[Intuitively, Λ should contain no independent noise not accoutned for by the Xi]
That condition doesn’t work, but here’s a few alternatives which do (you can pick any one of them):
Λ=(x↦P[X=x|Λ]) - most conceptually confusing at first, but most powerful/useful once you’re used to it; it’s using the trick from Minimal Map.
Require that Λ be a deterministic function of X, not just any latent variable.
H(Λ)=I(X,Λ)
(The latter two are always equivalent for any two variables X,Λ and are somewhat stronger than we need here, but they’re both equivalent to the first once we’ve already asserted the other natural latent conditions.)
That condition doesn’t work, but here’s a few alternatives which do (you can pick any one of them):
Λ=(x↦P[X=x|Λ]) - most conceptually confusing at first, but most powerful/useful once you’re used to it; it’s using the trick from Minimal Map.
Require that Λ be a deterministic function of X, not just any latent variable.
H(Λ)=I(X,Λ)
(The latter two are always equivalent for any two variables X,Λ and are somewhat stronger than we need here, but they’re both equivalent to the first once we’ve already asserted the other natural latent conditions.)