1996

Richard Jeffrey has labelled his philosophy of probability "radical probabilism" and qualified this position as "Bayesian", "nonfoundational" and "anti-rationalist". This paper explores the roots of radical probabilism, to be traced back to the work of Frank P. Ramsey and Bruno de Finetti.

Philosophy of Mathematics, 2009

This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.

EOLSS Publishers

The philosophy of probability is a well-established, yet still greatly expanding field within the philosophy of science, which focuses upon questions regarding the nature and interpretation of the notion of probability; the connections between probability and metaphysical chance; and the role that the notion of probability plays in statistical modelling practice across the sciences. This chapter provides a state of the art review of the philosophy of probability with an eye on the fundamental duality and irreducible pluralism of objective and subjective variants of chance. Throughout the essay I emphasize the different ways in which an appropriate articulation of the subjective dimension of probability has historically been facilitated by a proper regard for its objective dimension of proba bility, and viceversa.

Philosophical Books, 1983

Advances in Pure Mathematics, 2012

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.

2014

La vera logica di questo mondo il calcolo delle probabilità ... Questa branca della matematica, che di solito viene ritenuta favorire il gioco d'azzardo, quello dei dadi e delle scommesse, e quindi estremamente immorale,è la sola 'matematica per uomini pratici', quali noi dovremmo essere. Ebbene, come la conoscenza umana deriva dai sensi in modo tale che l'esistenza delle cose esterneè inferita solo dall'armoniosa (ma non uguale) testimonianza dei diversi sensi, la comprensione, che agisce per mezzo delle leggi del corretto ragionamento, assegnerà a diverse verità (o fatti, o testimonianze, o comunque li si voglia chiamare) diversi gradi di probabilità."

We show that the dominant definitions of probability are seriously flawed. While finite frequentism fails to define, infinite frequentism is not operational. Similarly, the Dutch Book arguments used to establish the existence of subjective probability fail to do so. An alternative definition of probability as a metaphor is offered. It is shown that this definition resolves several puzzles regarding the interpretation of common frequentist procedures and also some puzzles regarding the philosophy of science.

2014

Computer vision is an ever growing discipline whose ambitious goal is to enable machines with the intelligent visual skills humans and animals are provided by Nature, allowing them to interact effortlessly with complex, dynamic environments. Designing automated visual recognition and sensing systems typically involves tackling a number of challenging tasks, and requires an impressive variety of sophisticated mathematical tools. In most cases, the knowledge a machine has of its surroundings is at best incomplete – missing data is a common problem, and visual cues are affected by imprecision. The need for a coherent mathematical ‘language’ for the description of uncertain models and measurements then naturally arises from the solution of computer vision problems. The theory of evidence (sometimes referred to as ‘evidential reasoning’, ‘belief theory’ or ‘Dempster- Shafer theory’) is, perhaps, one of the most successful approaches to uncertainty modelling, as arguably the most straightforward and intuitive approaches to a generalized probability theory. Emerging in the last Sixties from a profound criticism of the more classical Bayesian theory of inference and modelling of uncertainty, it stimulated in the last decades an extensive discussion of the epistemic nature of both subjective ‘degrees of beliefs’ and frequentist ‘chances’ or relative frequencies. More recently, a renewed interest in belief functions, the mathematical generalization of probabilities which are the object of study of the theory of evidence, has seen a blossoming of applications to a variety of fields of applied science. In this Book we are going to show how, indeed, the fruitful interaction of computer vision and evidential reasoning is able stimulate a number of advances in both fields. From a methodological point of view, novel theoretical advances concerning the geometric and algebraic properties of belief functions as mathematical objects will be illustrated in some detail in Part II, with a focus on a perspective ‘geometric approach’ to uncertainty and an algebraic solution of the issue of conflicting evidence. In Part III we will illustrate how these new perspectives on the theory of belief functions arise from important computer vision problems, such as articulated object tracking, data association and object pose estimation, to which in turn the evidential formalism can give interesting new solutions. Finally, some initial steps towards a generalization of the notion of total probability to belief functions will be taken, in the perspective of endowing the theory of evidence with a complete battery of estimation and inference tools to the benefit of scientists and practitioners.

Springer eBooks, 2007

Mathematical Structures in Computer Science, 2014

In this paper, we discuss the crucial but little-known fact that, as Kolmogorov himself claimed, the mathematical theory of probabilities cannot be applied to factual probabilistic situations. This is because it is nowhere specified how, for any given particular random phenomenon, we should construct, effectively and without circularity, the specific and stable distribution law that gives the individual numerical probabilities for the set of possible outcomes. Furthermore, we do not even know what significance we should attach to the simple assertion that such a distribution law “exists”. We call this problem Kolmogorov's aporia†.We provide a solution to this aporia in this paper. To do this, we first propose a general interpretation of the concept of probability on the basis of an example, and then develop it into a non-circular and effective general algorithm of semantic integration for the factual probability law involved in a specific factual probabilistic situation. The dev...