Exponential family
What is important to note, and what characterizes all exponential family variants, is that the parameter(s) and the observation variable(s) must factorize (can be separated into products each of which involves only one type of variable), either directly or within either part (the base or exponent) of an exponentiation operation. Generally, this means that all of the factors constituting the density or mass function must be of one of the following forms:
Exponential field
A field is an algebraic structure composed of a set of elements, "F", two binary operations, addition (+) such that "F" forms an abelian group with identity 0 and multiplication (·), such that "F" excluding 0 forms an abelian group under multiplication with identity 1, and such that multiplication is distributive over addition, that is for any elements "a", "b", "c" in "F", one has . If there is also a function "E" that maps "F" into "F", and such that for every "a" and "b" in "F" one has
then "F" is called an exponential field, and the function "E" is called an exponential function on "F". Thus an exponential function on a field is a homomorphism between the additive group of "F" and its multiplicative group.