This track develops the mathematics of normal variance–mean mixtures, their mixing distributions, and estimation through latent variables. Begin with conditioning a mixture, an original calculation that fixes independence and moment assumptions explicitly.
The first theory batch provides the foundations:
Generalized inverse Gaussian defines the positive mixing density, its moments, boundary families, and exponential-family coordinates.
Generalized hyperbolic integrates the conditional Gaussian, relates joint and marginal laws, and derives the scale nonidentifiability and moment maps.
EM for GH derives the GIG posterior, the three mixing expectations, and the normal and mixing-law M-steps.
Exponential-family core connects classical, natural, and expectation coordinates through one log-partition function and its derivatives.
Mixture architecture explains the mathematical roles of joint, marginal, and posterior laws and fixes the sufficient-statistic order.
Why not gradient descent? separates structural optimization advantages from the upstream benchmark evidence and discusses their limits.
GH family tour follows gamma, inverse gamma, inverse Gaussian, and GIG mixing into their named normal mixtures.
Normal variance–mean mixtures develops observable moments, an exact multivariate example, and scalar CDF identities.
For a gentler route after the conditioning example, read the family tour and normal-mixtures note before the density and estimation derivations. The IG exponential-family entry and ordinary EM entry supply the general assumptions and geometry behind the calculations.
Continue after batch EM and the optimization comparison with the second batch.
Sequential and regularized EM¶
Online EM averages posterior sufficient statistics sequentially and derives a Bregman regret identity with its assumptions.
Shrinkage derives penalized EM toward a reference distribution and distinguishes joint-KL shrinkage from covariance-only updates.
EM as sufficient-statistic updates separates aggregation, update rules, and parameter recovery, then composes running averages with Bregman shrinkage.
Structured covariances¶
Factor analysis for GH develops low-rank plus diagonal dispersion, ten sufficient-statistic blocks, and a constrained M-step for a curved exponential family. It connects to the IG exponential-family entry without assuming that ambient moment matching fits a constrained model.
Information quantities¶
Entropy, varentropy, and Rényi entropy derives density-power formulas for exponential-family components and joint normal mixtures. The Fisher geometry entry explains the score covariance appearing in the varentropy formula; the duality entry supports the EM divergences.
Notation and implementation¶
Cross-track symbols follow the shared notation canon. Do not introduce a second name for a concept already fixed there.
The thirteen imported notes are static mathematical explanations, each retaining its pinned source and MIT notice. Their literature mixing variable is called ; it corresponds to in the conditioning introduction. Standard normal noise is called throughout the overlapping constructions.
The normix repository owns the JAX implementation, public API, package tests, and releases. Its package documentation remains the entry point for installation and supported operations. This site does not install or vendor that package. Later computational notes should use a tested upstream version and map their mathematical convention to its supported API.
The complete EM implementation framework, Bessel and solver design, and package API remain upstream references. Finance theory, remaining tutorials, and research notes are deferred; the later IG/GH research sequence remains planned.