A Formal Framework for Knowledge Acquisition: Going Beyond Machine Learning

Hössjer, Ola; Díaz–Pachón, Daniel Andrés; Rao, J. Sunil

doi:10.31234/osf.io/qt5kw

Cited by 4 publications

(8 citation statements)

References 22 publications

(25 reference statements)

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“…Remark 1 Each proof in this section is based on the supposition ǫ ≪ 1. This can be seen in ( 22), ( 30), ( 33), (35), and (42), since in all these equations the assumption was that ǫ was small enough to warrant that the prior density of X was constant over the life-permitting interval I X .…”

Section: C6 Summary Of Resultsmentioning

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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Section: C6 Summary Of Resultsmentioning

confidence: 99%

“…Despite the name, these types of constraints serve a good purpose. They mean knowledge -and since knowledge constrains randomness, knowledge also reduces uncertainty [42]. Step 2 of the procedure for finding tuning probabilities determines which probability distributions to use.…”

Section: Determining the Constraintsmentioning

confidence: 99%

Sometimes size does not matter

Díaz–Pachón¹,

Hössjer²,

Marks³

2022

Preprint

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“…(Evaluation of student test scores [ 44 ]). As a generalization of the example given in Section 1.2 , suppose that a number of students perform a test.…”

Section: Examplesmentioning

confidence: 99%

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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“…In fact, can be seen as the message sent, as the message received, and as the channel between them distorting the message. 7,8…”

Section: Estimatorsmentioning

confidence: 99%

“…) ∕𝑃 [𝑇 = 1] as the channel between them distorting the message. 7,8 Analogously to (7) with Proposition 1, the naive estimator of individuals with symptoms 𝑠, 𝑝 𝑠, * 𝑇 , and the naive estimator of individuals with infection status 𝑖, 𝑝 * ,𝑖 𝑇 , are defined as…”

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confidence: 99%

Correcting prevalence estimation for biased sampling with testing errors

Zhou

Díaz–Pachón

Zhao

et al. 2021

Preprint

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Surveillance studies for Covid-19 prevalence estimation are subject to sampling bias due to oversampling of symptomatic individuals and error-prone tests, particularly rapid antigen tests which are known to have high false negative rates for asymptomatic individuals. This results in naïve estimators which can be very far from the truth.In this work, we present a method that removes these two sources of error directly. Moreover, our procedure can be easily extended to the stratified error situation in which a test has very different error rate profiles for symptomatic and asymptomatic individuals as is the case for rapid antigen testing. The result is an easily understandable four-step algorithm that produces much more reliable prevalence estimates as demonstrated on data from the Israeli Ministry of Health. Thus it may re-open the debate about whether we are under-valuing rapid testing as a surveillance tool and may have policy implications in Third-World countries or disadvantaged communities where access to PCR testing may be less accessible.

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A Formal Framework for Knowledge Acquisition: Going Beyond Machine Learning

Cited by 4 publications

References 22 publications

Sometimes size does not matter

Sometimes size does not matter

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

Correcting prevalence estimation for biased sampling with testing errors

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