Fix a measurable space
has as its effective domain the natural parameter space
The potential
Property. Dual flatness
The natural coordinates
are affine for the exponential ( -) connection; the expectation coordinates are affine for the mixture ( -) connection, and is a diffeomorphism onto the open convex set . The free energy and its Legendre conjugate , (the negative entropy), satisfy
with Fisher metric.
Property. The maximum-entropy characterisation
Fixing the means
and maximising the entropy forces the Gibbs form . This is an exponential family whose sufficient statistics are exactly the constrained , with the Lagrange multipliers as the natural parameters. Matching the means, , is the first-order condition of the convex dual . Equivalently it is the minimum Kullback–Leibler (KL) distribution to under the constraints, and KL restricted to is the Bregman divergence of . Maximum entropy, exponential families, Bregman divergence, and dual-flat geometry are thus four readings of one convex-duality picture.
Property. The two intrinsic geodesics are closed for free
The dual affine charts give two deterministic interpolations intrinsic to
: the -geodesic (normalised geometric mean ) and the -geodesic (moment matching). Both stay in automatically because and are convex. This carries no content beyond dual flatness and should not be confused with a probabilistic operation on ; see the interpolation of distributions.
Example. Deformed (
) generalization Replacing
by a strictly increasing (a deformed logarithm) and requiring gives the -deformed exponential families, dually flat in the -geometry. The endpoints and are the exponential and mixture families; Tsallis / Kaniadakis deformations give the -/ -exponential families (Student- , -Gaussian).
Adding a dispersion index to a natural exponential family gives the exponential dispersion models, the reproductive/Tweedie convolution axis.