The theorem doesn’t actually specify a coarse-graining process. The proof would say:
We can construct a new variable T’ by sampling a temperature given mirror-state. By construction, mirror-state perfectly mediates between T’ and goosebumps.
There exists some pareto-optimal (under the errors of the natural latent conditions) latent which is pareto-as-good-as T’
Any pareto optimal latent which is pareto-as-good-as T’ can be perfectly coarse-grained, by graining together any values of the latent which give exactly the same distribution P[mirror|latent value].
Because the middle bullet is not constructive, we don’t technically specify a process. That said, one could specify a process straightforwardly by just starting from T’ and pareto-improving the latent in a specific direction until one hits an optimum.
In this case, the coarse-graining would probably just be roughly (temperatures at which the mirror fogs) and (temperatures at which it doesn’t), since that’s the only nontrivial coarse-graining allowed by the setup (because the coarse-grained value must be approximately determined by the mirror-state).
Yup, good example!
The theorem doesn’t actually specify a coarse-graining process. The proof would say:
We can construct a new variable T’ by sampling a temperature given mirror-state. By construction, mirror-state perfectly mediates between T’ and goosebumps.
There exists some pareto-optimal (under the errors of the natural latent conditions) latent which is pareto-as-good-as T’
Any pareto optimal latent which is pareto-as-good-as T’ can be perfectly coarse-grained, by graining together any values of the latent which give exactly the same distribution P[mirror|latent value].
Because the middle bullet is not constructive, we don’t technically specify a process. That said, one could specify a process straightforwardly by just starting from T’ and pareto-improving the latent in a specific direction until one hits an optimum.
In this case, the coarse-graining would probably just be roughly (temperatures at which the mirror fogs) and (temperatures at which it doesn’t), since that’s the only nontrivial coarse-graining allowed by the setup (because the coarse-grained value must be approximately determined by the mirror-state).