Von Mises–Fisher laws as tilts of the uniform sphere #
The natural parameter η weights uniform sphere probability by exp ⟪η, x⟫. The sphere is
bounded, so the weight is integrable, its normalizer is positive in positive dimension, and
all real linear observables have every exponential moment. Their moment-generating functions
are ratios of normalizers at η + t • θ and η.
The direction/concentration form uses a unit direction and a nonnegative concentration; negative concentration gives zero. Zero natural parameter or concentration gives the uniform law. The natural-parameter law on the empty sphere in dimension zero is zero.
This construction uses Mathlib's Measure.tilted and its density and integration APIs, and
TauCeti.measurable_tilted for joint parameter measurability. The explicit two-point laws in
one dimension are developed in VonMisesFisher.OneDimensional.
References #
- K. V. Mardia, P. E. Jupp, Directional Statistics, Wiley (2000), §9.3.
The von Mises–Fisher law with natural parameter η, relative to uniform sphere probability.
In dimension zero the sphere and this measure are empty.
Instances For
The natural-parameter law is the exponential tilt of uniform sphere probability.
The von Mises–Fisher law with unit direction m and concentration κ ≥ 0.
Negative concentrations give the zero measure.
Instances For
At nonnegative concentration, the natural parameter is the concentration times the direction.
Negative concentration gives the zero measure.
Every natural parameter defines a probability law in positive dimension.
With a unit direction and nonnegative concentration, the law is a probability measure.
The direction/concentration law is a probability measure exactly at nonnegative concentration.
The natural-parameter law has the normalized exponential density relative to the uniform sphere law, including the empty-sphere case.
The Radon–Nikodym derivative against uniform sphere probability is the normalized exponential weight.
Zero natural parameter gives the uniform sphere law.
In dimension zero the natural-parameter law is zero on the empty sphere.
Natural parameters form a Giry-measurable family, including dimension zero.
The direction/concentration family is jointly measurable, including negative concentrations.
The unit-sphere embedding has every finite moment and is essentially bounded under every natural-parameter von Mises–Fisher law.
Every linear observable of a natural-parameter sphere law has all finite moments.
The exact exponential-integrability domain of every linear observable is all of ℝ.
The directional mgf is the ratio of the sphere normalizers at η + t • θ and η.
In positive dimension the directional cgf is the difference of the log normalizers.