Context: At this weekend’s MIRI workshop on logical uncertainty, we were talking about Markus Mueller’s paper on priors over Turing machines and bitstrings (as those seem analogous to priors over logical statements).
Mueller’s original result examined a directed graph structure over prefix computers, where an edge is given by a prefix that causes one computer to simulate another. This gave rise to a Markov chain structure, but Mueller showed in two different ways that this Markov chain was not positive recurrent, and thus that it would not give rise to a non-arbitrary prior.
However, I noted that the Markov chain structure was set up to punish computers that were difficult to simulate, but not to punish computers that were bad at simulating others. Some natural ways to modify the transition probabilities defeated Mueller’s more direct counterexample.
However, in this draft Jacob and Janos showed that no stationary transition matrix (with nonzero coefficients) on that digraph of universal prefix computers could be positive recurrent, or even null recurrent (going beyond Mueller’s result).
Context: At this weekend’s MIRI workshop on logical uncertainty, we were talking about Markus Mueller’s paper on priors over Turing machines and bitstrings (as those seem analogous to priors over logical statements).
Mueller’s original result examined a directed graph structure over prefix computers, where an edge is given by a prefix that causes one computer to simulate another. This gave rise to a Markov chain structure, but Mueller showed in two different ways that this Markov chain was not positive recurrent, and thus that it would not give rise to a non-arbitrary prior.
However, I noted that the Markov chain structure was set up to punish computers that were difficult to simulate, but not to punish computers that were bad at simulating others. Some natural ways to modify the transition probabilities defeated Mueller’s more direct counterexample.
However, in this draft Jacob and Janos showed that no stationary transition matrix (with nonzero coefficients) on that digraph of universal prefix computers could be positive recurrent, or even null recurrent (going beyond Mueller’s result).