Where are you on the spectrum from "SSA and SIA are equally valid ways of reasoning" to "it's more and more likely that in some sense SIA is just true"? I feel like I've been at the latter position for a few years now.
Interesting! Can you write up the WLIC, here or in a separate post?
I thought Diffractor's result was pretty troubling for the logical induction criterion:
...the limit of a logical inductor, P_inf, is a constant distribution, and by this result, isn't a logical inductor! If you skip to the end and use the final, perfected probabilities of the limit, there's a trader that could rack up unboundedly high value!
But maybe understanding has changed since then? What's the current state?
Wait, can you describe the temporal inference in more detail? Maybe that's where I'm confused. I'm imagining something like this:
Check which variables look uncorrelated
Assume they are orthogonal
From that orthogonality database, prove "before" relationships
Which runs into the problem that if you let a thermodynamical system run for a long time, it becomes a "soup" where nothing is obviously correlated to anything else. Basically the final state would say "hey, I contain a whole lot of orthogonal variables!" and that would stop you from proving any reasonable "before" relationships. What am I missing?
I think your argument about entropy might have the same problem. Since classical physics is reversible, if we build something like a heat engine in your model, all randomness will be already contained in the initial state. Total "entropy" will stay constant, instead of growing as it's supposed to, and the final state will be just as good a factorization as the initial. Usually in physics you get time (and I suspect also causality) by pointing to a low probability macrostate and saying "this is the start", but your model doesn't talk about macrostates yet, so I'm not sure how much it can capture time or causality.
That said, I like really like how your model talks only about information, without postulating any magical arrows. Maybe it has a natural way to recover macrostates, and from them, time?
Thanks for the response! Part of my confusion went away, but some still remains.
In the game of life example, couldn't there be another factorization where a later step is "before" an earlier one? (Because the game is non-reversible and later steps contain less and less information.) And if we replace it with a reversible game, don't we run into the problem that the final state is just as good a factorization as the initial?
Not sure we disagree, maybe I'm just confused. In the post you show that if X is orthogonal to X XOR Y, then X is before Y, so you can "infer a temporal relationship" that Pearl can't. I'm trying to understand the meaning of the thing you're inferring - "X is before Y". In my example above, Bob tells Alice a lossy function of his knowledge, and Alice ends up with knowledge that is "before" Bob's. So in this case the "before" relationship doesn't agree with time, causality, or what can be computed from what. But then what conclusions can a scientist make from an inferred "before" relationship?
I feel that interpreting "strictly before" as causality is making me more confused.
For example, here's a scenario with a randomly changed message. Bob peeks at ten regular envelopes and a special envelope that gives him a random boolean. Then Bob tells Alice the contents of either the first three envelopes or the second three, depending on the boolean. Now Alice's knowledge depends on six out of ten regular envelopes and the special one, so it's still "strictly before" Bob's knowledge. And since Alice's knowledge can be computed from Bob's knowledge but not vice versa, in FFS terms that means the "cause" can be (and in fact is) computed from the "effect", but not vice versa. My causal intuition is just blinking at all this.
Here's another scenario. Alice gets three regular envelopes and accurately reports their contents to Bob, and a special envelope that she keeps to herself. Then Bob peeks at seven more envelopes. Now Alice's knowledge isn't "before" Bob's, but if later Alice predictably forgets the contents of her special envelope, her knowledge becomes "before" Bob's. Even though the special envelope had no effect on the information Alice gave to Bob, didn't affect the causal arrow in any possible world. And if we insist that FFS=causality, then by forgetting the envelope, Alice travels back in time to become the cause of Bob's knowledge in the past. That's pretty exotic.
I think the definition of history is the most natural way to recover something like causal structure in these models.
I'm not sure how much it's about causality. Imagine there's a bunch of envelopes with numbers inside, and one of the following happens:
Alice peeks at three envelopes. Bob peeks at ten, which include Alice's three.
Alice peeks at three envelopes and tells the results to Bob, who then peeks at seven more.
Bob peeks at ten envelopes, then tells Alice the contents of three of them.
Under the FFS definition, Alice's knowledge in each case is "strictly before" Bob's. So it seems less of a causal relationship and more like "depends on fewer basic facts".
Instant strong upvote. This post changed my view as much as the risk aversion post (which was also by you!)