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 tilting
In Monte Carlo Estimation, exponential tilting (ET), exponential twisting, or exponential change of measure (ECM) is a distribution shifting technique commonly used in rare-event simulation, and rejection and importance sampling in particular. Exponential tilting is also used in Esscher tilting, an indirect Edgeworth approximation technique. The earliest formalization of ECM is often attributed to Esscher with its use in importance sampling being attributed to David Siegmund. ET is known as the Esscher transform in mathematical finance and is used in such contexts as insurance futures pricing.