Class Of Probabilistic Models Research Articles

Transmuted Weibull Logistic Distribution

Most of the time real data does not follow any of the classical or standard probability models, so more effort has been expended in the development of large classes of standard probability distribution by generalizing the standard distributions . A complete study of the transmuted Weibull Logistic distribution is proposed, introducing some basic properties of this distribution, such as Quantile function, mode, characteristic function and entropy, as well as the derivation of maximum likelihood estimators of the parameters and the information matrix. Real life data is used as an application to this distribution with a comparison with other distributions to illustrate the flexibility and ability to model lifetime data. Also, a simulation study is conducted to demonstrate the effect of the sample on the estimates of the parameters.

Fully Nonparametric Regression for Bounded Data Using Dependent Bernstein Polynomials

ABSTRACTWe propose a novel class of probability models for sets of predictor-dependent probability distributions with bounded domain. The proposal extends the Dirichlet–Bernstein prior for single density estimation, by using dependent stick-breaking processes. A general model class and two simplified versions are discussed in detail. Appealing theoretical properties such as continuity, association structure, marginal distribution, large support, and consistency of the posterior distribution are established for all models. The behavior of the models is illustrated using simulated and real-life data. The simulated data are also used to compare the proposed methodology to existing methods. Supplementary materials for this article are available online.

Particle MCMC algorithms and architectures for accelerating inference in state-space models

Particle Markov Chain Monte Carlo (pMCMC) is a stochastic algorithm designed to generate samples from a probability distribution, when the density of the distribution does not admit a closed form expression. pMCMC is most commonly used to sample from the Bayesian posterior distribution in State-Space Models (SSMs), a class of probabilistic models used in numerous scientific applications. Nevertheless, this task is prohibitive when dealing with complex SSMs with massive data, due to the high computational cost of pMCMC and its poor performance when the posterior exhibits multi-modality. This paper aims to address both issues by: 1) Proposing a novel pMCMC algorithm (denoted ppMCMC), which uses multiple Markov chains (instead of the one used by pMCMC) to improve sampling efficiency for multi-modal posteriors, 2) Introducing custom, parallel hardware architectures, which are tailored for pMCMC and ppMCMC. The architectures are implemented on Field Programmable Gate Arrays (FPGAs), a type of hardware accelerator with massive parallelization capabilities. The new algorithm and the two FPGA architectures are evaluated using a large-scale case study from genetics. Results indicate that ppMCMC achieves 1.96x higher sampling efficiency than pMCMC when using sequential CPU implementations. The FPGA architecture of pMCMC is 12.1x and 10.1x faster than state-of-the-art, parallel CPU and GPU implementations of pMCMC and up to 53x more energy efficient; the FPGA architecture of ppMCMC increases these speedups to 34.9x and 41.8x respectively and is 173x more power efficient, bringing previously intractable SSM-based data analyses within reach.

Fit Adequacy of Dichotomous Logit Response Models of the Regressor Bernoulli and Binomial Probability Distributions

Logit models belong to the class of probability models that determine discrete probabilities over a limited number of possible outcomes. They are often called ‘Quantal Variables’ or ‘Stimulus and Response Models’ in Biological Literature. The conventional R2 measure of goodness-of-fit is problematic in logit models. This has therefore led to the proposal of several alternative goodness-of-fit measures. But researchers in this area have identified the base rate problem in using these several alternative goodness-of-fit measures. This research is an extension of work done by people in this area. Specifically, this research is aimed at investigating the goodness-of-fit performances of eight statistics using the Bernoulli and Binomial distributions as explanatory variables under various scenarios. The study will draw conclusions on the “best” fit. The data for the study was generated through simulation and analysed using the multiple correlation analysis. The findings clearly show that for the Bernoulli Distribution, the goodness-of-fit statistics to use are: RO2, RC2, RM2 and λp; and for the Binomial Distribution, the goodness-of-fit statistics to use are: and RN2 and λp. RO2 stood out as the “best” goodness-of-fit statistics.

Bell Could Become the Copernicus of Probability

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as the concrete example of the non-Kolmogorovian model of probability, similarly to the Lobachevskian model — the first example of the non-Euclidean model of geometry. This is the “probability model” interpretation of the violation of Bell’s inequality. We also criticize the standard interpretation—an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.

Elements of Quantum Finance Models

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend a classical probability model to a quantum probability model. Classical stochastic calculus is replaced by its quantum counterpart on a Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. By its very nature of coping with non-commutative random variables however, many essential elements of classic mathematical finance but in quantum form have not been recovered in this note. A couple that have been established are not in their full generality. For instance, existence of non-trivial self-financing portfolios and preserving of self-financing property under change of numeraire can only be reclaimed subject to technical conditions. This writing puts together a few helpful results for the purpose of future studies of this potentially new research subject.

Statistical and subjective interpretations of probability in quantum-like models of cognition and decision making

The paper starts with an introduction to the basic mathematical model of classical probability (CP), i.e. the Kolmogorov (1933) measure-theoretic model. Its two basic interpretations are discussed: statistical and subjective. We then present the probabilistic structure of quantum mechanics (QM) and discuss the problem of interpretation of a quantum state and the corresponding probability given by Born’s rule. Applications of quantum probability (QP) to modeling of cognition and decision making (DM) suffer from the same interpretational problems as QM. Here the situation is even more complicated than in physics. We analyze advantages and disadvantages of the use of subjective and statistical interpretations of QP. The subjective approach to QP was formalized in the framework of Quantum Bayesianism (QBism) as the result of efforts from C. Fuchs and his collaborators. The statistical approach to QP was presented in a variety of interpretations of QM, both in nonrealistic interpretations, e.g., the Copenhagen interpretation (with the latest version due to A. Plotnitsky), and in realistic interpretations (e.g., the recent Växjö interpretation). At present, we cannot make a definite choice in favor of any of the interpretations. Thus, quantum-like DM confronts the same interpretational problem as quantum physics does.

Text Mining Using Metadata for Generation of Side Information

Text Mining is knowledge discovery process from large database to find out unknown patterns. In many metadata based text mining applications, side information also known as metadata which is associated with the text document. There are different types side information containing large amount of data i.e. metadata, weblogs and non-textual data (image, video, etc.). The side information is difficult to estimate when it contains noisy data. To achieve this, there is scope of improvement in generating side information i.e. selecting efficient classification and clustering algorithms, providing security for clustered side information, document organization, exploring filtering approaches. In future, there is a scope to design an extended approach for clustering using classical partitioning and probabilistic model.

A Scale Invariant Distribution of the Prime Numbers

The irregular distribution of prime numbers amongst the integers has found multiple uses, from engineering applications of cryptography to quantum theory. The degree to which this distribution can be predicted thus has become a subject of current interest. Here, we present a computational analysis of the deviations between the actual positions of the prime numbers and their predicted positions from Riemann’s counting formula, focused on the variance function of these deviations from sequential enumerative bins. We show empirically that these deviations can be described by a class of probabilistic models known as the Tweedie exponential dispersion models that are characterized by a power law relationship between the variance and the mean, known by biologists as Taylor’s power law and by engineers as fluctuation scaling. This power law behavior of the prime number deviations is remarkable in that the same behavior has been found within the distribution of genes and single nucleotide polymorphisms (SNPs) within the human genome, the distribution of animals and plants within their habitats, as well as within many other biological and physical processes. We explain the common features of this behavior through a statistical convergence effect related to the central limit theorem that also generates 1/f noise.

A Novel Parameter Estimation Method for Boltzmann Machines.

We propose a novel estimator for a specific class of probabilistic models on discrete spaces such as the Boltzmann machine. The proposed estimator is derived from minimization of a convex risk function and can be constructed without calculating the normalization constant, whose computational cost is exponential order. We investigate statistical properties of the proposed estimator such as consistency and asymptotic normality in the framework of the estimating function. Small experiments show that the proposed estimator can attain comparable performance to the maximum likelihood expectation at a much lower computational cost and is applicable to high-dimensional data.

When Is the Probability of a Large Earthquake Too Small?

Classical probabilistic seismic-hazard models (Cornell, 1968), which typically refer to the homogeneous Poisson process for earthquake occurrence, are not able to model explicitly the space-time clustering of earthquakes. Clustering may be particularly evident in time windows of days and weeks (e.g., Kagan and Knopoff, 1987; Ogata, 1988), but it may be still appreciable in the medium term, because the time sequences to large earthquakes may last long (Kagan and Jackson, 1991; Parsons, 2002; Faenza et al. , 2003; Marzocchi and Lombardi, 2008). The modeling of such a space–time clustering is an important subject of seismological research (Jordan et al. , 2011). In fact, accounting for time–space clustering of earthquakes may provide additional information, not only to seismic-hazard assessment aimed at structural design (e.g., Iervolino et al. , 2014; Marzocchi and Taroni, 2014), but also to short-term seismic risk management. The latter issue has been explored by the International Commission for Earthquake Forecasting, established after the L’Aquila earthquake in 2009, which paves the way to the so-called operational earthquake forecasting (OEF). As defined by Jordan et al. (2011), OEF comprises procedures for gathering and disseminating authoritative information about the time dependence of seismic hazards to help communities prepare for potentially destructive earthquakes. Notwithstanding some recent earthquake sequences showing the importance of tracking the time evolution of seismic hazard (e.g., as for the recent Canterbury sequence in New Zealand; Wein and Becker, 2013), currently OEF represents a controversial issue in seismology. Most critics are not focused on debating the scientific credibility of the models used to describe short-term earthquake clustering, but they dispute the usefulness (if not the potential danger) of the information they provide, in particular, the probability of a damaging event in a short time frame. According to OEF models available in the literature, the weekly probability of …

Modeling failure of high performance fibers: on the prediction of long-term time-to-failure

Our objective is to predict the time-to-failure distribution of fibers at loads for which mean time-to-failure is comparable or longer than the fibers’ economic lifetime. We describe load induced time-to-failure of high performance fiber in terms of a classical probabilistic failure model developed by Coleman. Mimicking a series of time-to-failure measurements, using Monte Carlo simulations, we will show how to capture model parameters and their variability using the least squares method and maximum likelihood estimation. It is relatively easy to obtain an accurate prediction for the maximum allowable fiber load such that time-to-failure exceeds a predefined minimum time-to-failure with high probability. However, obtaining a reliable lower prediction limit for time-to-failure at a given, low fiber load is very difficult and requires an unfeasible extensive program of time-to-failure measurements.

A Quantum Finance Model

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and the probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend the classical probability model to a quantum probability model. The classical stochastic calculus is replaced by its quantum counterpart on the Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. As an application, prices of quantum European options are obtained.

Two-slit experiment: quantum and classical probabilities

Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).

“Caterpillar”-SSA and Box-Jenkins hybrid models and methods for time series forecasting

Trend and decomposition approaches to non-stationary time series forecasting are considered in the paper. According to them, various hybrid models for non-stationary time series forecasting, as well as identification methods for these models based on the combined use of the “Caterpillar”-SSA and Box-Jenkins methods were proposed. Hybrid mathematical models of the trend approach to forecasting, based on the “Caterpillar”-SSA and Box-Jenkins methods lie in modeling the process as deviation of actual time series values with respect to the trend component, which is represented in the proposed models by the linear recurrence formula (LRF) of the “Caterpillar”-SSA method and its approximation by the SARIMA model. The main goal of the decomposition approach to forecasting based on the “Caterpillar”-SSA and Box-Jenkins methods is the decomposition of the original time series into multiple time series with a simpler structure, considered independently of each other using the “Caterpillar”-SSA method; forecasting the data of decomposition components by SARIMA models and calculating the total forecast by combining forecasts of the constructed simplified models. The proposed models were tested on the electricity and natural gas consumption time series, and their forecasting results were compared with the results, obtained by classical probabilistic SARIMA models, generalized for the case of several seasonal components. The obtained results allow to conclude that for effective forecasts, it is necessary to carry out decomposition of the studied time series and combine different models, describing both statistical and deterministic time series components that provides the best forecasting quality.

Classical probability model for Bell inequality

We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles (θi,θ'j) fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the classical probabilistic framework. The crucial point is that in constructing the common probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles (θi, θ'j)). One may (but need not) say that randomness of "free will" has to be accounted for.

Split-complex numbers and Dirac <i>bra-kets</i>

We describe the real tessarines or \split-complex anddescribe a novel instance where they arise in biomedical informat-ics. We use the split-complex numbers to give a mathematical def-inition of a Hyperbolic Dirac Network (HDN) | a hyperbolic ana-logue of a classical probabilistic graphical model that uses quantumphysics as an underlying heuristic.The methods of theoretical physics should be applicable to all those branches ofthought in which the essential features are expressible in numbers.| P. A. M. Dirac, Nobel Prize Banquet Speech 1933

Asymptotics of Bayesian Inference for a Class of Probabilistic Models under Misspecification

Asymptotics of Bayesian Inference for a Class of Probabilistic Models under Misspecification

Quantum-State Dynamics as Linear Representation of Classical (Nonlinear) Stochastic Dynamics

We show that the basic equation of the theory of open systems, the Gorini–Kossakowski–Sudarshan–Lindblad equation, as well as its linear and nonlinear generalizations have a natural classical probabilistic interpretation – within the framework of prequantum classical statistical field theory. The latter gives an example of the classical probabilistic model (with random fields as subquantum variables) reproducing the basic probabilistic predictions of quantum mechanics.

Updated seismic hazard assessment of Tunisia

Modern earthquake loss models make use of earthquake catalogs relevant to the seismic hazard assessment upon seismicity and seismotectonic analysis. The main objective of this paper is to investigate a recently compiled catalog (National Institute of Meteorology or INM catalog: 412-2011) and to generate seismic hazard maps through classical probabilistic seismic hazard assessment (PSHA) and smoothed-gridded seismicity models for Tunisia. It is now established with the local earthquake bulletin that the recent seismicity of Tunisia is sparse and moderate. Therefore, efforts must be undertaken to elaborate a robust hazard analysis for risk assessment and seismic design purposes. These recommendations follow the recently published reports by the World Bank that describe the seismic risk in Tunis City as being beyond a tolerable level with an MSK intensity level of VII. Some attempts were made during the past two decades to assess the seismic hazard for Tunisia and they have mostly failed to properly investigate the historical and instrumental seismicity catalog. This limitation also exists for the key aspect of epistemic and random uncertainties impact on the final seismic hazard assessment. This study also investigates new ground motion prediction equations suitable for use in Tunisia. The methodology applied herein uses, for the first time in PSHA of Tunisia, seismicity parameters integrated in logic tree framework to capture epistemic uncertainties through three different seismic source models. It also makes use of the recently released version of OpenQuake engine; an open-source tool for seismic hazard and risk assessment developed in the framework of the Global Earthquake Model.