In case anyone’s wondering: You really do have to assume Alice has a mediator for the necessity on the translation theorem. That is you can’t derive it from being determined by all mediators—otherwise, her “latent” could be a constant. Intuitively: translation means that all her info is in all mediators that Bob could have, while the mediator condition is says she has “enough” info to be a natural latent instead of something stupid like a constant.
In case anyone’s wondering: You really do have to assume Alice has a mediator for the necessity on the translation theorem. That is you can’t derive it from being determined by all mediators—otherwise, her “latent” could be a constant. Intuitively: translation means that all her info is in all mediators that Bob could have, while the mediator condition is says she has “enough” info to be a natural latent instead of something stupid like a constant.