Low Complexity, Low Probability Patterns and Consequences for Algorithmic Probability Applications

Alaskandarani, Mohamed; Dingle, Kamaludin

doi:10.20944/preprints202207.0323.v1

Cited by 3 publications

(7 citation statements)

References 30 publications

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“…On the other hand, the ubiquity of low-complexity, low-probability outputs [31,62] suggests that for many y we may have Pfalse(xfalse→yfalse)≪2−aKfalse~false(yfalsefalse|xfalse)−b. Such phenotypes y are those that have low conditional complexity, yet at the same time appear with low probability due to map-specific constraints and biases.…”

Section: Simplicity Bias In Transitionsmentioning

confidence: 99%

“…The fact that the actual slope is flatter than the predicted one is presumably due to the following: the value of log10⁡false(2false)false(Kfalse~false(yjfalsefalse|xfalse)−Kfalse~false(yifalsefalse|xfalse)false) will be large when y j is (conditionally) complex and y i is simple. Therefore, y j is unlikely to be far from the upper bound, while very simple phenotypes can be very far from the upper bound, compared with the low-complexity, low-probability phenomenon [31,62]. Hence, the value of P ( x → y i )/ P ( x → y j ) is likely to be an underestimate, rather than an overestimate, which will tend to make the slope flatter.…”

Section: Empirical Phenotype Transition Probabilitiesmentioning

confidence: 99%

“…On the other hand, the ubiquity of low-complexity, lowprobability outputs [31,62] suggests that for many y we may have Pðx ! yÞ ( 2 Àa KðyjxÞÀb .…”

Section: The Bound Is Close With High Probabilitymentioning

confidence: 99%

“…Such phenotypes y are those that have low conditional complexity, yet at the same time appear with low probability due to map-specific constraints and biases. See [ 62 ] for an in-depth discussion of this low-complexity, low-probability phenomenon.…”

Section: Simplicity Bias In Transitionsmentioning

confidence: 99%

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Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Dingle

Novev

Ahnert

et al. 2022

J. R. Soc. Interface.

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Unravelling the structure of genotype–phenotype (GP) maps is an important problem in biology. Recently, arguments inspired by algorithmic information theory (AIT) and Kolmogorov complexity have been invoked to uncover simplicity bias in GP maps, an exponentially decaying upper bound in phenotype probability with the increasing phenotype descriptional complexity. This means that phenotypes with many genotypes assigned via the GP map must be simple, while complex phenotypes must have few genotypes assigned. Here, we use similar arguments to bound the probability P ( x → y ) that phenotype x , upon random genetic mutation, transitions to phenotype y . The bound is P ( x → y ) ≲ 2 − a K ~ ( y | x ) − b , where K ~ ( y | x ) is the estimated conditional complexity of y given x , quantifying how much extra information is required to make y given access to x . This upper bound is related to the conditional form of algorithmic probability from AIT. We demonstrate the practical applicability of our derived bound by predicting phenotype transition probabilities (and other related quantities) in simulations of RNA and protein secondary structures. Our work contributes to a general mathematical understanding of GP maps and may facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilizing detailed knowledge of the GP map.

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Section: Simplicity Bias In Transitionsmentioning

confidence: 99%

Section: Empirical Phenotype Transition Probabilitiesmentioning

confidence: 99%

“…On the other hand, the ubiquity of low-complexity, lowprobability outputs [31,62] suggests that for many y we may have Pðx ! yÞ ( 2 Àa KðyjxÞÀb .…”

Section: The Bound Is Close With High Probabilitymentioning

confidence: 99%

Section: Simplicity Bias In Transitionsmentioning

confidence: 99%

See 2 more Smart Citations

Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Dingle

Novev

Ahnert

et al. 2022

J. R. Soc. Interface.

Self Cite

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show abstract

“…Some simple geometries may have low or medium values with respect to the objective function. This is related to the occurrence of low complexity, low probability patterns in studies of simplicity bias [24,30].…”

Section: Resultsmentioning

confidence: 99%

Optima and Simplicity in Nature

Dingle¹

2022

Preprint

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Why are simple, regular, and symmetric shapes common in nature? Many natural shapes arise as solutions to energy minimisation or other optimisation problems, but is there a general relation between optima and simple, regular shapes and geometries? Here we argue from algorithmic information theory that for objective functions common in nature --- based on physics and engineering laws --- optimal geometries will be simple, regular, and symmetric. Further, we derive a null model prediction that if a given geometry is an optimal solution for one natural objective function, then it is a priori more likely to be optimal or close to optimal for another objective function.

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Low Complexity, Low Probability Patterns and Consequences for Algorithmic Probability Applications

Alaskandarani,

Dingle

2023

Complexity

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Developing new ways to estimate probabilities can be valuable for science, statistics, engineering, and other fields. By considering the information content of different output patterns, recent work invoking algorithmic information theory inspired arguments has shown that a priori probability predictions based on pattern complexities can be made in a broad class of input-output maps. These algorithmic probability predictions do not depend on a detailed knowledge of how output patterns were produced, or historical statistical data. Although quantitatively fairly accurate, a main weakness of these predictions is that they are given as an upper bound on the probability of a pattern, but many low complexity, low probability patterns occur, for which the upper bound has little predictive value. Here, we study this low complexity, low probability phenomenon by looking at example maps, namely a finite state transducer, natural time series data, RNA molecule structures, and polynomial curves. Some mechanisms causing low complexity, low probability behaviour are identified, and we argue this behaviour should be assumed as a default in the real-world algorithmic probability studies. Additionally, we examine some applications of algorithmic probability and discuss some implications of low complexity, low probability patterns for several research areas including simplicity in physics and biology, a priori probability predictions, Solomonoff induction and Occam’s razor, machine learning, and password guessing.

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Low Complexity, Low Probability Patterns and Consequences for Algorithmic Probability Applications

Cited by 3 publications

References 30 publications

Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Optima and Simplicity in Nature

Low Complexity, Low Probability Patterns and Consequences for Algorithmic Probability Applications

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