A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning

Abstract

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.

Is It Possible to Know Cosmological Fine-tuning?

“…where the target A ⊂ Ω is a subset of a search space Ω, whereas P and P 0 are probability distributions on Ω that represent searches of the programmer and blind search, respectively. Since its inception, active information has been used in the measurement of bias for machine-learning algorithms (Montañez 2017a(Montañez , 2017bMontañez et al 2019Montañez et al , 2021, hypothesis testing (Hössjer et al 2023;Zhou et al 2023), among others. In fact, active information can be used as a measure of FT if the search space Ω equals the sample space  of the physical parameter X, Equation (4) is large for A = ℓ X , with P 0 (A) = F 0 (A) as in Equation (1) and d = ( ) ( )…”

Correcting prevalence estimation for biased sampling with testing errors

“…as the channel between them distorting the message. 11,12 This interpretation, taken from Shannon's information diagram, is particularly important to analyze bias as a modification of the information inherent to the prevalence parameter in Appendix D. 13 Analogously to (9) with Proposition 1, the naive estimator of individuals with symptoms s, ps, * T , and the naive estimator of individuals with infection status i, p * ,i T , are defined as…”

“…Then $$$ {N}_s^{(i)} $$$, the population size of $$$ {I}_s^{(i)} $$$, disappears from the sample estimator, and () in the Appendix shows that all information in the sample about the group $$$ {I}_s^{(i)} $$$ comes from the sampling mechanism $$$ q\left({I}_s^{(i)}\right) $$$. In fact, $$$ {p}_s^{(i)} $$$ can be seen as the message sent, $$$ {\hat{\mathbf{p}}}_T^{s,i} $$$ as the message received, and $$$ q\left({I}_s^{(i)}\right)=P\left(T=1|{I}_s^{(i)}\right) $$$ as the channel between them distorting the message 11,12 . This interpretation, taken from Shannon's information diagram, is particularly important to analyze bias as a modification of the information inherent to the prevalence parameter in Appendix 13 …”