When can statistical theories be causally closed?

The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle (RCCP)

On the tension between physics and mathematics

Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The paper looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists. Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics

A categorial approach to relativistic locality

Relativistic locality is interpreted in this paper as a web of conditions expressing the compatibility of a physical theory with the underlying causal structure of spacetime. Four components of this web are distinguished: spatiotemporal locality, along with three distinct notions of causal locality, dubbed CL-Independence, CL-Dependence, and CL-Dynamic. These four conditions can be regimented using concepts from the categorical approach to quantum field theory initiated by Brunetti, Fredenhagen, and Verch (2003). A covariant functor representing a general quantum field theory is defined to be causally local if it satisfies the three CL conditions. Any such theory is viewed as fully compliant with relativistic locality. We survey current results indicating the extent to which an algebraic quantum field theory satisfying the Haag–Kastler axioms is causally local

Categorial subsystem independence as morphism co-possibility

This paper formulates a notion of independence of subobjects of an object in a general (i.e. not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C∗ - algebras with respect to operations (completely positive unit preserving linear maps on C∗ - algebras)as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory

General properties of Bayesian learning as statistical inference determined by conditional expectations

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization

Defusing Bertrand's Paradox

The classical interpretation of probability together with the Principle of Indifference are formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called Labeling Invariance is defined in the category of Haar probability spaces, it is shown that Labeling Invariance is violated and Bertrand's Paradox is interpreted as the very proof of violation of Labeling Invariance. It is shown that Bangu's attempt (Bangu 2010) to block the emergence of Bertrand's Paradox by requiring the re-labeling of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve Labeling Invariance is identified and it is argued that, under the interpretation of Bertrand's Paradox suggested in the paper, the paradox does not undermine either the Principle of Indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena; it is shown in particular that violation of Labeling Invariance does not entail that labeling of random events affects the probabilities of random events. It also is argued however that the content of the Principle of Indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful

The maxim of probabilism, with special regard to Reichenbach

It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations

The Bayes Blind Spot of a finite Bayesian Agent is a large set

The Bayes Blind Spot of a Bayesian Agent is the set of probability measures on a Boolean algebra that are absolutely continuous with respect to the background probability measure (prior) of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by conditionalizing no matter what (possibly uncertain) evidence he has about the elements in the Boolean algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has the same cardinality as the set of all probability measures (continuum); it has the same measure as the measure of the set of all probability measures (in the natural measure on the set of measures); and is a ``fat'' (second Baire category) set in topological sense in the set of all probability measures taken with its natural topology

Target Diameter Models for Leuce Poplar Stands Growing on Sandy Soils

The fact that certain ecological factors fundamentally influencing tree growth have become unfavourable in Hungary in recent years, has led to the more extensive use of white poplar (and its hybrids) in afforestation and forest regeneration. An intensive integrated research and development work has been carried out on the growth of Leuce poplars on sandy soils, including primarily the white poplar (Populus alba) and its natural hybrid the grey poplar (Populus x canescens). The research revealed several factors influencing stand growth. The study presents a new, simplified tending operation model for Leuce poplar stands, as well as age, growing space and target diameter models suitable for qualitaty log production and for mass assortments. The simplicity of these practice-oriented models may foster the qualitative development of Leuce poplar management in Hungary