I know very, very little about category theory, but some of this work regarding natural latents seem to absolutely smack of it. There seems to be a fairly important three-way relationship between causal models, finite factored sets, and Bayes nets.
To be precise, any causal model consisting of root sets B, downstream sets X, and functions mapping sets to downstream sets like f4:(B1⊗B3⊗X2)→X4 must, when equipped with a set of independent probability distributions over B, create a joint probability distribution compatible with the Bayes net that’s isomorphic to the causal model in the obvious way. (So in the previous example, there would be arrows from only B1, B3, and X2 to X4) The proof of this seems almost trivial but I don’t trust myself not to balls it up somehow when working with probability theory notation.
In the resulting Bayes net, one “minimal” natural latent which conditionally separates Xi and Xj is just the probabilities over just the root elements from B which both Xi and Xj depend on. It might be possible to show that this “minimal” construction of Λ satisfies a universal property, and so other Λ′ which is also “minimal” in this way must be isomorphic to Λ.
I know very, very little about category theory, but some of this work regarding natural latents seem to absolutely smack of it. There seems to be a fairly important three-way relationship between causal models, finite factored sets, and Bayes nets.
To be precise, any causal model consisting of root sets B, downstream sets X, and functions mapping sets to downstream sets like f4:(B1⊗B3⊗X2)→X4 must, when equipped with a set of independent probability distributions over B, create a joint probability distribution compatible with the Bayes net that’s isomorphic to the causal model in the obvious way. (So in the previous example, there would be arrows from only B1, B3, and X2 to X4) The proof of this seems almost trivial but I don’t trust myself not to balls it up somehow when working with probability theory notation.
In the resulting Bayes net, one “minimal” natural latent which conditionally separates Xi and Xj is just the probabilities over just the root elements from B which both Xi and Xj depend on. It might be possible to show that this “minimal” construction of Λ satisfies a universal property, and so other Λ′ which is also “minimal” in this way must be isomorphic to Λ.