Bayesian Evidence Synthesis

We exist in the world of uncertainties.
Any uncertainty always arises from a conflict of experience and chance, more precisely, from a conflict between the observer's experience and the chance observation. In other words, this indivisible pair, ``experience and chance'', is the source of any uncertainty.
Other sources of uncertainty simply do not exist.
A theory that describes this conflict strictly mathematically is the new theory, called theory of co~events, or the theory of experience and chance. Now we can say that this theory is a theory of uncertainty in its broadest sense.

The eventology approach gave impetus to the development of a theory that turned out to be broader than the theory of probabilities.
This new theory is a dual combination of two theories - Kolmogorov's theory of probabilities of ket-events (k-e.'s) and its dual reflection - a new theory of believabilities of bra-events (b-e.'s). Today, the theory of co~events has a strict axiomatics, in which Kolmogorov k-e.'s describing the future case of observation are dually reflected in b-e.'s describing the past experience of the observer, and a co~event is defined as a measurable binary relation on the Cartesian product: ``a set of bra-events X a set of terraced ket-events''.

Practitioners and construction management researchers lack believable and practical methods to assess the value proposition of emerging methods such as Virtual Design and Construction (VDC) including understanding how different levels of implementation affect its benefits. Furthermore, current methods of understanding VDC implementation and benefits cannot be updated easily to incorporate new data. This paper presents a Bayesian framework to predict benefits from application of Virtual Design and Construction (VDC) given data about its implementation. We analyzed data from 40 projects that performed some formal modeling of the project scope and/or the construction process. The analysis suggests that more extensive or higher levels of VDC implementation lead to higher project benefits. We explain the use of a Bayesian framework as an alternative to the application of classical probability theory to construction management research, how we used it to interpret data about VDC practice and outcomes, our finding that benefits have strong positive contingent correlation with the level of VDC implemented on projects, and our suggestion to use the method to update conclusions about benefits given changing data about implementation and outcomes.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

We introduce the set-theoretic language for the element-set labelling a Cartesian product by measurable binary relations intended for the labelling, or for the naming of parts and details of the construction that we are going to propose in the theory of experience and chance, or the theory of co~events that serve as mathematical models of events as dual pairs. Before you start working in a set-theoretic space whose objects of interest are simultaneously the elements of space, the sets of elements, and the sets of subsets of elements, it is necessary to stock up some system of «coordinates» suitable for a labelling the space itself, and its parts.

Keywords: eventology, theory of experience and chance, theory of co~events, measurable binary relation, event, co~event, experience, chance.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

The author has developed a general theory for measuring the statistical interconnections between events, called ``mathematical eventology'' \cite[2007]{Vorobyev2007}, \cite[2011]{Vorobyev2011em}. The paper represents fresh ``eventological'' ideas that radically rethinking the basis of previous approaches to measuring statistical interconnections of events beyond the traditional measurements in the probability theory and mathematical statistics, adding to the already well-known  methods of measuring association, or statistical dependency, another new group of methods of measuring agreement of a set of events.
It shows by useful ``total'' quantities (called agreement, association, covariance, set-distance, set-nearness, and set-closeness of a set of events) formulas in this context can be simplified and elegantly summarized for many applications, including for the correct generalization of the traditional indicators of statistical interconnections of events as Pearson correlation, Yule association, as well as a popular index of agreement --- Cohen kappa \cite[1960]{Cohen1960}.
Proposed new methods demonstrate the growing potential of ``eventological'' technologies in the study of structures of interconnections (agreement, association, or statistical dependency) of events, which applies to the measurement in the nominal (categorical), and the ordered and numeric (difference, ratio) scales.
The development and use in a variety of applications of the new concept of the set-distance for a set of events must have necessarily come into contact with the area of modern multivariate statistics that deals with measuring the commonality (agreement) of experts in estimates within categorical (nominal) scales. The results of this dramatic contact are set forth below. As the ``whipping boy'', the most popular group of such statistics is chosen: the kappa-indicators of the commonality (agreement) of experts in estimates within categorical scales.

A fixed company of players observes a person selected from a fixed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that person is bald, or on what is not. A denite formula of the gain is suggested, such that every time after bets the gain of each player from a given company are completely determined by this formula. However, before bets player's gain is an uncertain value. Is it possible for a given company of players and a given queue of people before bets to build a correct mathematical model of uncertaint gain of each player within the framework of Kolmogorov's probability theory? If not, what else do you need to add to the foundations of probability theory so that before bets to be able to use this model for decision making? The paper answers these questions within the framework of the new theory of experience and of chance (the certainty theory) [1] that consists of two dual halves: the believability theory and the probability theory, and that is intended for the mathematical description of experienced-random experiments, the uncertainty in outcomes of which is generated by the observer. A xed company of players observes a person selected from a xed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that the person is bald, or on what is not. If, as a result of the bets made, the player's choice is in the minority, the player loses the bet. If, on the other hand, in the majority, the player is returned his bet plus an equal share of the loser players' bets. With a draw, the equality of the number of betting on the bald and the number of betting on the not bald, all players remain at their own: they are returned their bets 1. Every time after the bets, the gain of each player from this company is fully certain. But before the bets, the gain is an uncertain value. Is it possible for a given company of players and a given queue of persons before bets to build a correct mathematical model of uncertain gain of each player within the framework of Kolmogorov's probability theory?
c ⃝ 2016 O.Yu.Vorobyev Oleg Vorobyev (ed.), Proc. XV FAMEMS'2016, Krasnoyarsk: SFU

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of erce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters., theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

This is a further formation of the theory of experience and chance. My previous works are devoted to the elements of this new theory, which can also be called the theory of co∼events, or the certainty theory. The theory is a synthesis of two dual theories: Kolmogorov's theory of probability and the believability theory based on the axiomatics, which is a non-trivial extension of Kolmogorov's axiomatics. In my paper " Theory of dual co∼event-based means " (2016), the concepts of two dual co∼event-based means of one co∼event are introduced (the bra-mean and the ket-mean). Here I first suggest the definition of the unifying mean of one co∼event (the bra-ket-mean), and then I enter the general definition of the bra-ket-mean for a finite set of co∼events. The mean co∼event-based characteristics of co∼events play the same role in the theory of experience and chance as the mean numerical characteristics of random variables — in the probability theory, or the mean characteristics of events — in the eventological theory.

In my recent works, I propose a new theory of experience and chance, or the theory of certainties, which is a synthesis of two dual theories: Kolmogorov's theory of probabilities and a new theory of believabilities. The theory of experience and chance is based on axiomatics, adding to the Kolmogorov axiomatics of probability theory an axiom of co~event, a new concept defined as a pair of dual objects, one of which serves as a Kolmogorov event in probability space and the other - as an event in the dualistic believability space. One of my papers gives a theory of dual co~event-based means and examples of its use for processing
statistics of ``approving voting'' in elections and in forestry. In this paper, I will go into more detail in examples of processing the statistics
of ``approving voting'' in forestry, where a group of forestry experts is actually engaged in ``approving voting'' when they label the trees of the
forest, labelling those that according to their opinions must be felling. My goal is to show in these examples that the theory of experience and
chance is an effective tool for constructing co~evnts models of experts' behavior in different fields.

"""The present work presents a general theoretical framework for the study of operators which merge partial probabilistic evidence from different sources which are individually coherent, but may be collectively incoherent. We consider a number of principles for such an operator to satisfy including a set of principles derived from those of Konieczny and Pino Perez [11] which were formulated for the different context of propositional merging.

Finally we investigate two specific such merging operators derived from the Kullback-Leibler notion of informational distance: the social entropy operator, and its dual, the linear entropy operator. The first of these is strongly related to both the multi-agent normalised  geometric mean pooling operator and the single agent maximum entropy inference process,
ME.  By contrast the linear entropy operator is similarly related to both the arithmetic mean pooling operator and the limit centre of mass inference process, CM-infinity."""

The commercialization of forensic scientific provision in the UK over the last two decades has had a major role in shaping a changing epistemic identity for forensic scientists working within this jurisdiction. Efforts to match the presumed epistemological standards of the ‘pure’ sciences have been brought together with concerns about value for money in a new approach to the interpretation of evidence, an activity that lies at the heart of criminal investigative practice. A study of the Case Assessment and Interpretation method developed by members of the UK Forensic Science Service is used to show how a technical innovation in the delivery of forensic science services to the police has instantiated these two recent social processes.

Background: Coral reefs are rapidly changing in response to local and global stressors. Research to better understand and inform the management of these stressors is burgeoning. However, in situ studies of coral reef ecology are constrained by complex logistics and limited resources. Many reef studies are also hampered by the scale-dependent nature of ecological patterns, and inferences made on causal relationships within coral reef systems are limited by the scales of observation. This is because most socio-ecological studies are conducted at scales relevant to the phenomenon of interest. However, management often occurs across a significantly broader, often geopolitical, range of scales. While there is a critical need for incisive coral reef management actions at relevant spatial and temporal scales, it remains unclear to what extent the scales of empirical study overlap with the scales at which management inferences and recommendations are made. This systematic map protocol will evaluate this potential scale mismatch with the goal of raising awareness about the significance of effectively addressing and reporting the scales at which researchers collect data and make assumptions. Methods: We will use the Collaboration for Environmental Evidence (CEE) systematic mapping guidelines to identify relevant studies using a framework-based synthesis to summarise the spatial and temporal scales of coral reef fish ecology research and the scales at which management inferences or recommendations are made. Using tested pre-defined terms, we will search for relevant published academic and grey literature, including bibliographic databases, web-based search engines, and organisational websites. Inclusion criteria for the evidence map are empirical studies that focus on coral reef fish ecological organisation and processes, those informing management interventions and policy decisions, and management documents that cite coral reef research for management decision-making. Study results will be displayed graphically using data matrices and heat maps. This is the first attempt to systematically assess and compare the scales of socio-ecological research conducted on coral reef systems with their management.

Very briefly on the new axiomatics of co∼events, it is intended for the rigorous mathematical description of an entanglement between observer and observation. Here we try to consider the new axiomatics in the normal human non-mathematical language, although it uses a new but quite natural terminology: ``a co∼event occurs (as an entanglement of observer and observation), a ket-event happens (as chance of observation), a bra-event is experienced (as an experience of observer) ''. And finally, it seems to me that the new theory in a not very rigorous mathematical language is most conveniently set out in the form of questions and answers that my students, graduate students, and colleagues often asked me during numerous internal and external discussions.

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the
Kolmogorov probability theory. The author's main idea was the natural inclusion of Kolmogorov's axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The analogy between the measure of a set and the probability of an event has become clear for a long time. This analogy also allows further evolution: the measure of a set is completely analogous to the believability of an event. In order to postulate the theory of experience and chance on the basis of this analogy, you just need to add to the Kolmogorov probability
theory its dual reflection --- the believability theory, so that the theory of experience and chance could be postulated as the certainty
(believability-probability)  theory on the Cartesian product of the probability and believability spaces, and the central concept of the theory is the new notion of co~event as a measurable binary relation on the Cartesian product of sets of elementary incomes and elementary outcomes. Attempts to build the foundations of the theory of experience and of chance from this general point of view are unknown to me, and the whole range of ideas presented here has not yet acquired popularity even in a narrow circle of specialists; in addition, there was still no complete system of the postulates of the theory of experience and chance free from unnecessary complications. Postulating the theory of experience and chance can be carried out in different ways, both in the choice of axioms and in the choice of basic concepts and relations. If one tries to achieve the possible simplicity of both the system of axioms and the theory constructed from it, then it is hardly possible to suggest anything other than axiomatization of concepts co~event and its certainty (believability-probability). The main result of this work is the \textbf{axiom of co~event}, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work, a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment.