# How to determine sample space, $\sigma$-algebra and probability measure from the exponential family?

The sample space of binomial distribution is the set $$\{0,1\}$$ and its $$\sigma$$-algebra is the power set of $$\{0,1\}$$ while the sample space of normal distribution is $$\mathbb R$$ and its $$\sigma$$-algebra is the Lebesgue measurable set. In these cases, the probability measure of binomial distribution is of the form of exponential family while the radon-nikodym derivative of the probability measure (i.e. pdf) of normal distribution is of the form of exponential family. Their natures are so different and why do people even call it a family?

My questions are:

1. How do we determine if exponential family gives us the probability measure or the probability density function?
2. How can we determine the sample space, $$\sigma$$-algebra and its probability measure from an exponential family?

Any help are welcome.

• The sample space of a Binomial $\mathcal B(n,p)$ is $\{0,1,\ldots,n\}$. Jul 29, 2021 at 3:14

• The pmf for a distribution with a countable sample space is its density wrt to the counting measure.
• Exponential families are made of families of parametric distributions whose density wrt to a given dominating measure $$\text d\lambda(x)$$ write as $$f_\theta(x)=\exp\{R(\theta)^\text{T}S(x)-\psi(\theta)\}$$, where $$\psi(\theta)$$ is the log-normalising constant.$$^1$$ The Binomial and the Normal distributions belong to different exponential families.$$^2$$
• the sample space, $$\mathfrak X$$, its σ-algebra $$\mathfrak B(\mathfrak X)$$ and the dominating measure $$\text d\lambda(x)$$ are components of the definition of a given exponential family, just as much as the functions $$R(\cdot)$$ and $$S(\cdot)$$ above. If they are not provided, the associated exponential family is not defined.$$^3$$

$$^1$$I purposedly did not include the traditional $$h(x)$$ in the representation of $$f_\theta(x)$$, i.e., $$f_\theta(x)=h(x)\exp\{R(\theta)^\text{T}S(x)-\psi(\theta)\}$$ to stress the fact that $$h(\cdot)$$ can as well be part of the dominating measure. Changing $$h(\cdot)$$ thus amounts to changing the dominating measure.

$$^2$$In particular, the Binomial distribution is absolutely continuous wrt the counting measure on $$\mathfrak X=\mathbb N$$ and the Normal distribution is absolutely continuous wrt the Lebesgue measure on $$\mathbb R$$

$$^3$$In other words, a probability distribution does not exist independently of its dominating measure, which itself does not exist independently of its σ-algebra $$\mathfrak B(\mathfrak X)$$, which itself requires a sample space $$\mathfrak X$$

• I understand what a measure and a probability measure are, but a 'dominating' measure is new to me. What is it? Jul 29, 2021 at 3:27
• If you invoke the Radon-Nykodim theorem, you are using dominating measures. Jul 29, 2021 at 3:35
• Jul 29, 2021 at 3:41
• Except for the fact that the PKD theorem is about exponential families, I do not see the relation. From its definition, an exponential family distribution necessarily has a sample space that does not vary with the associated parameter. Jul 29, 2021 at 4:57
• Thank you from your clarification. The traditional $h(x)$ bothered me a lot. Jul 29, 2021 at 11:45