Bayesian Statistics

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1.3 Families of random variables

Definition 1.3.1 We use the term family (of distributions) with parameter space $\Pi \sw \R ^d$ to mean that each $\theta \in \Pi $ corresponds to a random variable $M_\theta $, with parameters given by $\theta $. We require that all the random variables within a given family have the same range $R$, which we call the range of the family.

For example:

• The Beta family refers to the distributions $\Beta (\alpha ,\beta )$ where the parameter $\theta =(\alpha ,\beta )$ takes values in parameter space $\Pi =(0,\infty )^2$. It has range $[0,1]$.
• The Binomial family refers to the distributions $\Bin (n,p)$ where the parameter $\theta =(n,p)$ takes values in parameter space $\Pi =\N \times [0,1]$. It has range $\N $.

We say that a family is discrete if it is made up of (exclusively) discrete random variables, and continuous if is it made up of (exclusively) continuous random variables. So the Beta family is an continuous family and the Binomial family is a discrete family.

We will use this term for statistical models, written $(M_\theta )_{\theta \in \Pi }$ with parameter $\theta $. For this reason we will often refer to $(M_\theta )$ as a model family.

Assumption 1.3.2 For our model families, we require that $\theta \mapsto \P [M_\theta \in A]$ is a continuous function, for all $A\sw R$.

The purpose of this assumption is that if we change $\theta $ slightly then we only change the distribution of $M_\theta $ slightly. This will be necessary for our inference methods later on. This condition holds for most common families of random variables, including all of those listed on the reference sheets in Appendix A.