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:
Exponentiation
A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties. The real numbers, for example, form a field, as do the complex numbers and rational numbers. Unlike these familiar examples of fields, which are all infinite sets, some fields have only finitely many elements. The simplest example is the field with two elements formula_146 with addition defined by formula_147 and formula_148, and multiplication formula_149 and formula_150.