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:
Ordered exponential
where the term "n" = 0 is equal to 1 and
where formula_3 is a higher-order operation that ensures the exponential is time-ordered: any product of "a"("t") that occurs in the expansion of the exponential must be ordered such that the value of "t" is increasing from right to left of the product; a schematic example:
This restriction is necessary as products in the algebra are not necessarily commutative.