In the Alice and Bob example aren’t the rules of the game part of the natural latent? Like Carol also needs to remember that neither Alice nor Bob will draw any [color that isn’t purple] lines and this information isn’t in the green lines.
Conditional on the rules, the green lines serve as a natural latent between the pictures, so knowing the rules induce a phase change where the green lines become a natural latent. But without knowing the rules, you can’t infer the rules from only one picture, and so the rules cannot serve as a natural latent, even when combined with the green lines.
In the Alice and Bob example aren’t the rules of the game part of the natural latent? Like Carol also needs to remember that neither Alice nor Bob will draw any [color that isn’t purple] lines and this information isn’t in the green lines.
Conditional on the rules, the green lines serve as a natural latent between the pictures, so knowing the rules induce a phase change where the green lines become a natural latent. But without knowing the rules, you can’t infer the rules from only one picture, and so the rules cannot serve as a natural latent, even when combined with the green lines.