Variants of Solomonoff induction

This page lists some variants of Solomonoff induction.

For determinism, I think "deterministic" is the same as "Solomonoff prior" and "stochastic" is the same as "universal mixture". Sterkenburg calls deterministic versions a "bottom-up approach" whereas the universal mixture is a "top-down approach" (p. 30). For deterministic variants, the type of universal machine must be specified. With universal mixtures, one must specify two things: the weighting to use, and the class of distributions to consider.

For discrete vs continuous, I think this just means whether the prior we define is over finite strings or over infinite sequences (where we want to know the probability of an infinite sequence starting with a given finite string).

Source	Formula	Determinism	Type of machine used	Discrete vs continuous	Notes
LessWrong Wiki	$m(y_{0})=\sum _{p\in {\mathcal {P}}:U(p)=y_{0}}2^{-\ell (p)}$ where ${\mathcal {P}}$ is the set of self-delimiting programs	Deterministic	Page doesn't say, but uses self-delimiting programs and it's discrete, so prefix Turing machine?	Discrete because of the $U(p)=y_{0}$ rather than $U(p)=y_{0}*$
Scholarpedia discrete universal a priori probability	$m(x)=\sum _{p:U(p)=x}2^{-\ell (p)}$ where the sum is over halting programs	deterministic?	prefix Turing machine	discrete
Scholarpedia continuous universal a priori probability	$M(x)=\sum _{p:U(p)=x*}2^{-\ell (p)}$ where the sum is over minimal programs	deterministic?	Monotone Turing machine	Continuous
Sterkenburg (p. 22)	$P_{\mathrm {I} }(\sigma )=\lim _{n\to \infty }{\frac {|T_{\sigma ,n}|}{|T_{n}|}}$ where $\sigma$ is a finite string, $T_{n}$ is the set of all halting (valid) inputs of length $n$ to the reference machine $U$, $T_{\sigma ,n}$ is the set of all halting (valid) inputs of length $n$ that output something starting with $\sigma$	deterministic	universal Turing machine (no restrictions on prefix-free-ness)	discrete?
Sterkenburg (p. 24)	$P'_{\mathrm {II} }(\sigma )=2^{-|\tau _{\mathrm {min} }|}$ where $\tau _{\mathrm {min} }$ is the shortest program $\tau$ such that $U(\tau )=\sigma$ (i.e. the shortest program that causes the reference machine to output $\sigma$ and halt)	deterministic	universal Turing machine, universal prefix machine (to get a probability distribution; see remark on p. 27)	discrete?	this formula does not define a probability distribution over strings $\sigma$ because the sum of probabilities does not converge
Sterkenburg (p. 25)	$P''_{\mathrm {II} }(\sigma )=\lim _{n\to \infty }\sum _{\tau \in T_{\sigma ,n}}2^{-|\tau |}$ where $T_{\sigma ,n}$ is the set of all programs $\tau$ of length $n$ such that $U(\tau )$ begins with $\sigma$	deterministic	universal Turing machine		$P''_{\mathrm {II} }(\sigma )$ is divergent even for a single $\sigma$, so this is not actually a workable version, but is intended as a stepping stone
Sterkenburg (p. 26)	$P_{\mathrm {II} }(\sigma )=\lim _{\epsilon \to 0}\lim _{n\to \infty }\sum _{\tau \in T_{\sigma ,n}}\left({\frac {1-\epsilon }{2}}\right)^{|\tau |}$	deterministic	universal Turing machine, universal prefix machine (to get a probability distribution; see remark on p. 27)		The use of the $\epsilon$ is a hack to get the sum to converge
Sterkenburg (p. 29)	$Q_{U}(\sigma )=\sum _{\tau \in T_{\sigma }}2^{-|\tau |}$ where $T_{\sigma }$ is the set of minimal descriptions of $\sigma$ (i.e. set of programs that output something starting with $\sigma$ such that if one removes one bit from the end of the program, it no longer outputs something starting with $\sigma$)	deterministic	universal monotone machine	continuous?
Sterkenburg (p. 31)	$P_{\mathrm {IV} }(\sigma )=\lim _{n\to \infty }\sum _{i}f_{i,n}P_{i}(\sigma )$	Stochastic
Sterkenburg (p. 33)	$\xi _{w}(\sigma )=\sum _{i}w(\mu _{i})\mu _{i}(\sigma )$	Stochastic