Finally, rather than have an environment E that (when combined with a policy π) generates a world history (oa)*, you could have the state s directly be the world history (oa)*, _without_ including the policy π. In normal Bayesianism, using (oa)* as states would be equivalent to using environments E as states (since we could construct a belief over E that implies the given belief over (oa)*), but in the case of infra-Bayesianism it is not. (Roughly speaking, the differences occur when you use a “belief” that isn’t just a claim about reality, but also a claim about which parts of reality you “care about”.) This ends up allowing some but not all flavors of acausal influence, and so the authors call this setup “pseudocausal”.

Re:

In normal bayesianism, you do not have a pseudocausal-causal equivalence. Every ordinary environment is straight-up causal.

What I meant was that if you define a Bayesian belief over world-histories (oa)*, that is equivalent to having a Bayesian belief over environments E, which I think you agree with. I’ve edited slightly to make this clearer.

Re: the dispute over normal bayesianism: For me, “environment” denotes “thingy that can freely interact with any policy in order to produce a probability distribution over histories”. This is a different type signature than a probability distribution over histories, which doesn’t have a degree of freedom corresponding to which policy you pick.

But for infra-bayes, we can associate a classical environment with the set of probability distributions over histories (for various possible choices of policy), and then the two distinct notions become the same sort of thing (set of probability distributions over histories, some of which can be made to be inconsistent by how you act), so you can compare them.

Thanks for checking! I’ve changed point 3 to:

Re:

What I meant was that if you define a Bayesian belief over world-histories (oa)*, that is equivalent to having a Bayesian belief over environments E, which I think you agree with. I’ve edited slightly to make this clearer.

Looks good.

Re: the dispute over normal bayesianism: For me, “environment” denotes “thingy that can freely interact with any policy in order to produce a probability distribution over histories”. This is a different type signature than a probability distribution over histories, which doesn’t have a degree of freedom corresponding to which policy you pick.

But for infra-bayes, we can associate a classical environment with the

setof probability distributions over histories (for various possible choices of policy), and then the two distinct notions become the same sort of thing (set of probability distributions over histories, some of which can be made to be inconsistent by how you act), so you can compare them.Ah right, that makes sense. That was a mistake on my part, my bad.