Your natural latents seem to be quite related to the common construction IID variables conditional on a latent—in fact, all of your examples are IID variables (or “bundles” of IID variables) conditional on that latent. Can you give me an interesting example of a natural latent that is not basically the conditionally IID case?
(I was wondering if the extensive literature on the correspondence between De Finetti type symmetries and conditional IID representations is of any help to your problem. I’m not entirely sure if it is, given that mostly addresses the issue of getting from a symmetry to a conditional independence, whereas you want to get from one conditional independence to another, but it’s plausible some of the methods are applicable)
A natural latent is, by definition, a latent which satisfies two properties. The first is that the observables are IID conditional on the latent, i.e. the common construction you’re talking about. That property by itself doesn’t buy us much of interest, for our purposes, but in combination with the other property required for a natural latent, it buys quite a lot.
Your natural latents seem to be quite related to the common construction IID variables conditional on a latent—in fact, all of your examples are IID variables (or “bundles” of IID variables) conditional on that latent. Can you give me an interesting example of a natural latent that is not basically the conditionally IID case?
(I was wondering if the extensive literature on the correspondence between De Finetti type symmetries and conditional IID representations is of any help to your problem. I’m not entirely sure if it is, given that mostly addresses the issue of getting from a symmetry to a conditional independence, whereas you want to get from one conditional independence to another, but it’s plausible some of the methods are applicable)
A natural latent is, by definition, a latent which satisfies two properties. The first is that the observables are IID conditional on the latent, i.e. the common construction you’re talking about. That property by itself doesn’t buy us much of interest, for our purposes, but in combination with the other property required for a natural latent, it buys quite a lot.
Wait, I thought the first property was just independence, not also identically distributed.
In principle I could have e.g. two biased coins with their biases different but deterministically dependent.
Oh, you’re right. Man, I was really not paying attention before bed last night! Apologies, you deserve somewhat less tired-brain responses than that.