This was all clear to me, but only from reading the text; my comment is just to say that the graphical statement doesn’t show ΛA being a mediator in the premises, so in isolation it gives the wrong idea; this led to a little confusion.
To be clear, I am talking about the reverse direction, as pictured here:
I understand that you have already set up ΛA as a mediator immediately above the image. Your text is perfectly clear:
In other words, we want to show: if Alice’ latent ΛA satisfies Mediation, and for any latent ΛB Bob could choose (i.e. any other mediator) we have ΛA←ΛB→ΛA, then Alice’ latent must be natural.
The other problem is that the image has only a single B, but the actual theorem proves necessity of Alice’s being a redund from the requirement that hers is determined by all possible Bob’s (that are mediators and agree on observables). Without the for all, you can’t sub in X_1 and X_2 for his latent.
This was all clear to me, but only from reading the text; my comment is just to say that the graphical statement doesn’t show ΛA being a mediator in the premises, so in isolation it gives the wrong idea; this led to a little confusion.
To be clear, I am talking about the reverse direction, as pictured here:
I understand that you have already set up ΛA as a mediator immediately above the image. Your text is perfectly clear:
The other problem is that the image has only a single B, but the actual theorem proves necessity of Alice’s being a redund from the requirement that hers is determined by all possible Bob’s (that are mediators and agree on observables). Without the for all, you can’t sub in X_1 and X_2 for his latent.