Exponential families can be handled compactly
Exponential families! The secret magic at the heart of traditional statistics.
Exponential families are probability distributions that just work, in the sense that and the things we would hope we can do with them, we can. Thus these are the distributions we are taught to handle in statistics classes, and which lead us to undue optimism about statistics more generally, all of which falls apart later. Often, though, we can approximate intractable families by exponential ones or cunning combinations thereof, e.g. in variational inference, so this is not a complete waste of time.
Natural exponential families
a.k.a. NEFs. The simplest case. Suppose that \(\mathbf{x} \in \mathcal {X} \subseteq \mathbb{R} ^{p}.\) Then, a natural exponential family of order p has density or mass function of the form: \[ {\displaystyle f_{X}(\mathbf {x} \mid {\boldsymbol {\theta }})=h(\mathbf {x} )\ \exp {\Big (}{\boldsymbol {\theta }}^{\rm {T}}\mathbf {x} -A({\boldsymbol {\theta }})\ {\Big )}\,\!,} \] where in this case the parameter \(\boldsymbol {\theta }\in \mathbb {R}^{p}.\)
Important members of this sub-family: Gamma, Gaussian, negative binomial, Poisson and binomial.
I mention this family first because it is a good intuition pump. More commonly we consider a more general family.
(Full-blown) exponential families
In the more general case we allow the natural statistics and the parameters to not be in natural form but rather related by some \(\mathbb{R} ^{p}\to\mathbb{R} ^{p}\) functions \(T\) and \(\eta.\) The non-trivial part is the \(T\) function — we can always redefine the \(\eta(\theta)\) to tbe the real parameters rather than \(\theta\) and in fact we frequently do, calling it the canonical parameterisation. \[ \displaystyle f_{X}\!\left(\,\mathbf {x} \mid {\boldsymbol {\theta }}\,\right)=h(\mathbf {x} )\,\exp \!{\Big (}\,{\boldsymbol {\eta }}({\boldsymbol {\theta }})\cdot \mathbf {T} (\mathbf {x} )-A({\boldsymbol {\theta }})\,{\Big )}. \] I.e. these are nonlinear transformation of NEFs. We call \(\eta\) the natural parameter, and \(\mathbf {T}\) the sufficient statistic, and \(A\) the log-partition function.
Natural parameters and sufficient statistics
One of the neat things about the exponential families is that the partition function, natural statistics and natural parameters are informative about each other.
The cumulant-generating function is simply \(K(u|\eta)=A(\eta+u)-A(\eta)\).
For the natural exponential families, \(T\) and \(\eta\) are identities, the mean vector and covariance matrix are \[ \operatorname {E} [X]=\nabla A({\boldsymbol {\theta }}){\text{ and }}\operatorname {Cov} [X]=\nabla \nabla ^{\rm {T}}A({\boldsymbol {\theta }})\] where \(\nabla\) is the gradient and \(\nabla \nabla ^{\top}\) is the Hessian matrix.
Natural exponential families with quadratic variance functions
A special case with even nicer properties (Morris 1982, 1983; Morris and Lock 2009).
Morris (1982):
The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance property, including infinite divisibility, cumulants, orthogonal polynomials, large deviations, and limits in distribution.
Conjugate priors
TBD. Diaconis and Ylvisaker (1979)
PCA
PCA is famous for Gaussian data. I gather there is some sense in which it can be generalised to all exponential families as the Exponential Family PCA (Collins, Dasgupta, and Schapire 2001; Jun Li and Dacheng Tao 2013; Liu, Dobriban, and Singer 2017; Mohamed, Ghahramani, and Heller 2008).
For random graphs
Exponential random graph models. TBD
In graphical models
Curved exponential families
A generalisation I occasionally see is that of curved exponential families. I do not know how these work of if they have enough features to benefit me.
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