Bayesian Methods (MCMC)

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

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages.

Resumen El objetivo de este trabajo es evaluar el efecto de algunas variables latentes influyentes sobre la competencia lectora y la alfabetización matemática en una muestra de estudiantes costarricenses participantes en la prueba PISA 2009. Se estable un modelo de medida para los constructos latentes y un modelo estructural que establece la relación entre constructos, basado en la teoría y en la opinión de expertos en la temática. Se estima un modelo de ecuaciones estructurales, tanto desde un enfoque clásico o frecuentista como desde una perspectiva Bayesiana. En ambos enfoques las estimaciones resultan muy similares en el modelo de medida y en el modelo estructural, resultados que dan evidencia empírica a favor de las hipótesis planteadas.

The aim of this paper is to evaluate the effect of some influential latent variables on reading competence and mathematical literacy in a sample of Costa Rican students participating  in  the  PISA  2009  test.  A  measurement  model  is  established  for  the  latent  variables,  and  a  structural  model  that  establishes  the  relationship  between  latent  variables,  based  on  the  theory  and  the  opinion  of  experts  in  the  subject.  A  structural  equations  model  is  estimated,  both  from  a  classical  or  frequentist  approach  and  from  a  Bayesian  perspective.  In  both  approaches,  the  estimates  are  very similar in the measurement model and in the structural model, results that give empirical evidence in favor of the hypotheses proposed.

A two-phase Monte Carlo simulation (TPMCS) uncertainty analysis framework is used to analyze epistemic and aleatory uncertainty associated with simulated exceedances of an in-stream fecal coliform (FC) water quality criterion when using the Hydrological Simulation Program–FORTRAN (HSPF). The TPMCS framework is compared with a single-phase or standard Monte Carlo simulation (SPMCS) analysis. Both techniques are used to assess two total maximum daily load (TMDL) pollutant allocation scenarios. The application of TPMCS illustrates that cattle directly depositing FC in the stream is a greater source of epistemic uncertainty than FC loading from cropland overland runoff, the two sources specifically targeted for reduction in the allocation scenario. This distinction is not possible using SPMCS. Although applying the TPMCS framework involves subjective decisions about how selected model parameters are considered within the framework, this uncertainty analysis approach is transparent and the results provide information that can be used by decision makers when considering pollution control measure implementation alternatives, including quantifying the level of confidence in achieving applicable water quality standards.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.

Resumen La necesidad de generar efi ciencias en las compras por volumen, o de mejorar la exactitud de los pronósticos de venta, crea un esfuerzo de integración en las organizaciones que buscan tener pre-sencia en el canal comercial para ganar cuota de mercado. La planeación de largo plazo queda restringida a una planeación estratégica de lineamientos, escenarios prospectivos o políticas comerciales que debido a la incertidumbre sobre el comportamiento de las diversas variables infl uyentes en un plan de nivel táctico reducen la posibilidad de incrementar los horizontes de planeación. En el presente artículo se describe el aporte del enfoque bayesiano para mejorar la exactitud de los pronósticos ante un incremento de la periodicidad de los consumos, en un entorno altamente dinámico debido a la infl uencia de los cambios de estrategias comerciales de mediano y corto plazo, como ocurre usualmente en el retail marketing. Palabras clave: cadena de abastecimiento / gestión de demanda / marketing cuantitativo / pronósticos bayesianos Summary The need to generate effi ciencies in volume purchases or improve the accuracy of sales forecasts is based on integration eff orts within organizations competing in retail channel pursuing gain market share. Long-term planning is usually restricted to a strategic planning guidelines, foresight scenarios or trade policies where the uncertainty of diff erent variables generates little infl uence on tactical level planning. This article discusses the contribution of the Bayesian approach used to improve tactical planning in a highly dynamic environment due to the infl uence of changes in business strategies of medium and short term as usually occurs in retail marketing environment.

Homo naledi is a recently discovered species of fossil hominin from South Africa. A considerable amount is already known about H. naledi but some important questions remain unanswered. Here we report a study that addressed two of them: &quot; Where does H. naledi fit in the hominin evolutionary tree? &quot; and &quot; How old is it? &quot; We used a large supermatrix of craniodental characters for both early and late hominin species and Bayesian phylogenetic techniques to carry out three analyses. First, we performed a dated Bayesian analysis to generate estimates of the evolutionary relationships of fossil hominins including H. naledi. Then we employed Bayes factor tests to compare the strength of support for hypotheses about the relationships of H. naledi suggested by the best-estimate trees. Lastly, we carried out a resampling analysis to assess the accuracy of the age estimate for H. naledi yielded by the dated Bayesian analysis. The analyses strongly supported the hypothesis th...

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

Statistical asteroid-orbit-computation methods have proven important for applications such as computing the collision probability, performing dynamical classification, identifying asteroids, and aiding the recovery of lost objects. These methods will be of major importance during the Gaia mission (to be launched in 2012). Here we discuss these methods and some of their applications and potentials. As earlier, we examine the Bayesian a posteriori probability density of the orbital elements using Monte-Carlo methods that map the discretized volume of solutions in the orbital-element phase space. In particular, we use the Markov-chain Monte-Carlo (MCMC) method to map the phase space. The marginal orbital-element distributions then serve as a base for applications as collision probability computation, dynamical classification and others. A number of recent applications are presented. Importance of the method for the Gaia mission is also described.

Morphological integration predicts that correlated characters will coevolve; thus, each distinct suite of correlated characters might be expected to evolve according to a separate clock or ‘pacemaker’. Characters in a large morphological dataset for mammals were found to be evolving according to seven separate clocks, each distinct from the molecular clock. Total-evidence tip-dating using these multiple clocks inflated divergence time estimates, but potentially improved topological inference. In particular, single-clock analyses placed several meridiungulates and condylarths in a heterodox position as stem placentals, but multi-clock analyses retrieved a more plausible and orthodox position within crown placentals. Several shortcomings (including uneven character sampling) currently impact upon the accuracy of total-evidence dating, but this study suggests that
when sufficiently large and appropriately constructed phenotypic datasets become more commonplace, multi-clock approaches are feasible and can affect both divergence dates and phylogenetic relationships.

According to both domain expert knowledge and empirical evidence, wavelet coefficients of real signals tend to exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). A wavelet de-noising approach that takes into account such a feature of the signal may in practice outperform other, more vanilla methods, both in terms of the estimation error and visual appearance of the estimates. Motivated by this observation, we present a Bayesian approach to wavelet de-noising, where dependencies between neighbouring wavelet coefficients are a priori modelled via a Markov chain-based prior, that we term the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method (due to Johnstone and Silverman). We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre–Veronese configuration, an explicit form of a Gröbner basis consisting of binomials of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.

This work presents the current state-of-the-art in techniques for tracking a number of objects moving in a coordinated and interacting fashion. Groups are structured objects characterized with particular motion patterns. The group can be comprised of a small number of interacting objects (e.g. pedestrians, sport players, convoy of cars) or of hundreds or thousands of components such as crowds of people. The group object tracking is closely linked with extended object tracking but at the same time has particular features which differentiate it from extended objects. Extended objects, such as in maritime surveillance, are characterized by their kinematic states and their size or volume. Both group and extended objects give rise to a varying number of measurements and require trajectory maintenance. An emphasis is given here to sequential Monte Carlo (SMC) methods and their variants. Methods for small groups and for large groups are presented, including Markov Chain Monte Carlo (MCMC) methods, the random matrices approach and Random Finite Set Statistics methods. Efficient real-time implementations are discussed which are able to deal with the high dimensionality and provide high accuracy. Future trends and avenues are traced.

Recently, Markov Chain Monte Carlo (MCMC) sampling methods have evolved as new promising solutions to both multiuser and multiple-input multiple-output (MIMO) detection problems. Approaches based on Gibbs sampling as a special type of MCMC methods are well suited due to their good trade-off between performance and complexity. However, it is known that detection methods based on Gibbs sampling show a

We examine some Markov chain Monte Carlo (MCMC) methods for a generalized non-linear regression model, the Logit model. It is first shown that MCMC algorithms may be used since the posterior is proper under the choice of non-informative priors. Then two non-standard MCMC methods are compared: a Metropolis–Hastings algorithm with a bivariate normal proposal resulting from an approximation, and a Metropolis–Hastings algorithm with an adaptive proposal. The results presented here are illustrated by simulations, and show the good behavior of both methods, and superior performances of the method with an adaptive proposal in terms of convergence to the stationary distribution and exploration of the posterior distribution surface.

Information on stock status is available only for a few of the species forming the catch assemblage of rapido fishery of the North-central Adriatic Sea (Mediterranean Sea). Species that are caught almost exclusively by this gear, either as target (such as Pectinidae) or accessory catches (such as flatfishes apart from the common sole), remain unassessed mainly due to the lack of data and biological information. Based on cluster analysis, the catch assemblage of this fishery was identified and assessed using CMSY model. The results of this data-poor methodology showed that, among the species analyzed, no one is sustainably exploited. The single-species CMSY results were used as input to an extension of the same model, to test the effect of four different harvest control rule (HCR) scenarios on the entire catch assemblage, through 15-years forecasts. The analysis showed that the percentage of the stocks that will reach B msy at the end of the projections will depend on the HCR applied. Forecasts showed that a reduction of 20% of fishing effort may permit to most of the target and accessory species of the rapido trawl fishery in the Adriatic Sea to recover to B msy levels within 15 years, also providing a slight increase in the expected catches.

The paper proposes Bayesian analysis as an alternative approach for the conventional frequentist approach in analyzing social data. A step-by-step protocol of how to implement Bayesian multilevel model analysis with social data and how to interpret the result is presented. The article used a dataset regarding religious teachings and behaviors of lying and violence as an example. An analysis is performed using R statistical software and a bayesvl R package, which offers a network-structured model construction and visualization power to diagnose and estimate results.
•The paper provides guidance for conducting a Bayesian multilevel analysis in social sciences through constructing directed acyclic graphs (DAGs, or "relationship trees") for different models, basic and more complex ones.
•The method also illustrates how to visualize Bayesian diagnoses and simulated posterior.
•The interpretations of visualized diagnoses and simulated posteriors of Bayesian inference are also discussed.

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

This paper presents the application of a population Markov Chain Monte Carlo (MCMC) technique to generate history-matched models. The technique has been developed and successfully adopted in challenging domains such as computational biology but has not yet seen application in reservoir modelling. In population MCMC, multiple Markov chains are run on a set of response surfaces that form a bridge from the prior to posterior. These response surfaces are formed from the product of the prior with the likelihood raised to a varying power less than one. The chains exchange positions, with the probability of a swap being governed by a standard Metropolis accept/reject step, which allows for large steps to be taken with high probability. We show results of Population MCMC on the IC Fault Model—a simple three-parameter model that is known to have a highly irregular misfit surface and hence be difficult to match. Our results show that population MCMC is able to generate samples from the complex, multi-modal posterior probability distribution of the IC Fault model very effectively. By comparison, previous results from stochastic sampling algorithms often focus on only part of the region of high posterior probability depending on algorithm settings and starting points.