*1.2. Truncated Exponential Families with Nested Supports*

In this paper, we shall consider truncated exponential families [7] with nested supports. A truncated exponential family is a set of parametric probability distributions obtained by truncation of the support of an exponential family. Truncated exponential families are exponential families but their statistical inference is more subtle [8,9]. Let ETrunc = {*qθ*} be a truncated exponential family of E = {*pθ*} with nested supports supp(*q<sup>θ</sup>* ) ⊂ supp(*p<sup>θ</sup>* ). The canonical decompositions of densities *p<sup>θ</sup>* and *q<sup>θ</sup>* have the following expressions:

$$p\_{\theta}(\mathbf{x}) \quad = \ \exp\left(\theta^{\top}t(\mathbf{x}) + k(\mathbf{x}) - F(\theta)\right),\tag{4}$$

$$q\_{\boldsymbol{\theta}}(\mathbf{x}) \;=\; \frac{p\_{\boldsymbol{\theta}}(\mathbf{x})}{Z^{X\_{\text{Trunc}}}(\boldsymbol{\theta})} = \exp\left(\boldsymbol{\theta}^{\top}\boldsymbol{t}(\mathbf{x}) + k(\mathbf{x}) - F\_{\text{Trunc}}(\boldsymbol{\theta})\right),\tag{5}$$

where the log-normalizer of the truncated exponential family is:

$$F\_{\text{Trunc}}(\theta) = F(\theta) + \log Z^{X\_{\text{Trunc}}}(\theta),\tag{6}$$

where *Z*XTrunc (*θ*) is a normalizing term that takes into account the truncated support XTrunc. These equations show that densities of truncated exponential families only differ by their log-normalizer functions. Let XTrunc denote the support of the distributions of ETrunc = supp(*q<sup>θ</sup>* ) and X = supp(*p<sup>θ</sup>* ) the support of E. Family ETrunc is a truncated exponential family of E that can be notationally written as EXTrunc . Family E can also be interpreted as the (un)truncated exponential family EX with densities *p*<sup>X</sup> *<sup>θ</sup>* = *pθ*. A truncated exponential family EXTrunc of E is said to have nested support when XTrunc ⊂ X . For example, the family of half-normal distributions defined on the support XTrunc = [0, ∞) is a nested truncated exponential family of the family of normal distributions defined on the support X = (−∞, ∞).
