Good point. In that case the Drake equation must be modified to include panspermia probabilities and the variance in time-to-civilization among our sister lineages. I’m curious what kind of Bayesian update we get on those...
The observation can provide all sorts of information about the universe, including whether exploration occurs. The exact set of possible observations depends on the decision problem.
E and O can have any relationship, but the most interesting case is when one can infer E from O with certainty.
Thanks, I made this change to the post.
Yeah, I think the fact that Elo only models the macrostate makes this an imperfect analogy. I think a better analogy would involve a hybrid model, which assigns a probability to a chess game based on whether each move is plausible (using a policy network), and whether the higher-rated player won.
I don’t think the distinction between near-exact and nonexact models is essential here. I bet we could introduce extra entropy into the short-term gas model and the rollout would still be superior for predicting the microstate than the Boltzmann distribution.
Sure: If we can predict the next move in the chess game, we can predict the next move, then the next, then the next. By iterating, we can predict the whole game. If we have a probability for each next move, we multiply them to get the probability of the game.
Conversely, if we have a probability for an entire game, then we can get a probability for just the next move by adding up all the probabilities of all games that can follow from that move.
Thanks, I didn’t know that about the partition function.
In the post I was thinking about a situation where we know the microstate to some precision, so the simulation is accurate. I realize this isn’t realistic.
The sum isn’t over i, though, it’s over all possible tuples of length n−1. Any ideas for how to make that more clear?
I’m having trouble following this step of the proof of Theorem 4: “Obviously, the first conditional probability is 1”. Since the COD isn’t necessarily reflective, couldn’t the conditional be anything?
The linchpin discovery is probably February 2016.
Ok. I think that’s the way I should have written it, then.
The definition involving the permutation is a generalization of the example earlier in the post: ϕ(T) is the identity and ϕ(H) swaps heads and tails. And X=ϕ(A)−1(C). In general, if you observe A=a and C=c, then the counterfactual statement is that if you had observed A=a′, then you would have also observed C=ϕ(a′)(ϕ(a)−1(c)).
I just learned about probability kernels thanks to user Diffractor. I might be using them wrong.
Oh, interesting. Would your interpretation be different if the guess occurred well after the coinflip (but before we get to see the coinflip)?
That sounds about right to me. I think people have taken stabs at looking for causality-like structure in logic, but they haven’t found anything useful.
What predictions can we get out of this model? If humans use counterfactual reasoning to initialize MCMC, does that imply that humans’ implicit world models don’t match their explicit counterfactual reasoning?
I agree exploration is a hack. I think exploration vs. other sources of non-dogmatism is orthogonal to the question of counterfactuals, so I’m happy to rely on exploration for now.
“Programmatically Interpretable Reinforcement Learning” (Verma et al.) seems related. It would be great to see modular, understandable glosses of neural networks.