Active Information, Learning, and Knowledge Acquisition

Philosophers frequently define knowledge as justified, true belief. In this paper we build a mathematical framework that makes possible to define learning (increased degree of true belief) and knowledge of an agent in precise ways. This is achieved by phrasing belief in terms of epistemic probabilities, defined from Bayes' Rule. The degree of true belief is then quantified by means of active information $I^+$, that is, 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 if 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, of which one is true and the others are counterfactuals. We also generalize the framework of learning and knowledge acquisition to a sequential setting, where information and data is updated over time. The theory is illustrated using examples of coin tossing, historical events, future events, replication of studies, and causal inference.

Quantum-Like Sampling

Probability theory is built around Kolmogorov’s axioms. To each event, a numerical degree of belief between 0 and 1 is assigned, which provides a way of summarizing the uncertainty. Kolmogorov’s probabilities of events are added, the sum of all possible events is one. The numerical degrees of belief can be estimated from a sample by its true fraction. The frequency of an event in a sample is counted and normalized resulting in a linear relation. We introduce quantum-like sampling. The resulting Kolmogorov’s probabilities are in a sigmoid relation. The sigmoid relation offers a better importability since it induces the bell-shaped distribution, it leads also to less uncertainty when computing the Shannon’s entropy. Additionally, we conducted 100 empirical experiments by quantum-like sampling 100 times a random training sets and validation sets out of the Titanic data set using the Naïve Bayes classifier. In the mean the accuracy increased from 78.84% to 79.46%.

Taking a Gamble

Gambling is an ancient human activity. We indulge ourselves by allowing ourselves to experience the dangers and thrills of chance in a somewhat controlled way. The history of lotteries and related games is explored. The chances of drawing various poker hands are laid out. The role of probability in horse racing is described, and how the odds quoted are not strictly statements of probability, but terms on which business is to be done. Political prediction betting markets give us a further interpretation of probability as a way of expressing strength of opinion in a quantifiable, albeit flawed way. Wagers encourage boasters to put their money where their mouth is, and so to quantify their degree of belief.

Truth and Uncertainty. A critical discussion of the error concept versus the uncertainty concept

Contrary to the claims put forward in “Evaluation of measurement data – Guide to the expression of uncertainty in measurement”, issued by the Joint Committee for Guides in Metrology, the error concept and the uncertainty concept are the same. Arguments in favour of the contrary have been analyzed and were found not compelling. Neither was any evidence presented in this document that “errors” and “uncertainties” define a different relation between the measured and the true value of the variable of interest, nor does this document refer to a Bayesian account of uncertainty beyond the mere endorsement of a degree-of-belief-type conception of probability.

Practical reasoning and degrees of outright belief

According to a suggestion by Williamson (Knowledge and its limits, Oxford University Press, 2000, p. 99), outright belief comes in degrees: one has a high/low degree of belief iff one is willing to rely on the content of one’s belief in high/low-stakes practical reasoning. This paper develops an epistemic norm for degrees of outright belief so construed. Starting from the assumption that outright belief aims at knowledge, it is argued that degrees of belief aim at various levels of strong knowledge, that is, knowledge which satisfies particularly high epistemic standards. This account is contrasted with and shown to be superior to an alternative proposal according to which higher degrees of outright belief aim at higher-order knowledge. In an “Appendix”, it is indicated that the logic of degrees of outright belief is closely linked to ranking theory.

Two Senses of “Probability”

The word “probability” has long been used in (at least) two distinct senses. One sense has to do with a rational agent’s degree of belief, commonly called credence in the philosophical literature. The other sort of probability is thought to be characteristic of a physical system, such as a roulette wheel; these are “in the world” rather than in our heads. This concept is called chance. It would be a mistake to think of these as rivals for the title of the single correct interpretation of probability. Rather, they are both useful concepts, with different roles to play. This chapter is an introduction to these concepts and their relations. It includes a discussion of the proper formulation and justification of a principle that links the two concepts, the Principal Principle. It is argued that neither of these concepts is dispensable. This raises the question of whether there is a notion of probability that can play the role of objective chance and is compatible with deterministic laws of physics.

A Modified Supervaluationist Framework for Decision-Making

How strongly an agent beliefs in a proposition can be represented by her degree of belief in that proposition. According to the orthodox Bayesian picture, an agent's degree of belief is best represented by a single probability function. On an alternative account, an agent’s beliefs are modeled based on a set of probability functions, called imprecise probabilities. Recently, however, imprecise probabilities have come under attack. Adam Elga claims that there is no adequate account of the way they can be manifested in decision-making. In response to Elga, more elaborate accounts of the imprecise framework have been developed. One of them is based on supervaluationism, originally, a semantic approach to vague predicates. Still, Seamus Bradley shows that some of those accounts that solve Elga’s problem, have a more severe defect: they undermine a central motivation for introducing imprecise probabilities in the first place. In this paper, I modify the supervaluationist approach in such a way that it accounts for both Elga’s and Bradley’s challenges to the imprecise framework.

Probability functions, belief functions and infinite regresses

In a recent paper Ronald Meester and Timber Kerkvliet argue by example that infinite epistemic regresses have different solutions depending on whether they are analyzed with probability functions or with belief functions. Meester and Kerkvliet give two examples, each of which aims to show that an analysis based on belief functions yields a different numerical outcome for the agent’s degree of rational belief than one based on probability functions. In the present paper we however show that the outcomes are the same. The only way in which probability functions and belief functions can yield different solutions for the agent’s degree of belief is if they are applied to different examples, i.e. to different situations in which the agent finds himself.