Is cosmological tuning fine or coarse?

Díaz–Pachón, Daniel Andrés; Hössjer, Ola; Marks, Robert J.

doi:10.1088/1475-7516/2021/07/020

Cited by 7 publications

(11 citation statements)

References 36 publications

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“…The active information

is then a measure of how much background knowledge this agent has infused. Following [ 26 , 27 ], we conclude that X is finely tuned when the lower bound

is large enough. That is, FT corresponds to infusing a high degree of background knowledge into a problem.…”

Section: Introductionmentioning

confidence: 78%

Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information

Díaz–Pachón

Hössjer

2022

Entropy

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A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function f quantifies specificity for each possible outcome x of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target as intended than by chance. The distribution of a random outcome X of the algorithm involves a parameter θ that quantifies how much background information has been infused. A simple choice of this parameter is to use θf in order to exponentially tilt the distribution of the outcome of the search algorithm under the null distribution of no tuning, so that an exponential family of distributions is obtained. Such algorithms are obtained by iterating a Metropolis–Hastings type of Markov chain, which makes it possible to compute their active information under the equilibrium and non-equilibrium of the Markov chain, with or without stopping when the targeted set of fine-tuned states has been reached. Other choices of tuning parameters θ are discussed as well. Nonparametric and parametric estimators of active information and tests of fine-tuning are developed when repeated and independent outcomes of the algorithm are available. The theory is illustrated with examples from cosmology, student learning, reinforcement learning, a Moran type model of population genetics, and evolutionary programming.

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“…The active information

is then a measure of how much background knowledge this agent has infused. Following [ 26 , 27 ], we conclude that X is finely tuned when the lower bound

is large enough. That is, FT corresponds to infusing a high degree of background knowledge into a problem.…”

Section: Introductionmentioning

confidence: 78%

Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information

Díaz–Pachón

Hössjer

2022

Entropy

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“…A fourth theoretical extension is to consider the case

. In this case, instead of the (discrete or continuous) uniform distribution given by ( 2 ), it will be necessary to consider more general maximum entropy distributions

, subject to some restrictions, in order to measure learning and knowledge acquisition [ 20 , 64 , 65 , 66 ].…”

Section: Discussionmentioning

confidence: 99%

A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning

Hössjer

Díaz–Pachón

Rao

2022

Entropy

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Philosophers frequently define knowledge as justified, true belief. We built a mathematical framework that makes it possible to define learning (increasing number of true beliefs) and knowledge of an agent in precise ways, by phrasing belief in terms of epistemic probabilities, defined from Bayes’ rule. The degree of true belief is quantified by means of active information I+: a comparison between the degree of belief of the agent and a completely ignorant person. Learning has occurred when either the agent’s strength of belief in a true proposition has increased in comparison with the ignorant person (I+>0), or the strength of belief in a false proposition has decreased (I+<0). Knowledge additionally requires that learning occurs for the right reason, and in this context we introduce a framework of parallel worlds that correspond to parameters of a statistical model. This makes it possible to interpret learning as a hypothesis test for such a model, whereas knowledge acquisition additionally requires estimation of a true world parameter. Our framework of learning and knowledge acquisition is a hybrid between frequentism and Bayesianism. It can be generalized to a sequential setting, where information and data are updated over time. The theory is illustrated using examples of coin tossing, historical and future events, replication of studies, and causal inference. It can also be used to pinpoint shortcomings of machine learning, where typically learning rather than knowledge acquisition is in focus.

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“…However, this interval spans over more than one order of magnitude and for this reason the theory from Appendix C for small ǫ does not apply (see Remark 1 on Appendix C.6). In fact, under the assumption that the amplitude of the primordial fluctuations has an exponential distribution (corresponding to the scale family of distributions whose sample space is R + ), TP max ≈ 0.7 [28]. Therefore, surprisingly, fine-tuning is not detected in this case.…”

Section: Amplitude Of Primordial Fluctuationsmentioning

confidence: 92%

“…With these considerations, the following procedure to measure the tuning probabilities was proposed in [28]:…”

Section: Back To Fine-tuningmentioning

confidence: 99%

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Sometimes size does not matter

Díaz–Pachón¹,

Hössjer²,

Marks³

2022

Preprint

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Cosmological fine-tuning has traditionally been associated with the narrowness of the intervals in which the parameters of the physical models must be located to make life possible. A more thorough approach focuses on the probability of the interval, not on its size. Most attempts to measure the probability of the life-permitting interval for a given parameter rely on a Bayesian statistical approach for which the prior distribution of the parameter is uniform. However, the parameters in these models often take values in spaces of infinite size, so that a uniformity assumption is not possible. This is known as the normalization problem. This paper explains a framework to measure tuning that, among others, deals with normalization, assuming that the prior distribution belongs to a class of maximum entropy (maxent) distributions. By analyzing an upper bound of the tuning probability for this class of distributions the method solves the so-called weak anthropic principle, and offer a solution, at least in this context, to the well-known lack of invariance of maxent distributions. The implication of this approach is that, since all mathematical models need parameters, tuning is not only a question of natural science, but also a problem of mathematical modeling. Cosmological tuning is thus a particular instantiation of a more general scenario. Therefore, whenever a mathematical model is used to describe nature, not only in physics but Size does not matter in all of science, tuning is present. And the question of whether the tuning is fine or coarse for a given parameter -if the interval in which the parameter is located has low or high probability, respectively -depends crucially not only on the interval but also on the assumed class of prior distributions. Novel upper bounds for tuning probabilities are presented.

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Is cosmological tuning fine or coarse?

Cited by 7 publications

References 36 publications

Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information

Assessing, Testing and Estimating the Amount of Fine-Tuning by Means of Active Information

A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning

Sometimes size does not matter

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