Bayesian Point Of View Research Articles

Expected goals under a Bayesian viewpoint: uncertainty quantification and online learning

Abstract While the use of expected goals (xG) as a metric for assessing soccer performance is increasingly prevalent, the uncertainty associated with their estimates is often overlooked. This work bridges this gap by providing easy-to-implement methods for uncertainty quantification in xG estimates derived from Bayesian models. Based on a convenient posterior approximation, we devise an online prior-to-posterior update scheme, aligning with the typical in-season model training in soccer. Additionally, we present a novel framework to assess and compare the performance dynamics of two teams during a match, while accounting for evolving match scores. Our approach is well-suited for graphical representation and improves interpretability. We validate the accuracy of our methods through simulations, and provide a real-world illustration using data from the Italian Serie A league.

Extending global fits of 4D Composite Higgs Models with partially composite leptons

We perform the first convergent Bayesian global fits of 4D Composite Higgs Models with partially-composite third generation quarks and leptons based on the minimal SO(5) → SO(4) symmetry breaking pattern. We consider two models with the τ lepton and its associated neutrino in different representations of SO(5). Fitting each model with a wide array of experimental constraints allows us to analyse the Bayesian evidence and currently-observed fine-tuning of each model by calculating the Kullback-Leibler divergence between their respective priors and posteriors. Notably both models are found to be capable of satisfying all constraints simultaneously at the 3σ level at scales of < 5 TeV. From a Bayesian viewpoint of naturalness the model with leptons in the 14 and 10 representations is preferred over those in the 5 representation due to its lower fine-tuning. Finally, we consider the experimental signatures for the preferred parameters in these models, including lepton partner decay signatures and gluon-fusion produced Higgs signal strengths, and discuss their potential phenomenology at future high-luminosity LHC runs.

Streaming variational inference-empowered Bayesian nonparametric clustering for online structural damage detection with transmissibility function

Transmissibility function (TF) is widely applied in damage detection due to its sensitivity to damage and robustness to external excitations, but its application in online damage detection is rarely reported due to challenges in handling data streams. This study proposes a new TF-based online damage detection method that integrates a truncation-free variational inference-based full Dirichlet process Gaussian mixture model (VI-FDPGMM) within a streaming variational inference (SVI) paradigm. As an improved Bayesian nonparametric approach, the truncation-free VI-FDPGMM addresses the issue of truncating mixing components in traditional VI-DPGMM for online learning with increasing data by strategically setting the variational distributions of parameters for the components without assigned data (i.e., inactivated components) to their prior distributions based on the Bayesian viewpoint, which enables computing the probabilities to assign data points to these components and determining the creation of new components. As a result, the truncation-free VI-FDPGMM allows dynamically adding components to the mixture model, providing the flexibility to automatically adapt the number of components for arbitrary amounts of data. This characteristic enables its intuitive integration into the SVI paradigm featured as the variational posterior conditioned on the previous data as the prior when new data are observed, facilitating continuous refinement of the mixture model without repeatedly making inference of previous data. Therefore, the proposed method is highly efficient and well-suited for online damage detection. The proposed method is validated using two case studies, demonstrating its capability to dynamically generate new clusters as new data are available online to indicate the emergence of new damage patterns during the monitoring process, which enables it to perform structural anomaly detection tasks in a semi-supervised manner. Furthermore, the method outperforms some state-of-the-art methods due to its capability for continuous model refinement and robustness in interpreting and capturing uncertainties.

How and when should a Bayesian point of view be employed?

How and when should a Bayesian point of view be employed?

Maximal entropy prior for the simple step‐stress accelerated test

AbstractThe step‐stress procedure is a popular accelerated test used to analyze the lifetime of highly reliable components. This paper considers a simple step‐stress accelerated test assuming a cumulative exposure model with uncensored lifetime data following a Weibull distribution. The maximum likelihood approach is often used to analyze accelerated stress test data. Another approach is to use the Bayesian inference, which is useful when there is limited data available. In this paper, the parameters of the model are estimated based on the objective Bayesian viewpoint using non‐informative priors. Our main aim is to propose the maximal data information prior (MDIP) presented by Zellner (1984) as an alternative prior to the conventional independent gamma priors for the unknown parameters, in situations where there is little or no a priori knowledge about the parameters. We also obtain the Bayes estimators based on both classes of priors, assuming three different loss functions: square error loss function (SELF), linear‐exponential loss function (LINEX), and generalized entropy loss function (GELF). The proposed MDIP prior is compared with the gamma priors via Monte Carlo simulations by examining their biases and mean square errors under the three loss functions, and coverage probability. Additionally, we employ the Markov Chain Monte Carlo (MCMC) algorithm to extract characteristics of marginal posterior distributions, such as the Bayes estimator and credible intervals. Finally, a real lifetime data is presented to illustrate the proposed methodology.

Analysis of a new extension version of the exponential model using improved adaptive progressive censored data and its applications

This article covers the issue of evaluating the two shape parameters and reliability metrics of a novel Kumaraswamy-exponential lifetime distribution, whose density exhibits a left-skewed, right-skewed, or symmetric shape, through a type-II improved adaptive progressively censored sample. Both conventional and Bayesian viewpoints are used to evaluate the various parameters, which include point and interval estimations. While the estimation of one of the shape parameters requires a numerical solution, the other shape parameter estimation can be carried out in closed form by the classical method. Besides, the likelihood method’s asymptotic traits are employed to provide interval estimations for all parameters. Leveraging the Markov chain Monte Carlo process, the symmetric squared loss function and independent gamma priors are taken into account for calculating Bayes points and the highest posterior density interval estimations. To illustrate the accuracy, compare estimation methods, and show the applicability of the various suggested methods, a simulation examination and a pair of applications are looked at. In the end, four accuracy indicators are taken into consideration to figure out the best progressive censoring pattern. The numerical results indicate that when collecting samples using the suggested censored procedure, it is advisable to use the Bayesian estimation approach for evaluating the Kumaraswamy-exponential distribution.

Model Falsification from a Bayesian Viewpoint with Applications to Parameter Inference and Model Selection of Dynamical Systems

Model Falsification from a Bayesian Viewpoint with Applications to Parameter Inference and Model Selection of Dynamical Systems

On Estimation of Shannon’s Entropy of Maxwell Distribution Based on Progressively First-Failure Censored Data

Shannon’s entropy is a fundamental concept in information theory that quantifies the uncertainty or information in a random variable or data set. This article addresses the estimation of Shannon’s entropy for the Maxwell lifetime model based on progressively first-failure-censored data from both classical and Bayesian points of view. In the classical perspective, the entropy is estimated using maximum likelihood estimation and bootstrap methods. For Bayesian estimation, two approximation techniques, including the Tierney-Kadane (T-K) approximation and the Markov Chain Monte Carlo (MCMC) method, are used to compute the Bayes estimate of Shannon’s entropy under the linear exponential (LINEX) loss function. We also obtained the highest posterior density (HPD) credible interval of Shannon’s entropy using the MCMC technique. A Monte Carlo simulation study is performed to investigate the performance of the estimation procedures and methodologies studied in this manuscript. A numerical example is used to illustrate the methodologies. This paper aims to provide practical values in applied statistics, especially in the areas of reliability and lifetime data analysis.

Increasing Stability in an Inverse Boundary Value Problem—Bayesian Viewpoint

Increasing Stability in an Inverse Boundary Value Problem—Bayesian Viewpoint

IP Promotion and Integrated Media Communication of Regional Culture in Second- and Third-Tier Cities Based on Bayesian Viewpoint Evolution Modeling

Abstract With the rapid development of Internet media in recent years, the main position of IP promotion and dissemination of urban regional culture has shifted from traditional media to new media mainly based on mobile Internet. This paper takes the viewpoint evolution model as a research tool, decides the best choice for decision-making based on the quality value of the main evidence, and adjusts its viewpoint using Bayes’ rule. The analysis includes both positive and negative perspectives on the issue of promoting regional cultural communication in second and third-tier cities through IP promotion and integrated media. Finally, we discuss the mechanism of the communication of the regional culture of second and third-tier cities through the integrated media and analyze the influence of the integrated media communication effect on the regional culture of second- and third-tier cities through the Bayesian model of improving the evolution of viewpoints. The results show that in the process of evolution, the group of people who agree with the positive view is 0.483, while the group of people who hold the negative view is only 0.223. In addition, ‘Culture X’s’ enhancement through the melting media has resulted in a nearly 1.8 million increase in the number of cultural program broadcasts on the entire network. The best way to combine the advantages of both is to promote regional culture in second and third-tier cities through integrated media, as suggested by this. This study has the potential to provide an optimization strategy for promoting and disseminating regional culture in second and third-tier cities, which has significant strategic value.

Reliability Analysis and Its Applications for a Newly Improved Type-II Adaptive Progressive Alpha Power Exponential Censored Sample

Recently, a newly improved Type-II adaptive progressive censoring plan was devised, which can successfully ensure that the test length will not surpass a particular threshold period. In this study, we explore the statistical inference of the alpha power exponential distribution in the context of improved adaptive progressive Type-II censored data. The parameters, reliability, and hazard functions were estimated from both classical and Bayesian viewpoints using this censoring plan. To begin, we applied the maximum likelihood estimation approach to obtain parameter, reliability, and hazard function estimators. Following that, the approximate confidence intervals for the aforementioned metrics were derived, assuming the asymptotic normality traits of the maximum likelihood estimators. Additionally, by employing the Bayesian method via the Markov chain Monte Carlo technique, the point estimators and highest posterior density intervals of various parameters were created based on the symmetric squared error loss. A simulation study that incorporates numerous scenarios was used to assess the effectiveness of various estimation methodologies. The optimal progressive censorship plans are then discussed based on a set of criteria. Finally, three applications from the engineering and medical domains have been offered as examples.

Application of Sparse Regularization in Spherical Radial Basis Functions-Based Regional Geoid Modeling in Colorado

Spherical radial basis function (SRBF) is an effective method for calculating regional gravity field models. Calculating gravity field models with high accuracy and resolution requires dense basis functions, resulting in complex models. This study investigated the application of sparse regularization in SRBFs-based regional gravity field modeling. L1-norm regularization, also known as the least absolute shrinkage selection operator (LASSO), was employed in the parameter estimation procedure. LASSO differs from L2-norm regularization in that the solution obtained by LASSO is sparse, specifically with a portion of the parameters being zero. A sparse model would be advantageous for improving the numerical efficiency by reducing the number of SRBFs. The optimization problem of the LASSO was solved using the fast iterative shrinkage threshold algorithm, which is known for its high efficiency. The regularization parameter was selected using the Akaike information criterion. It was specifically tailored to the L1-norm regularization problem. An approximate covariance matrix of the estimated parameters in the sparse solution was analytically constructed from a Bayesian viewpoint. Based on the remove–compute–restore technique, a regional geoid model of Colorado (USA) was calculated. The numerical results suggest that the LASSO adopted in this study provided competitive results compared to Tikhonov regularization; however, the number of basis functions in the final model was less than 25% of the Tikhonov regularization. Without significantly reducing model accuracy, the LASSO solution provides a very simple model. This is the first study to apply the LASSO to SRBFs-based modeling of the regional gravity field in real gravity observation data.

A Data-Driven Factor Graph Model for Anchor-Based Positioning.

This work presents a data-driven factor graph (FG) model designed to perform anchor-based positioning. The system makes use of the FG to compute the target position, given the distance measurements to the anchor node that know its own position.The aim was to design a hybrid structure (that involves data and modeling approaches) to address positioning models from a Bayesian point of view, customizing them for each technology and scenario. The weighted geometric dilution of precision (WGDOP) metric, which measures the effect on the positioning solution of distance error to the corresponding anchor node and network geometry of the anchor nodes, was taken into account. The presented algorithms were tested with simulated data and also with real-life data collected from IEEE 802.15.4-compliant sensor network nodes with a physical layer based on ultra-wide band (UWB) technology, in scenarios with one target node, three and four anchor nodes, and a time-of-arrival-based range technique. The results showed that the presented algorithm based on the FG technique provided better positioning results than the least squares-based algorithms and even UWB-based commercial systems in various scenarios, with different setups in terms of geometries and propagation conditions.

Regularized Pairwise Relationship based Analytics for Structured Data

In line with the increasing machine learning model inference accuracy, deep learning (DL) models have been increasingly applied to structured data for a wide spectrum of real-world applications, including product recommendations, online advertisement, healthcare analytics and risk analysis. However, unlike unstructured data, structured data is high-dimensional and sparse and therefore engenders a large number of parameters in DL, making DL models more prone to overfitting. To alleviate the overfitting problem, various regularization methods have been designed to constrain the model parameters as a means to control the model complexity. Unfortunately, these methods are often restricted to regularizing the parameter values directly without considering the intrinsic correlations and dependencies between attribute fields of structured data which is however key to effective structured data modeling. In this paper, we re-examine DL for structured data from a new perspective of attribute interactions. In particular, we seek to explicitly model and regularize the pairwise relationships between attribute fields of structured data, in a field-adaptive manner, via a proposed attentive and interpretable framework called ATT-Reg. Specifically, in this framework, a set of attentive weight matrices are introduced to each attribute field for modeling obviously different relationships with its neighboring attribute fields. Further, we derive from the Bayesian viewpoint a novel Attentive Regularization method for imposing adaptive regularization strengths on different pairs of attribute fields, based on the informativeness of their relationship, which is calculated using both data-driven information and functional dependency (FD) knowledge. Such adaptive regularization facilitates each attribute field to learn discriminative and diversified representations for more effective predictive analytics. We also develop a feature attribution method for supporting more interpretable predictions We validate the effectiveness of our ATT-Reg on six real-world datasets. Extensive experimental results show that ATT-Reg achieves significant improvement over state-of-the-art graph models, attentive models as well as regularization methods and supports an excellent degree of interpretation.

Aerobic bioreactors: A Bayesian point of view applied to hydrodynamic characterization and experimental evaluation of tracers

This work describes the hydrodynamic diagnosis of an aerobic fixed-bed reactor, obtained by estimating parameters via the Markov chain Monte Carlo (MCMC) method. Five hydrodynamic models are evaluated. Of these models, four were classical (large dispersion, small dispersion, N-CSTR in series and plug flow with mass exchange), while the fifth is a proposal for the characterization of the short-circuit anomaly, the N-CSTR in parallel model. Due to the three-phase peculiarities of the reactor involving adsorption and dispersion phenomena, three tracers were evaluated through the stimulus–response test: eosin Y, bromophenol blue and bromocresol green. Green bromocresol is the most appropriate tracer for this type of analysis, with the hydrodynamic characterization showing that the reactor is mainly influenced by short-circuit anomaly. Parameter estimation obtained via MCMC, together with the model selection carried out by Bayesian criteria, pointed at the proposed model, n-CSTR in Parallel, as the best fit to the experimental data.

A Bayesian viewpoint on the price formation process

We introduce a simple framework in which market participants update their prior about an efficient price with a model-based learning process. We show that exponential intensities for the arrival of aggressive orders arise naturally in this setting. Our approach allows us to fully describe market dynamics in the case with Brownian efficient price and informed market takers. We are also able to revisit the emergence of market impact due to meta-order splitting, making several connections with existing literature.

A multi-dimensional non-homogeneous Markov chain of order K to jointly study multi-pollutant exceedances

In this work we consider a multivariate non-homogeneous Markov chain of order K ge 0 to study the occurrences of exceedances of environmental thresholds. In the model, d ge 1 pollutants may be observed and, according to their respective environmental thresholds, a pollutant’s concentration measurement may be considered an exceedance or not. The parameters of the model are the order of the chain, and its initial and transition distributions. These parameters are estimated under the Bayesian point of view with the maximum a posteriori and leave-one-out cross validation methods used to estimate the order. In the case of the initial and transition probabilities, the estimation is made through samples generated using their respective posterior distributions. Once these parameters are obtained, we may estimate the probability of having no, one or more pollutants exceeding the associated environmental thresholds. This is made using the Markov property as well as a recurrence formula. Results are applied to the case where d = 2 which will correspond to ozone and particulate matter with diameter smaller than 10 microns (PM_{10}) measurements obtained from the Mexico City monitoring network.

Risk measures under model uncertainty: A Bayesian viewpoint

Risk measures under model uncertainty: A Bayesian viewpoint

Bayesian Framework for a MIMO Volterra Tensor Network

This paper proposes a Bayesian Volterra tensor network (TN) to solve high-order discrete nonlinear multiple-input multiple-output (MIMO) Volterra system identification problems. Using a low-rank tensor network to compress all Volterra kernels at once, we avoid the exponential growth of monomials with respect to the order of the Volterra kernel. Our contribution is to introduce a Bayesian framework for the low-rank Volterra TN. Compared to the least squares solution for Volterra TNs, we include prior assumptions explicitly in the model. In particular, we show for the first time how a zero-mean prior with diagonal covariance matrix corresponds to implementing a Tikhonov regularization for the MIMO Volterra TN. Furthermore, adopting a Bayesian viewpoint enables simulations with Bayesian uncertainty bounds based on noise and prior assumptions. In addition, we demonstrate via numerical experiments how Tikhonov regularization prevents overfitting in the case of higher-rank TNs.

Incorporating ignorance within game theory: An imprecise probability approach

Ignorance within non-cooperative games, reflected as a player's uncertain preferences towards a game's outcome, is examined from a Bayesian point of view. This topic has had scarce treatment in the literature, which emphasises exogenous uncertainties caused by other players or nature and not by players themselves. That is primarily because a player's endogenous uncertainty over an outcome poses significant challenges and complex sequences of reciprocal expectations. Therefore, it is often ignored, and preferences are either assumed from a continuous domain or set using introspection, resulting in non-optimal models. We here explore a solution concept based on recent research in imprecise probabilities and de Finetti's approach to defining subjective probabilities, which utilises bets to assess beliefs. The resulting model allows players to be ignorant about their initial preferences and learn about them in repeated games. Furthermore, it permits improving the value of information in these situations. This model is proposed as a possible solution to the problem of utility inference in game-theoretic settings that include uncertainty over outcomes. We demonstrate it through motivating repeated-game problems modified to have uncertainty and through a simulation over a case of extreme ignorance.