A few quick observations (each with like 90% confidence; I won’t provide detailed arguments atm, but feel free to LW-msg me for more details):
Any finite number of iterates just gives you the solomonoff distribution up to at most a const multiplicative difference (with the const depending on how many iterates you do). My other points will be about the limit as we iterate many times.
The quines will have mass at least their prior, upweighted by some const because of programs which do not produce an infinite output string. They will generally have more mass than that, and some will gain mass by a larger multiplicative factor than others, but idk how to say something nice about this further.
As you do more and more iterates, there’s not convergence to a stationary distribution, at least in total variation distance. One reason is that you can write a quine which adds a string to itself (and then adds the same string again next time, and so on)[1], creating “a way for a finite chunk of probability to escape to infinity”. So yes, some mass diverges.
Quine-cycles imply (or at least very strongly suggest) probabilities also do not converge pointwise.
What about pointwise convergence when we also average over the number of iterates? It seems plausible you get convergence then, but not sure (and not sure if this would be an interesting claim). It would be true if we could somehow think of the problem as living on a directed graph with countably many vertices, but idk how to do that atm.
There are many different stationary distributions — e.g. you could choose any distribution on the quines.
A few quick observations (each with like 90% confidence; I won’t provide detailed arguments atm, but feel free to LW-msg me for more details):
Any finite number of iterates just gives you the solomonoff distribution up to at most a const multiplicative difference (with the const depending on how many iterates you do). My other points will be about the limit as we iterate many times.
The quines will have mass at least their prior, upweighted by some const because of programs which do not produce an infinite output string. They will generally have more mass than that, and some will gain mass by a larger multiplicative factor than others, but idk how to say something nice about this further.
Yes, you can have quine-cycles. Relevant tho not exactly this: https://github.com/mame/quine-relay
As you do more and more iterates, there’s not convergence to a stationary distribution, at least in total variation distance. One reason is that you can write a quine which adds a string to itself (and then adds the same string again next time, and so on)[1], creating “a way for a finite chunk of probability to escape to infinity”. So yes, some mass diverges.
Quine-cycles imply (or at least very strongly suggest) probabilities also do not converge pointwise.
What about pointwise convergence when we also average over the number of iterates? It seems plausible you get convergence then, but not sure (and not sure if this would be an interesting claim). It would be true if we could somehow think of the problem as living on a directed graph with countably many vertices, but idk how to do that atm.
There are many different stationary distributions — e.g. you could choose any distribution on the quines.
a construction from o3-mini-high: https://colab.research.google.com/drive/1kIGCiDzWT3guCskgmjX5oNoYxsImQre-?usp=sharing