As a category theorist, I am confused by the diagram that you say you included to mess with me; I’m not even sure what I was supposed to think it means (where is the cone for Λ∗? why does the direction of the arrow between Λ∗ and Λ seem inconsistent?).
I think a “minimal latent,” as you have defined it equationally, is a categorical product (of the Xi) in the coslice category Ω↓Stoch where Stoch is the category of Markov kernels and Ω is the implicit sample space with respect to which all the random variables are defined.
As a category theorist, I am confused by the diagram that you say you included to mess with me; I’m not even sure what I was supposed to think it means (where is the cone for Λ∗? why does the direction of the arrow between Λ∗ and Λ seem inconsistent?).
I think a “minimal latent,” as you have defined it equationally, is a categorical product (of the Xi) in the coslice category Ω↓Stoch where Stoch is the category of Markov kernels and Ω is the implicit sample space with respect to which all the random variables are defined.
Are you sure? Wouldn’t the categorical product need to make the Xi independent not just from each other but also from Ω?