Distributions

normix implements the Generalized Hyperbolic (GH) family and its relatives as a single, consistent set of exponential-family distributions. This guide is a map of what is available and how to choose.

The family at a glance

Every multivariate distribution here is a normal variance-mean mixture,

\[ X \mid Y \sim \mathcal{N}(\mu + \gamma\, Y,\; \Sigma\, Y), \qquad Y \sim \text{subordinator}, \]

and the choice of positive subordinator \(Y\) names the member:

Distribution	Subordinator	Subordinator parameters
VarianceGamma	Gamma	alpha, beta
NormalInverseGamma	Inverse Gamma	alpha, beta
NormalInverseGaussian	Inverse Gaussian	mu_ig, lam
GeneralizedHyperbolic	GIG	p, a, b

The shared location/shape parameters are always \(\mu\) (mu), \(\gamma\) (gamma), and the covariance \(\Sigma\) given through its Cholesky factor L_Sigma. Because the GIG nests the Gamma, Inverse Gamma, and Inverse Gaussian as limits, GeneralizedHyperbolic nests all the others — see A tour of the GH family.

Subordinators and the Gaussian core

The building blocks are exponential families in their own right:

Distribution	Parameters	Notes
Gamma	alpha, beta	closed-form moments and MLE
InverseGamma	alpha, beta	closed-form moments and MLE
InverseGaussian	mu, lam	closed-form moments and MLE
GIG / GeneralizedInverseGaussian	p, a, b	Bessel-valued log-partition
MultivariateNormal	mu, L_Sigma	the Gaussian core

See Univariate positive distributions, The Generalized Inverse Gaussian, and The multivariate normal.

Three layers per mixture

Each mixture comes in several layers; reach for the one that matches your task:

• Marginal (e.g. NormalInverseGaussian) — the distribution of \(X\). This is what you usually fit and evaluate: pdf, log_prob, mean, cov, rvs.

• Joint (e.g. JointNormalInverseGaussian, via model.joint) — the pair \((X, Y)\) with the latent subordinator, used by the EM E-step and accessible for joint.rvs and joint.conditional_expectations.

• Univariate (e.g. UnivariateNormalInverseGaussian) — the \(d = 1\) case, which adds a scipy-style cdf and ppf for tail calculations.

• Factor (e.g. FactorNormalInverseGaussian) — a high-dimensional variant with a low-rank-plus-diagonal covariance \(\Sigma = F F^\top + \operatorname{diag}(D)\); see Factor mixtures for high dimensions.

Choosing a distribution

• Light-to-moderate tails, symmetric or skewed: VarianceGamma or NormalInverseGamma.

• Heavy tails (financial returns): NormalInverseGaussian is a robust default; GeneralizedHyperbolic adds a third shape parameter for the heaviest cases.

• Unsure / want the most flexible model: GeneralizedHyperbolic — it contains the others as special cases.

• Many assets / dimensions: the corresponding Factor* variant.

• One-dimensional with CDF/quantile needs: the Univariate* variant.

A common interface

Whatever you pick, the API is the same:

from normix import NormalInverseGaussian

model = NormalInverseGaussian.from_classical(
    mu=mu, gamma=gamma, sigma=Sigma, mu_ig=1.0, lam=1.5)

model.pdf(x)            # density at one point (vmap to batch)
model.mean(); model.cov()
model.rvs(1000, seed=0)
result = model.fit(X)   # EM; result.model is the fit

Construction also works from natural parameters (from_natural) and from expectation parameters (from_expectation) — the three parametrizations are explained in Exponential-family structure.