Variants of Solomonoff induction: Difference between revisions

Revision as of 03:30, 31 March 2019

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".

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^[1]	$m(y_{0})=\sum _{p\in {\mathcal {P}}:U(p)=y_{0}}2^{-\ell (p)}$ 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^[2]	$m(x)=\sum _{p:U(p)=x}2^{-\ell (p)}$ where the sum is over halting programs	deterministic?	prefix Turing machine	discrete
Scholarpedia continuous universal a priori probability^[2]	$M(x)=\sum _{p:U(p)=x*}2^{-\ell (p)}$ where the sum is over minimal programs	deterministic?	Monotone Turing machine	Continuous
Sterkenburg (p. 22)^[3]	$P_{\mathrm {I} }(\sigma )=\lim _{n\to \infty }{\frac {\|T_{\sigma ,n}\|}{\|T_{n}\|}}$ 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)^[3]	$P'_{\mathrm {II} }(\sigma )=2^{-\|\tau _{\mathrm {min} }\|}$ 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)^[3]	$P''_{\mathrm {II} }(\sigma )=\lim _{n\to \infty }\sum _{\tau \in T_{\sigma ,n}}2^{-\|\tau \|}$ 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)^[3]	$P_{\mathrm {II} }(\sigma )=\lim _{\epsilon \to 0}\lim _{n\to \infty }\sum _{\tau \in T_{\sigma ,n}}\left({\frac {1-\epsilon }{2}}\right)^{\|\tau \|}$	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

References

↑ https://wiki.lesswrong.com/wiki/Solomonoff_induction
↑ ^2.0 ^2.1 Marcus Hutter; Shane Legg; Paul M.B. Vitanyi. "Algorithmic probability". Scholarpedia. 2007.
↑ ^3.0 ^3.1 ^3.2 ^3.3 Tom Florian Sterkenburg. "The Foundations of Solomonoff Prediction". February 2013.

[1] ttps://wiki.lesswrong.com/wiki/Solomonoff_induction

[scholarpedia-2] 2.0 ^2.1 Marcus Hutter; Shane Legg; Paul M.B. Vitanyi. "Algorithmic probability". Scholarpedia. 2007.

[sterkenburg-3] 3.0 ^3.1 ^3.2 ^3.3 Tom Florian Sterkenburg. "The Foundations of Solomonoff Prediction". February 2013.

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@@ Line 17: / Line 17: @@
 | Sterkenburg (p. 22)<ref name="sterkenburg">Tom Florian Sterkenburg. "The Foundations of Solomonoff Prediction". February 2013.</ref> || <math>P_{\mathrm{I}}(\sigma) = \lim_{n\to\infty} \frac{|T_{\sigma,n}|}{|T_n|}</math> where <math>\sigma</math> is a finite string, <math>T_n</math> is the set of all halting (valid) inputs of length <math>n</math> to the reference machine <math>U</math>, <math>T_{\sigma,n}</math> is the set of all halting (valid) inputs of length <math>n</math> that output something starting with <math>\sigma</math> || deterministic || universal Turing machine (no restrictions on prefix-free-ness) || discrete?
 |-
-| Sterkenburg (p. 24)<ref name="sterkenburg"/> || <math>P'_{\mathrm{II}}(\sigma) = 2^{-|\tau_\mathrm{min}|}</math> where <math>\tau_\mathrm{min}</math> is the shortest program <math>\tau</math> such that <math>U(\tau) = \sigma</math> (i.e. the shortest program that causes the reference machine to output <math>\sigma</math> and halt) || deterministic || universal Turing machine, universal prefix machine (to get a probability distribution) || discrete? || this formula does not define a probability distribution over strings <math>\sigma</math> because the sum of probabilities does not converge
+| Sterkenburg (p. 24)<ref name="sterkenburg"/> || <math>P'_{\mathrm{II}}(\sigma) = 2^{-|\tau_\mathrm{min}|}</math> where <math>\tau_\mathrm{min}</math> is the shortest program <math>\tau</math> such that <math>U(\tau) = \sigma</math> (i.e. the shortest program that causes the reference machine to output <math>\sigma</math> 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 <math>\sigma</math> because the sum of probabilities does not converge
 |-
 | Sterkenburg (p. 25)<ref name="sterkenburg"/> || <math>P''_{\mathrm{II}}(\sigma) = \lim_{n\to\infty} \sum_{\tau \in T_{\sigma,n}} 2^{-|\tau|}</math> where <math>T_{\sigma,n}</math> is the set of all programs <math>\tau</math> of length <math>n</math> such that <math>U(\tau)</math> begins with <math>\sigma</math> || deterministic || universal Turing machine || || <math>P''_{\mathrm{II}}(\sigma)</math> is divergent even for a single <math>\sigma</math>, so this is not actually a workable version, but is intended as a stepping stone
 |-
-| Sterkenburg (p. 26)<ref name="sterkenburg"/> || <math>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|}</math> || deterministic || universal Turing machine, universal prefix machine (to get a probability distribution) || || The use of the <math>\epsilon</math> is a hack to get the sum to converge
+| Sterkenburg (p. 26)<ref name="sterkenburg"/> || <math>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|}</math> || deterministic || universal Turing machine, universal prefix machine (to get a probability distribution; see remark on p. 27) || || The use of the <math>\epsilon</math> is a hack to get the sum to converge
 |}