Vikrant Varma

Research Engineer at DeepMind.

2yΩ01

Thanks for this sequence!

I don't understand why the computer case is a counterexample for mutual information, doesn't it depend on your priors (which don't know anything about the other background noise interacting with photons)?

Taking the example of a one-time pad, given two random bit strings A and B, if C = A ⊕ B, learning C doesn't tell you anything about A unless you already have some information about B. So I(C; A) = 0 when B is uniform and independent of A.

Over time, the photons bouncing off the object being sought and striking other objects will leave an imprint in every one of those objects that will have high mutual information with the position of the object being sought.

If our prior was very certain about any factors that could interact with photons, then indeed the resulting imprints would have high mutual information, but it seems like you can rescue mutual information here by saying that our prior is uncertain about these other factors so the resulting imprints are noisy as well.

On the other hand, it seems correct that an entity that did have a more certain prior over interacting factors would see photon imprints as accumulating knowledge (for example photographic film).

To add some more concrete counter-examples: