Satya Benson comments on Natural Latents: Latent Variables Stable Across Ontologies

Satya Benson 8 Oct 2025 14:52 UTC
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Graphical statement of Theorem 2
I find this picture pretty misleading, because it seems to say that if $Λ^{A}$ is determined by $Λ^{B}$ , then $Λ^{A}$ is a mediator, when really this is false, and it’s stated explicitly in the text above that Alice’s latent satisfying mediation is assumed.
- johnswentworth 8 Oct 2025 16:22 UTC
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  I think you might have misread something? The graphical statement of theorem 2 does not say that if $Λ^{A}$ is determined by $Λ^{B}$ , then $Λ^{A}$ is a mediator; that would indeed be false in general. It says that:
  - If $Λ^{B}$ is a mediator and we have agreement on observables, then...
  - … naturality of $Λ^{A}$ implies that $Λ^{A}$ is determined by $Λ^{B}$ .
  In particular, the theorem says that under some conditions $Λ^{A}$ is determined by $Λ^{B}$ . Determination is in the conclusion, not the premises. On the flip side, $Λ^{A}$ being a mediator is in the premises, not the conclusion.
  - Satya Benson 9 Oct 2025 12:52 UTC
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    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 $Λ^{A}$ 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 $Λ^{A}$ as a mediator immediately above the image. Your text is perfectly clear:
    In other words, we want to show: if Alice’ latent $Λ^{A}$ satisfies Mediation, and for any latent $Λ^{B}$ Bob could choose (i.e. any other mediator) we have $Λ^{A} \leftarrow Λ^{B} \to Λ^{A}$ , then Alice’ latent must be natural.