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
On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see ). One may choose one of these values, called the principal value, but there is no choice of the principal value for which an identity such as
is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.