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Universal Convergence of Semimeasures on Individual Random Sequences
[chapter]

2004
*
Lecture Notes in Computer Science
*

Solomonoff's central result

doi:10.1007/978-3-540-30215-5_19
fatcat:gmlx75srpff6rdtqtkowth2up4
*on*induction is that the posterior*of*a*universal**semimeasure*M*converges*rapidly and with probability 1 to the true*sequence*generating posterior µ, if the latter is computable ... We show that W*converges*to D and D to µ*on*all*random**sequences*. The Hellinger distance measuring closeness*of*two distributions plays a central role. ... -*random**sequence*and show the existence*of*a*universal**semimeasure*which does not*converge**on*this*sequence*, hence answering the open question negatively for some M . ...##
###
Universal Convergence of Semimeasures on Individual Random Sequences
[article]

2004
*
arXiv
*
pre-print

Solomonoff's central result

arXiv:cs/0407057v1
fatcat:khwjqxwkczh2nafiul2zjpn64q
*on*induction is that the posterior*of*a*universal**semimeasure*M*converges*rapidly and with probability 1 to the true*sequence*generating posterior mu, if the latter is computable ... We show that W*converges*to D and D to mu*on*all*random**sequences*. The Hellinger distance measuring closeness*of*two distributions plays a central role. ... -*random**sequence*and show the existence*of*a*universal**semimeasure*which does not*converge**on*this*sequence*, hence answering the open question negatively for some M. ...##
###
On semimeasures predicting Martin-Löf random sequences

2007
*
Theoretical Computer Science
*

We show that there are

doi:10.1016/j.tcs.2007.03.040
fatcat:smm3ujv6k5cxlbhjlqm77tuqma
*universal**semimeasures*M which do not*converge*to µ*on*all µ-*random**sequences*, i.e. we give a partial negative answer to the open problem. ... Solomonoff's central result*on*induction is that the prediction*of*a*universal**semimeasure*M*converges*rapidly and with probability 1 to the true*sequence*generating predictor µ, if the latter is computable ... A shorter version appeared in the proceedings*of*the ALT 2004 conference [1] . ...##
###
On Semimeasures Predicting Martin-Loef Random Sequences
[article]

2007
*
arXiv
*
pre-print

Solomonoff's central result

arXiv:0708.2319v1
fatcat:6ew7f75zejgftnwvjwwa74c3oa
*on*induction is that the posterior*of*a*universal**semimeasure*M*converges*rapidly and with probability 1 to the true*sequence*generating posterior mu, if the latter is computable ... We show that W*converges*to D and D to mu*on*all*random**sequences*. The Hellinger distance measuring closeness*of*two distributions plays a central role. ... In Section 4 we investigate whether*convergence*for all Martin-Löf*random**sequences*hold. We construct a µ-M.L.-*random**sequence**on*which some*universal**semimeasures*M do not*converge*to µ. ...##
###
On the Existence and Convergence Computable Universal Priors
[article]

2003
*
arXiv
*
pre-print

We introduce a generalized concept

arXiv:cs/0305052v1
fatcat:gvyokczehzakpfnb5xgp5dpu7q
*of**randomness*for*individual**sequences*and use it to exhibit difficulties regarding these issues. ... His central result is that the posterior*of*his*universal**semimeasure*M*converges*rapidly to the true*sequence*generating posterior mu, if the latter is computable. ... (iii) uses Martin-Löf's notion*of**randomness**of**individual**sequences*to define*convergence*M.L. ...##
###
On the Existence and Convergence of Computable Universal Priors
[chapter]

2003
*
Lecture Notes in Computer Science
*

We introduce a generalized concept

doi:10.1007/978-3-540-39624-6_24
fatcat:3dle7qlw6jayvairul3ry36pty
*of**randomness*for*individual**sequences*and use it to exhibit difficulties regarding these issues. ... His central result is that the posterior*of*his*universal**semimeasure*M*converges*rapidly to the true*sequence*generating posterior µ, if the latter is computable. ... (iii) uses Martin-Löf's notion*of**randomness**of**individual**sequences*to define*convergence*M.L. ...##
###
On Martin-Löf Convergence of Solomonoff's Mixture
[chapter]

2013
*
Lecture Notes in Computer Science
*

We study the

doi:10.1007/978-3-642-38236-9_20
fatcat:xsljijw47zhnrhymaoeb4k4y7m
*convergence**of*Solomonoff's*universal*mixture*on**individual*Martin-Löf*random**sequences*. ... A new result is presented extending the work*of*Hutter and Muchnik (2004) by showing that there does not exist a*universal*mixture that*converges**on*all Martin-Löf*random**sequences*. ... Martin-Löf*randomness*is the usual characterisation*of*the*randomness**of**individual**sequences*[6] . ...##
###
On Generalized Computable Universal Priors and their Convergence
[article]

2005
*
arXiv
*
pre-print

In particular, we show that

arXiv:cs/0503026v1
fatcat:5zya6ovf5jf6vept2po7kpkjtq
*convergence*fails (holds)*on*generalized-*random**sequences*in gappy (dense) Bernoulli classes. ... We introduce a generalized concept*of**randomness*for*individual**sequences*and use it to exhibit difficulties regarding these issues. ... (iii) uses Martin-Löf's notion*of**randomness**of**individual**sequences*to define*convergence*M.L. ...##
###
On generalized computable universal priors and their convergence

2006
*
Theoretical Computer Science
*

In particular, we show that

doi:10.1016/j.tcs.2006.07.039
fatcat:zidqw3nhhrhujpscobtrwd6wiu
*convergence*fails (holds)*on*generalized-*random**sequences*in gappy (dense) Bernoulli classes. ... We introduce a generalized concept*of**randomness*for*individual**sequences*and use it to exhibit difficulties regarding these issues. ... s notion*of**randomness**of**individual**sequences*to define*convergence*M.L. ...##
###
Ray Solomonoff, Founding Father of Algorithmic Information Theory

2010
*
Algorithms
*

Suppose we have an infinite binary

doi:10.3390/a3030260
fatcat:ypsejgoql5bgni4qksom72alxi
*sequence*every odd bit*of*which is uniformly*random*and every even bit is a bit*of*pi = 3.1415... written in binary. ... For prediction*one*uses not the*universal*a priori probability which is a probability mass function, but a*semimeasure*which is a weak form*of*a measure. ...##
###
Sequential predictions based on algorithmic complexity

2006
*
Journal of computer and system sciences (Print)
*

This paper studies

doi:10.1016/j.jcss.2005.07.001
fatcat:tpvox6g6jbakjlfnu6ti2aw2vy
*sequence*prediction based*on*the monotone Kolmogorov complexity Km = − log m, i.e. based*on**universal*deterministic/*one*-part MDL. m is extremely close to Solomonoff's*universal*prior ... We show that for deterministic computable environments, the "posterior" and losses*of*m*converge*, but rapid*convergence*could only be shown*on*-*sequence*; the off-*sequence**convergence*can be slow. ... M can be used to characterize*randomness**of**individual**sequences*: a*sequence*x 1:∞ is (Martin-Löf) -*random*, iff ∃c : M(x 1:n ) c · (x 1:n ) ∀n. ...##
###
Sequential Predictions based on Algorithmic Complexity
[article]

2005
*
arXiv
*
pre-print

This paper studies

arXiv:cs/0508043v1
fatcat:psltsxgxeba5pk6bb7af3l27fm
*sequence*prediction based*on*the monotone Kolmogorov complexity Km=-log m, i.e. based*on**universal*deterministic/*one*-part MDL. m is extremely close to Solomonoff's*universal*prior M, ... We show that for deterministic computable environments, the "posterior" and losses*of*m*converge*, but rapid*convergence*could only be shown*on*-*sequence*; the off-*sequence**convergence*can be slow. ... M can be used to characterize*randomness**of**individual**sequences*: A*sequence*x 1:∞ is (Martin-Löf) µ-*random*, iff ∃c : M(x 1:n ) ≤ c·µ(x 1:n )∀n. ...##
###
Ray Solomonoff, Founding Father of Algorithmic Information Theory

2010
*
Algorithms
*

Suppose we have an infinite binary

doi:10.3390/algor3030260
fatcat:7c3nfi3myjeqzkvygfvys36uxu
*sequence*every odd bit*of*which is uniformly*random*and every even bit is a bit*of*pi = 3.1415... written in binary. ... For prediction*one*uses not the*universal*a priori probability which is a probability mass function, but a*semimeasure*which is a weak form*of*a measure. ...##
###
Sequence Prediction based on Monotone Complexity
[article]

2003
*
arXiv
*
pre-print

We show that for deterministic computable environments, the "posterior" and losses

arXiv:cs/0306036v1
fatcat:qhcwtzup4vhs5lpuisbfph6gx4
*of*m*converge*, but rapid*convergence*could only be shown*on*-*sequence*; the off-*sequence*behavior is unclear. ... This paper studies*sequence*prediction based*on*the monotone Kolmogorov complexity Km=-log m, i.e. based*on**universal*deterministic/*one*-part MDL. m is extremely close to Solomonoff's prior M, the latter ... M can be used to characterize*randomness**of**individual**sequences*: A*sequence*x 1:∞ is (Martin-Löf) µ-*random*, iff ∃c : M(x 1:n ) ≤ c·µ(x 1:n )∀n. ...##
###
On prediction by data compression
[chapter]

1997
*
Lecture Notes in Computer Science
*

Making these ideas rigorous involves the length

doi:10.1007/3-540-62858-4_69
fatcat:wikhwtgwlnd7hjpnpy675akfmi
*of*the shortest effective description*of*an*individual*object: its Kolmogorov complexity. ... In a previous paper we have shown that optimal compression is almost always a best strategy in hypotheses identification (an ideal form*of*the minimum description length (MDL) principle). ... In fact, classical probability theory cannot express the notion*of**randomness**of*an*individual**sequence*. ...
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