In this article we consider a simple step stress set up under the cumulative exposure model assumption. At each stress level the lifetime distribution of the experimental units are assumed to follow the generalized exponential distribution. We provide the order restricted Bayesian inference of the model parameters by considering the fact that the expected lifetime of the experimental units are larger in lower stress level. Analysis and the related results are extended to different censoring schemes also. The Bayes estimates and the associated credible intervals of the unknown parameters are constructed using importance sampling technique. We perform extensive simulation experiments both for the complete and censored samples to see the performances of the proposed estimators. We analyze two simulated and one real data sets for illustrative purposes. An optimal value of the stress changing time is obtained by minimizing the total posterior coefficient of variations of the unknown parameters.
In this article, we proposed an inverse Lindley distribution and studied its fundamental properties such as quantiles, mode, stochastic ordering and entropy measure. The proposed distribution is observed to be a heavy-tailed distribution and has a upside-down bathtub shape for its failure rate. Further, we proposed its applicability as a stress-strength reliability model for survival data analysis. The estimation of stress-strength parameters and $R=P[X>Y]$, the stress-strength reliability has been approached by both classical and Bayesian paradigms. Under Bayesian set-up, non-informative (Jef…
In this paper, a Bayesian accelerated life testing model is presented. The Weibull distribution is used as the life distribution and the generalised Eyring model as the time transformation function. This is a model that allows for the use of more than one stressor, whereas other commonly used acceleration models, such as the Arrhenius and power law models, allow for only one stressor. The generalised Eyring-Weibull model is used in an application, where MCMC methods are utilised to generate samples for posterior inference.
This article revisits the problem of Bayesian shape-restricted inference in the light of a recently developed approximate Gaussian process that admits an equivalent formulation of the shape constraints in terms of the basis coefficients. We propose a strategy to efficiently sample from the resulting constrained posterior by absorbing a smooth relaxation of the constraint in the likelihood and using circulant embedding techniques to sample from the unconstrained modified prior. We additionally pay careful attention to mitigate the computational complexity arising from updating hyperparameters w…
In this paper, we consider the multicollinearity problem in the gamma regression model when model parameters are linearly restricted. The linear restrictions are available from prior information to ensure the validity of scientific theories or structural consistency based on physical phenomena. In order to make relevant statistical inference for a model any available knowledge and prior information on the model parameters should be taken into account. This paper proposes therefore an algorithm to acquire Bayesian estimator for the parameters of a gamma regression model subjected to some linear…
Squared error loss remains the most commonly used loss function for constructing a Bayes estimator of the parameter of interest. However, it can lead to sub-optimal solutions when a parameter is defined in a restricted space. It can also be an inappropriate choice in the context when an extreme overestimation and/or underestimation results in severe consequences and a more conservative estimator is preferred. We advocate a class of loss functions for parameters defined on restricted spaces which infinitely penalize boundary decisions like the squared error loss does on the real line. We also r…
For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a s…
In this paper, we investigate the statistical convergence rate of a Bayesian low-rank tensor estimator. Our problem setting is the regression problem where a tensor structure underlying the data is estimated. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a near optimal rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adapt…
Wood products that are subjected to sustained stress over a period of long duration may weaken, and this effect must be considered in models for the long-term reliability of lumber. The damage accumulation approach has been widely used for this purpose to set engineering standards. In this article, we revisit an accumulated damage model and propose a Bayesian framework for analysis. For parameter estimation and uncertainty quantification, we adopt approximation Bayesian computation (ABC) techniques to handle the complexities of the model. We demonstrate the effectiveness of our approach using …
A major problem for the learning of Bayesian networks (BNs) is the exponential number of parameters needed for conditional probability tables. Recent research reduces this complexity by modeling local structure in the probability tables. We examine the use of log-linear local models. While log-linear models in this context are not new (Whittaker, 1990; Buntine, 1991; Neal, 1992; Heckerman and Meek, 1997), for structure learning they are generally subsumed under a naive Bayes model. We describe an alternative interpretation, and use a Minimum Message Length (MML) (Wallace, 1987) metric for stru…
Solarillion Foundation（Solarillion基金会）；Global AI Accelerator, Ericsson（爱立信全球人工智能加速器）
In the recent past, psychological stress has been increasingly observed in humans, and early detection is crucial to prevent health risks. Stress detection using on-device deep learning algorithms has been on the rise owing to advancements in pervasive computing. However, an important challenge that needs to be addressed is handling unlabeled data in real-time via suitable ground truthing techniques (like Active Learning), which should help establish affective states (labels) while also selecting only the most informative data points to query from an oracle. In this paper, we propose a framewo…
The log-Lindley distribution was recently introduced in the literature as a viable alternative to the Beta distribution. This distribution has a simple structure and possesses useful theoretical properties relevant in insurance. Classical estimation methods have been well studied. We introduce estimation of parameters from Bayesian point of view for this distribution. Explicit structure of stress-strength reliability and its inference under both classical and Bayesian set-up is addressed. Extensive simulation studies show marked improvement with Bayesian approach over classical given reasonabl…
Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and t…
RIKEN Center for Advanced Intelligence Project（日本理化学研究所先进智能项目中心）；Institute for Physics of Intelligence（智能物理研究所）；The University of Tokyo（东京大学）
Restricted Boltzmann machines (RBMs) with low-precision synapses are much appealing with high energy efficiency. However, training RBMs with binary synapses is challenging due to the discrete nature of synapses. Recently Huang proposed one efficient method to train RBMs with binary synapses by using a combination of gradient ascent and the message passing algorithm under the variational inference framework. However, additional heuristic clipping operation is needed. In this technical note, inspired from Huang's work , we propose one alternative optimization method using the Bayesian learning r…
Moderation effects and elasticities in compositional regression with a total. Application to Bayesian spatiotemporal modelling of all-cause mortality from environmental stressors
The recently introduced class of simultaneous graphical dynamic linear models (SGDLMs) defines an ability to scale on-line Bayesian analysis and forecasting to higher-dimensional time series. This paper advances the methodology of SGDLMs, developing and embedding a novel, adaptive method of simultaneous predictor selection in forward filtering for on-line learning and forecasting. The advances include developments in Bayesian computation for scalability, and a case study in exploring the resulting potential for improved short-term forecasting of large-scale volatility matrices. A case study co…
This paper considers reparameterization invariant Bayesian point estimates and credible regions of model parameters for scientific inference and communication. The effect of intrinsic loss function choice in Bayesian intrinsic estimates and regions is studied with the following findings. A particular intrinsic loss function, using Kullback-Leibler divergence from the full model to the restricted model, has strong connection to a Bayesian predictive criterion, which produces point estimates with the best predictive performance. An alternative intrinsic loss function, using Kullback-Leibler dive…
Duke University（杜克大学）；Iowa State University（爱荷华州立大学）
A restricted Boltzmann machine (RBM) is an undirected graphical model constructed for discrete or continuous random variables, with two layers, one hidden and one visible, and no conditional dependency within a layer. In recent years, RBMs have risen to prominence due to their connection to deep learning. By treating a hidden layer of one RBM as the visible layer in a second RBM, a deep architecture can be created. RBMs are thought to thereby have the ability to encode very complex and rich structures in data, making them attractive for supervised learning. However, the generative behavior of …
Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers $π$, $e$, $\sqrt2$, and $\ln{2}$) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four…
In many applications, it is of interest to assess the dependence structure in multivariate longitudinal data. Discovering such dependence is challenging due to the dimensionality involved. By concatenating the random effects from component models for each response, dependence within and across longitudinal responses can be characterized through a large random effects covariance matrix. Motivated by the common problems in estimating this matrix, especially the off-diagonal elements, we propose a Bayesian approach that relies on shrinkage priors for parameters in a modified Cholesky decompositio…
In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (Lindley's paradox). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are "estimated" by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding r…
For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-α)%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the restricted parameter space. Namely, they establish the attractive lower bound of $\frac{1-α}{1+α}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-α}{1+α}$ is applicable for a substantially more general setting with underlying distributional symmetry, and obtain various other properties. The deriv…
In this paper, we are interested in evaluating the resilience of financial portfolios under extreme economic conditions. Therefore, we use empirical measures to characterize the transmission process of macroeconomic shocks to risk parameters. We propose the use of an extensive family of models, called General Transfer Function Models, which condense well the characteristics of the transmission described by the impact measures. The procedure for estimating the parameters of these models is described employing the Bayesian approach and using the prior information provided by the impact measures.…
Graz University of Technology（格拉茨工业大学）；Graz Center for Computational Engineering（格拉茨计算工程中心）；University Medicine Essen（埃森大学医学院）；Norwegian University of Science and Technology (NTNU)（挪威科技大学）
Inhomogeneities in the aortic wall can lead to localized stress accumulations, possibly initiating dissection. In many cases, a dissection results from pathological changes such as fragmentation or loss of elastic fibers. But it has been shown that even the healthy aortic wall has an inherent heterogeneous microstructure. Some parts of the aorta are particularly susceptible to the development of inhomogeneities due to pathological changes, however, the distribution in the aortic wall and the spatial extent, such as size, shape, and type, are difficult to predict. Motivated by this observation,…
We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian …
Under adaptive progressive Type-II censoring schemes, order restricted inference based on competing risks data is discussed in this article. The latent failure lifetimes for the competing causes are assumed to follow Weibull distributions, with an order restriction on the scale parameters of the distributions. The practical implication of this order restriction is that one of the risk factors is dominant, as often observed in competing risks scenarios. In this setting, likelihood estimation for the model parameters, along with bootstrap based techniques for constructing asymptotic confidence i…
In this paper, we propose a general framework for combining evidence of varying quality to estimate underlying binary latent variables in the presence of restrictions imposed to respect the scientific context. The resulting algorithms cluster the multivariate binary data in a manner partly guided by prior knowledge. The primary model assumptions are that 1) subjects belong to classes defined by unobserved binary states, such as the true presence or absence of pathogens in epidemiology, or of antibodies in medicine, or the "ability" to correctly answer test questions in psychology, 2) a binary …
Using a state-space system, I forecasted the US Treasury yields by employing frequentist and Bayesian methods after first decomposing the yields of varying maturities into its unobserved term structure factors. Then, I exploited the structure of the state-space model to forecast the Treasury yields and compared the forecast performance of each model using mean squared forecast error. Among the frequentist methods, I applied the two-step Diebold-Li, two-step principal components, and one-step Kalman filter approaches. Likewise, I imposed the five different priors in Bayesian VARs: Diffuse, Minn…
Heckman selection model is the most popular econometric model in analysis of data with sample selection. However, selection models with Normal errors cannot accommodate heavy tails in the error distribution. Recently, Marchenko and Genton proposed a selection-t model to perform frequentist' robust analysis of sample selection. Instead of using their maximum likelihood estimates, our paper develops new Bayesian procedures for the selection-t models with either continuous or binary outcomes. By exploiting the Normal mixture representation of the t distribution, we can use data augmentation to im…
We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically in this paper, we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between different number of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split-merge propo…
In this paper, we show how a complete and exact Bayesian analysis of a parametric mixture model is possible in some cases when components of the mixture are taken from exponential families and when conjugate priors are used. This restricted set-up allows us to show the relevance of the Bayesian approach as well as to exhibit the limitations of a complete analysis, namely that it is impossible to conduct this analysis when the sample size is too large, when the data are not from an exponential family, or when priors that are more complex than conjugate priors are used.
Computer experiments can emulate the physical systems, help computational investigations, and yield analytic solutions. They have been widely employed with many engineering applications (e.g., aerospace, automotive, energy systems. Conventional Bayesian optimization did not incorporate the nested structures in computer experiments. This paper proposes a novel nested Bayesian optimization for complex computer experiments with multi-step or hierarchical characteristics. We prove the theoretical properties of nested outputs given two cases: Gaussian or non-Gaussian. The closed forms of nested exp…
The R package BiDAG implements Markov chain Monte Carlo (MCMC) methods for structure learning and sampling of Bayesian networks. The package includes tools to search for a maximum a posteriori (MAP) graph and to sample graphs from the posterior distribution given the data. A new hybrid approach to structure learning enables inference in large graphs. In the first step, we define a reduced search space by means of the PC algorithm or based on prior knowledge. In the second step, an iterative order MCMC scheme proceeds to optimize within the restricted search space and estimate the MAP graph. Sa…
We consider the problem of shape restricted nonparametric regression on a closed set X ?\in R; where it is reasonable to assume the function has no more than H local extrema interior to X: Following a Bayesian approach we develop a nonparametric prior over a novel class of local extrema splines. This approach is shown to be consistent when modeling any continuously differentiable function within the class of functions considered, and is used to develop methods for hypothesis testing on the shape of the curve. Sampling algorithms are developed, and the method is applied in simulation studies an…
The equations of a physical constitutive model for material stress within tantalum grains were solved numerically using a tetrahedrally meshed volume. The resulting output included a scalar vonMises stress for each of the more than 94,000 tetrahedra within the finite element discretization. In this paper, we define an intricate statistical model for the spatial field of vonMises stress which uses the given grain geometry in a fundamental way. Our model relates the three-dimensional field to integrals of latent stochastic processes defined on the vertices of the one- and two-dimensional grain b…
