\theta(x)$	Score
$g_\theta$ , $I(\theta)$	Fisher metric / Fisher information (per observation when stated)
$t(x)$	Sufficient statistic in an exponential family
$P_\theta(x,z)$ , $p_\theta(x)$ , $r_\theta(z\mid x)$	Joint density, observable marginal, and posterior in a latent model
$D$ , $\mathrm{KL}$	Kullback–Leibler divergence (direction stated at use)

Smoothness, common support, integrability, and nonsingular Fisher information are assumed only where a note states them. Boundary laws and redundant parameterizations need separate treatment.

Incerto / fat tails¶

Symbol	Meaning
$F$ , $\bar F=1-F$	Distribution function and survival function
$\alpha$	Pareto / regularly varying tail index (power exponent)
$\xi$	Extreme-value index
$u$ , $x_m$	Threshold; Pareto scale (minimum) when that model is used
$e(u)$	Mean excess function above $u$
$L$	Slowly varying function in a regular-variation representation

A finite-sample exceedance count is not the same object as a model probability $\bar F(u)$ . Keep both visible when a note moves between data and model.

Normix / mixtures¶

Symbol	Meaning
$Y$	Positive mixing variable in Normix theory notes (GIG/GH constructions)
$W$	Mixing variable in the original conditioning example; map $W\leftrightarrow Y$ when linking
$\operatorname{GIG}(p,a,b)$	Generalized inverse Gaussian with parameters $(p,a,b)$
GH	Generalized hyperbolic law obtained by normal variance–mean mixing with a GIG
$\mu$ , $\gamma$ , $\delta$ , $\omega$	Location / skewness / scale conventions as defined in the GH notes

Package API documentation at normix may use implementation names; mathematical notes here follow this shared canon and map to the API when needed.

How to extend this page¶

Add a row when the same concept is about to be named in more than one track. Prefer one short shared definition over three local glossaries. Track-specific indexes (for example Incerto concept hubs) may point here for overlapping symbols.