# What is a Random Variable in the formulation of McCullagh's “What is a Statistical Model?”

I'm relatively new to posting here so am not sure if this is the type of question I can ask, but I have been trying to understand for some time this paper by Peter McCullagh (it's not as long as it looks): https://pdfs.semanticscholar.org/4a01/7dd1ace17979828bb9f57e26ccf9c91f0b3b.pdf

I have read some basic category theory so am able to follow along with most of the definitions. However, I am confused about how one would model something with no covariates, say, parametric estimation for a sequence of i.i.d. random variables, in this formulation, since it seems to be necessary to have a nonempty covariate space in order to construct the parameter space. Furthermore, we couldn't even have arrows from $U$ to $\Omega$ if $\Omega$ were empty ("a design is a map associating with each unit $u \in U$ a point $x_u \in \Omega$"p.1234). This is where I begin to question whether I understand this formulation at all. Would love some clarification on these definitions if anyone has read this paper (or feels inclined to skim through it). The construction of the definition of a statistical model is on p.1235.

edit:

This is the basic setup:

There are three main building blocks with three different categories

• $cat_U$ the category with sets of statistical units as objects, injective maps as morphisms
• $cat_\Omega$ the category of covaraite spaces
• $cat_V$ the category of response scales

Then we define a design to be a map $x:U \rightarrow \Omega$ and the set of all such designs to be a category $cat_D$. This is all on p.1234-1236.

A choice of response scale $V$ determines an object $V^\Omega$. In the linear model our parameter space $\Theta_\Omega$ is a subspace $\subset V^\Omega$. A linear model is determined by a subrepresentation $\Theta_\Omega \subset V^\Omega$, together with a design pullback $\psi^*$: $$U \overset{\psi}{\rightarrow} \Omega \ \ \ \ \ \mathcal{S} = V^U \overset{\psi^*}{\leftarrow}\Theta_\Omega$$

When we add a dispersion parameter to the parameter space, $$U \overset{\psi}{\rightarrow} \Omega \ \ \ \ \ \mathcal{P}(V^U) \overset{P_\psi}{\leftarrow}\Theta_\Omega$$

we get the probability distributions on the sample space.

• My impression is that $\Omega$ indexes the experimental conditions. If you have any data at all, you have (conceptually) performed at least one "experiment" so $\Omega$ would at least be a singleton set. Though it wouldn't be in the spirit of the paper, as far as I can tell, you could have $cat_\Omega$ be the category with one object and one arrow. $\mathcal U$ would likely index the random variables, and $cat_\mathcal U$ would be the category of injective maps on finite sets or similar. Naturality would then mean your model should not depend too much on the number of random variables. – Derek Elkins left SE Apr 20 '18 at 18:40