References on tilting distributions

Question

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent):

Consider the measure space $(\Omega, \mathscr{F}, \mathbb{P}$) and the non-negative measurable function $g$ such that $E(g(X)) < \infty $. Then we can define a new measure $\overline{\mathbb{P}}$ where we define the Radon-Nikodym derivative $\frac{d \overline{\mathbb{P}}}{d \mathbb{P}} = \frac{g(X)}{E(g(X))}$. The distribution of $X$ under this new measure $\overline{\mathbb{P}}$ is biased by the function $g$.

I have been looking for a source that discusses this particular idea, however, in my research most of the material online seems to be focused on specific distributions. For example, the vast majority of search results are references looking at specific treatments of Exponential Tilting - which is not what I am after.

I would be grateful for any resources that I could be redirected towards.

Answer 1

Size-bias (with $g(x)=x$ for $x\ge0$) arises in connection with the so-called "waiting time paradox" and Stein's method.

The survey paper Size bias for one and all by Arratia, Goldstein, and Kochman (AGK) does begin (after the Prologue) with a general definition of tilting (called bias in that paper and many other papers concerning Stein's method) -- see Section 2.1 on p. 4 there. At the bottom of the same page, there is a brief discussion of the obvious fact that "any two biasings commute, because multiplication is commutative."

Also on p. 4 of the AGK survey, at the end of the half-page Section 2.1, the following is stated:

The class of exponential functions, $h(x)=e^{\beta \, x}$ for various choices of $\beta \in (-\infty,\infty)$, is very important. This class is central to exponential families and large deviation theory, but no single value $\beta$ plays a special role. The family of power functions $h(x)=x^\beta$ for $\beta>0$ might be viewed as runner up, behind the family of exponential functions, but here the choice $\beta=1$ is truly special. We believe that $h(x)=x$ for $x \ge 0$ is the most important example of bias.

(The AGK survey uses the symbol $h$ where you use $g$.)

It appears that hardly anything interesting can be said about biasing in general except what is said on p. 4 of the AGK survey.