I am a PhD student in computer science at the University of Waterloo, supervised by Professor Ming Li and advised by Professor Marcus Hutter.
My current research is related to applications of algorithmic probability to sequential decision theory (universal artificial intelligence). Recently I have been trying to start a dialogue between the computational cognitive science and UAI communities (if this includes you, consider contacting me about the reading group). Sometimes I build robots, professionally or otherwise. Another hobby (and a personal favorite of my posts here) is the Sherlockian abduction master list, which is a crowdsourced project seeking to make “Sherlock Holmes” style inference feasible by compiling observational cues. Give it a read and see if you can contribute!
See my personal website colewyeth.com for an overview of my interests and work.
Technically the connection between the computability levels of AIT (estimability, lower/upper semi-computability, approximability) and the Turing degrees has not been worked out properly. See chapter 6 of Leike’s thesis, though there is a small error in the inequalities of section 6.1.2. It is necessary to connect the computability of real valued functions (type two theory of effectivity) to the arithmetic hierarchy—as far as I know this hasn’t been done, but maybe I’ll share some notes in a few months.
Roughly, most classes don’t have a universal distribution because they are not computably enumerable, but perhaps there are various reasons. There’s a nice table in Marcus Hutter’s original book, page 50.
It says that (negative log) universal probability is about the same as the (monotone) Kolmogorov complexity—in the discrete case up to a constant multiple. Basically, the Bayesian prediction is closely connected to the shortest explanation. See Li and Vitanyi’s “An Introduction to Kolmogorov Complexity and its Applications.”
Last question is a longer story I guess. Basically, the conditionals of the universal distribution are not lower semi-computable, and it gets even worse when you have to compare the expected values of different outcomes because of tie-breaking. But a good approximation of AIXI can still be computed in the limit.