Graphical models are based on the idea of representing certain types of conditional independences in a (joint) distribution via a graph, and are an active research area.

As argued (correctly I believe too) here, there is, hypothetically, no reason why one could not make a completely analogous theory of graph-based representations for marginal independences. Yet this does not appear to be done widely, if at all.

Question: Why is representing marginal independences via a graph a much less important idea (seemingly) than representing conditional independences via a graph?

Or is it just a historical accident that the former theory has not been heavily developed, while the latter one has not been developed much at all?

It seems that it could be important to have both theories, since conditional independences and marginal independences are incomparable, i.e. neither one implies the other. Hence we seemingly cannot derive one theory as a consequence of the other theory.

Also (see below) it seems that there would potentially be a large use for marginal independence graphs for those creating and using multivariate response regression models.

Note 1: Searching quickly, one argument I found for not studying marginal independence using graphs, and only conditional independence, can be found here. Specifically, the argument seems to go that while situations where marginal independence can be (correctly, fruitfully) applied as a model assumption are rare in practice, analogous are frequent or even common for conditional independence. So perhaps it is just an issue of prioritizing the more frequently applied notion.

This also raises (for me) the question: why would opportunities to apply conditional independence be more common in practice than opportunities to apply marginal independence? Is there a heuristic explanation for why one might reasonably expect this to be the case?

Note 2: Professor J. Whittaker says on p. 60 of his book, Graphical Models in Applied Multivariate Mathematical Statistics, the following:

"... there is no suitable theory for graphs constructed from pairwise marginal independences".

Indeed, this quote is the reason for my question. Saying that there is "no suitable theory" (and presumably implicitly that no such theory could exist) seems like a very strong statement, and one for which I would like to see justification. But he does not appear to mention any.

Later in the book (p. 325), where he is explaining how linear regression can be understood via conditional independence, he says this about the multivariate response case:

... though analyses can be put into direct correspondence, the usual hypotheses of multivariate regression are not those of the graphical modeller: while the latter examines the joint distribution of the response variables, the regression modeller focuses on the marginal distribution of each response.

This very strongly suggests that there is a population of statisticians who would be interested in a theory of representing marginal independences via a graph, namely those who are interested in multivariate response regression. This seems to very strongly contradict the claim given on p. 60, since the number of people (as far as I can tell) who use multivariate response regression methods is vast, indeed probably far vaster than those who use graphical model methods.

Further support for my interpretation that a theory of marginal independence graphs would be considered useful for those using regression is the Example 10.5.2. he gives on p. 327, where he displays a graph whose edges are defined according to marginal independence, not conditional independence, explaining that questions about whether certain regression coefficients are zero or non-zero correspond to questions about edge inclusion in the marginal independence graph.


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