Bayesian Point Of View Research Articles

Bayesian forecasting of temporal gene expression by using an autoregressive panel data approach.

We propose and evaluate a novel approach for forecasting gene expression over non-observed times in longitudinal trials under a Bayesian viewpoint. One of the aims is to cluster genes that share similar expression patterns over time and then use this similarity to predict relative expression at time points of interest. Expression values of 106 genes expressed during the cell cycle of Saccharomyces cerevisiae were used and genes were partitioned into five distinct clusters of sizes 33, 32, 21, 16, and 4. After removing the last observed time point, the agreements of signals (upregulated or downregulated) considering the predicted expression level were 72.7, 81.3, 76.2, 68.8, and 50.0%, respectively, for each cluster. The percentage of credibility intervals that contained the true values of gene expression for a future time was ~90%. The methodology performed well, providing a valid forecast of gene expression values by fitting an autoregressive panel data model. This approach is easily implemented with other time-series models and when Poisson and negative binomial probability distributions are assumed for the gene expression data.

A Sparse Bayesian Approach to the Identification of Nonlinear State-Space Systems

This technical note considers the identification of nonlinear discrete-time systems with additive process noise but without measurement noise. In particular, we propose a method and its associated algorithm to identify the system nonlinear functional forms and their associated parameters from a limited number of time-series data points. For this, we cast this identification problem as a sparse linear regression problem and take a Bayesian viewpoint to solve it. As such, this approach typically leads to nonconvex optimisations. We propose a convexification procedure relying on an efficient iterative re-weighted $\ell_1$-minimisation algorithm that uses general sparsity inducing priors on the parameters of the system and marginal likelihood maximisation. Using this approach, we also show how convex constraints on the parameters can be easily added to our proposed iterative re-weighted $\ell_1$-minimisation algorithm. In the supplementary material \cite{appendix}, we illustrate the effectiveness of the proposed identification method on two classical systems in biology and physics, namely, a genetic repressilator network and a large scale network of interconnected Kuramoto oscillators.

Relaxed Poisson cure rate models.

The purpose of this article is to make the standard promotion cure rate model (Yakovlev and Tsodikov, ) more flexible by assuming that the number of lesions or altered cells after a treatment follows a fractional Poisson distribution (Laskin, ). It is proved that the well-known Mittag-Leffler relaxation function (Berberan-Santos, ) is a simple way to obtain a new cure rate model that is a compromise between the promotion and geometric cure rate models allowing for superdispersion. So, the relaxed cure rate model developed here can be considered as a natural and less restrictive extension of the popular Poisson cure rate model at the cost of an additional parameter, but a competitor to negative-binomial cure rate models (Rodrigues etal., ). Some mathematical properties of a proper relaxed Poisson density are explored. A simulation study and an illustration of the proposed cure rate model from the Bayesian point of view are finally presented.

Bayesian factor analysis with uncertain functional constraints about factor loadings

Factor analysis with uncertain functional constraints about factor loading matrix is considered from a Bayesian viewpoint, in which the uncertain prior information is incorporated in the analysis. We propose a hierarchical screened scale mixture of normal factor ( HSMF ) model for flexible inference of the constrained factor loadings, factor scores, and specific variances as well as the covariance matrix of the factors. The proposed model makes provisions for robust factor analysis with uncertainty about the functional constraints. A number of inferential aspects of the proposed model are investigated in order to render the proposed analysis optimal. These include the closure properties of a class of rectangle-screened scale mixture of multivariate normal ( R S M N ) distributions which is useful for statistical inference of the HSMF model, eliciting the prior and posterior evolutions of the uncertainly constrained factor loadings, and providing the efficient Bayesian estimation procedure by using the MCMC methods. Empirical analysis for Bayesian factor models with synthetic data and real data applications is given to illustrate the usefulness of the proposed model.

Towards a hierarchical parametrization to address prior uncertainty in ensemble-based data assimilation

Ensemble-based methods are becoming popular assisted history matching techniques with a growing number of field applications. These methods use an ensemble of model realizations, typically constructed by means of geostatistics, to represent the prior uncertainty. The performance of the history matching is very dependent on the quality of the initial ensemble. However, there is a significant level of uncertainty in the parameters used to define the geostatistical model. From a Bayesian viewpoint, the uncertainty in the geostatistical modeling can be represented by a hyper-prior in a hierarchical formulation. This paper presents the first steps towards a general parametrization to address the problem of uncertainty in the prior modeling. The proposed parametrization is inspired in Gaussian mixtures, where the uncertainty in the prior mean and prior covariance is accounted by defining weights for combining multiple Gaussian ensembles, which are estimated during the data assimilation. The parametrization was successfully tested in a simple reservoir problem where the orientation of the major anisotropic direction of the permeability field was unknown.

Tolerance limits for minimal repair times of a series system with Rayleigh distributed component lifetimes

Assuming that component lifetimes are Rayleigh distributed, guaranteed-coverage and expected-coverage tolerance limits are computed for minimal repair times of series systems. In the frequentist setting, the empirical information about the unknown Rayleigh parameter is provided by a type-II right censored sample of component lifetimes. A Bayesian approach for constructing tolerance limits is also proposed, which may be viewed as a generalization of the frequentist viewpoint since classical and Bayesian perspectives harmonize in the noninformative case. Moreover, the accuracy of a given tolerance limit and the number of failures needed to attain a desired accuracy level are derived. Bayesian tolerance limits are shown to be more accurate than the corresponding frequentist ones, especially when prior knowledge is relevant. Optimal sample sizes for constructing tolerance limits are determined by minimizing appropriate cost functions. In general, the Bayesian point of view allows the reliability engineer to shorten test duration and save testing costs. A numerical example is presented for illustrative and comparative purposes.

Bayesian estimation and prediction based on generalized Type-II hybrid censored sample

ABSTRACTIn this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.

A Bayesian Predictive Discriminant Analysis with Screened Data

In the application of discriminant analysis, a situation sometimes arises where individual measurements are screened by a multidimensional screening scheme. For this situation, a discriminant analysis with screened populations is considered from a Bayesian viewpoint, and an optimal predictive rule for the analysis is proposed. In order to establish a flexible method to incorporate the prior information of the screening mechanism, we propose a hierarchical screened scale mixture of normal (HSSMN) model, which makes provision for flexible modeling of the screened observations. An Markov chain Monte Carlo (MCMC) method using the Gibbs sampler and the Metropolis–Hastings algorithm within the Gibbs sampler is used to perform a Bayesian inference on the HSSMN models and to approximate the optimal predictive rule. A simulation study is given to demonstrate the performance of the proposed predictive discrimination procedure.

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.

Thalamocortical Dysrhythmia: A Theoretical Update in Tinnitus

Tinnitus is the perception of a sound in the absence of a corresponding external sound source. Pathophysiologically it has been attributed to bottom-up deafferentation and/or top-down noise-cancelling deficit. Both mechanisms are proposed to alter auditory ­thalamocortical signal transmission, resulting in thalamocortical dysrhythmia (TCD). In deafferentation, TCD is characterized by a slowing down of resting state alpha to theta activity associated with an increase in surrounding gamma activity, resulting in persisting cross-frequency coupling between theta and gamma activity. Theta burst-firing increases network synchrony and recruitment, a mechanism, which might enable long-range synchrony, which in turn could represent a means for finding the missing thalamocortical information and for gaining access to consciousness. Theta oscillations could function as a carrier wave to integrate the tinnitus-related focal auditory gamma activity in a consciousness enabling network, as envisioned by the global workspace model. This model suggests that focal activity in the brain does not reach consciousness, except if the focal activity becomes functionally coupled to a consciousness enabling network, aka the global workspace. In limited deafferentation, the missing information can be retrieved from the auditory cortical neighborhood, decreasing surround inhibition, resulting in TCD. When the deafferentation is too wide in bandwidth, it is hypothesized that the missing information is retrieved from theta-mediated parahippocampal auditory memory. This suggests that based on the amount of deafferentation TCD might change to parahippocampocortical persisting and thus pathological theta–gamma rhythm. From a Bayesian point of view, in which the brain is conceived as a prediction machine that updates its memory-based predictions through sensory updating, tinnitus is the result of a prediction error between the predicted and sensed auditory input. The decrease in sensory updating is reflected by decreased alpha activity and the prediction error results in theta–gamma and beta–gamma coupling. Thus, TCD can be considered as an adaptive mechanism to retrieve missing auditory input in tinnitus.

Tracking rapid intracellular movements: A Bayesian random set approach

We focus on the biological problem of tracking organelles as they move through cells. In the past, most intracellular movements were recorded manually, however, the results are too incomplete to capture the full complexity of organelle motions. An automated tracking algorithm promises to provide a complete analysis of noisy microscopy data. In this paper, we adopt statistical techniques from a Bayesian random set point of view. Instead of considering each individual organelle, we examine a random set whose members are the organelle states and we establish a Bayesian filtering algorithm involving such set states. The propagated multi-object densities are approximated using a Gaussian mixture scheme. Our algorithm is applied to synthetic and experimental data.

On Bayesian Identification of Autoregressive Processes

The main objective of the current study is to handle the identification problem of autoregressive processes from the Bayesian point of view. Two Bayesian identification approaches are considered. They are referred to as the direct and the indirect approaches. The two approaches are employed to solve the Bayesian identification problem of autoregressive processes using three well known priors. These priors are the G prior, the Natural-Conjugate prior and Jeffrey's prior. The theoretical derivations related to the two Bayesian identification approaches are conducted using the above mentioned priors. Moreover, the performance of the two techniques, using each of the three priors, is investigated via comprehensive simulation studies. Simulation results show that the two techniques are adequate to solve the identification problem of autoregressive processes. The increase in the time series length leads to better performance for each technique. The use of different priors doesn't affect the previous results.

Admissible minimax estimators for the shape parameter of Topp–Leone distribution

The shape parameter of Topp–Leone distribution is estimated in this article from the Bayesian viewpoint under the assumption of known scale parameter. Bayes and empirical Bayes estimates of the unknown parameter are proposed under non informative and suitable conjugate priors. These estimates are derived under the assumption of squared and linear-exponential error loss functions. The risk functions of the proposed estimates are derived in analytical forms. It is shown that the proposed estimates are minimax and admissible. The consistency of the proposed estimates under the squared error loss function is also proved. Numerical examples are provided.

Weighted exponential distribution: properties and different methods of estimation

In this article, we investigate various properties and methods of estimation of the Weighted Exponential distribution. Although, our main focus is on estimation (from both frequentist and Bayesian point of view) yet, the stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics are derived first time for the said distribution. Different types of loss functions are considered for Bayesian estimation. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using Gibbs sampling. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and two real data sets have been analysed for illustrative purposes.

A Probabilistic Approach for Spring Recession Flows Analysis

Spring recession flows are analyzed from a Bayesian point of view. Two general equations are derived and it is shown that the classical formulas of recession flow are particular cases of both equations. It is shown that most of the recession equations reflect a non-Markovian process. That means that the groundwater storage exhibits a memory effect and that there is a nonlinear relationship between flow and storage. The Bayesian approach presented in this paper makes it possible to give a probabilistic meaning to recession flow equations derived according to a physical approach and can be an alternative to the study of complex reservoir for which the physical processes governing recession flow are unclear. Twelve spring recession flow series are analysed in order to validate the probabilistic approach presented in this paper and a conceptual model of storage-outflow is proposed.

Comparison of the Sampling Efficiency in Spatial Autoregressive Model

A random walk Metropolis-Hastings algorithm has been widely used in sampling the parameter of spatial interaction in spatial autoregressive model from a Bayesian point of view. In addition, as an alternative approach, the griddy Gibbs sampler is proposed by [1] and utilized by [2]. This paper proposes an acceptance-rejection Metropolis-Hastings algorithm as a third approach, and compares these three algorithms through Monte Carlo experiments. The experimental results show that the griddy Gibbs sampler is the most efficient algorithm among the algorithms whether the number of observations is small or not in terms of the computation time and the inefficiency factors. Moreover, it seems to work well when the size of grid is 100.

국면전환 임계 자기회귀 분석을 위한 베이지안 방법 비교연구

Autoregressive models are used to analyze an univariate time series data; however, these methods can be inappropriate when a structural break appears in a time series since they assume that a trend is consistent. Threshold autoregressive models (popular regime-switching models) have been proposed to address this problem. Recently, the models have been extended to two regime-switching models with delay parameter. We discuss two regime-switching threshold autoregressive models from a Bayesian point of view. For a Bayesian analysis, we consider a parametric threshold autoregressive model and a nonparametric threshold autoregressive model using Dirichlet process prior. The posterior distributions are derived and the posterior inferences is performed via Markov chain Monte Carlo method and based on two Bayesian threshold autoregressive models. We present a simulation study to compare the performance of the models. We also apply models to gross domestic product data of U.S.A and South Korea.

Bayesian Sequential Track Formation

This paper presents a theoretical framework for track building in multiple-target scenarios from the Bayesian point of view. It is assumed that the number of targets is fixed and known. We propose two optimal methods for building tracks sequentially. The first one uses the labelling of the current multitarget state estimate that minimizes the mean-square labeled optimal subpattern assignment error. This method requires knowledge of the posterior density of the vector-valued state. The second assigns the labeling that maximizes the probability that the current multi-target state estimate is optimally linked with the available tracks at the previous time step. In this case, we only require knowledge of the random finite-set posterior density without labels.

Varying dimensional Bayesian acoustic waveform inversion for 1D semi-infinite heterogeneous media

This paper introduces a methodology to infer the spatial variation of the acoustic characteristics of a 1D vertical elastic heterogeneous earth model via a Bayesian calibration approach, given a prescribed sequence of loading and the corresponding time history response registered at the ground level. This involves solving an inverse problem that maps the ground seismic response onto a random profile of the ground stratigraphy (i.e. a 1D continuous spatial random field). From a Bayesian point of view, the solution to an inverse problem is fully characterized by a posterior density function of the forward model random parameters, which explicitly overcomes the solution's non-uniqueness. This subsurface earth model is parameterized using a Bayesian partition model, where the number of soil layers, the location of the layers' interfaces, and their corresponding mechanical characteristics are defined as random variables. The partition model approach to an inverse problem is closely related to a Bayesian model selection problem, where the likely dimensionality of the inverse problem (number of unknowns) is inferred conditioned on the experimental observations. The main benefit of the proposed approach is that the explicit regularization of the inverted profile by global damping procedures is not required. A Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm is used to sample the target posterior of varying dimension, dependent on the number of layers. A synthetic case study is provided to indicate the applicability of the proposed methodology.

A non-homogeneous poisson model with spatial anisotropy applied to ozone data from Mexico City

In this work we consider a non-homogenous Poisson model to study the behaviour of the number of times that a pollutant’s concentration surpasses a given threshold of interest. Spatial dependence is imposed on the parameters of the Poisson intensity function in order to account for the possible correlation between measurements in different sites. An anisotropic model is used due to the nature of the region of interest. Estimation of the parameters of the model is performed using the Bayesian point of view via Markov chain Monte Carlo (MCMC) algorithms. We also consider prediction of the days in which exceedances of the threshold might occur at sites where measurements cannot be taken. This is obtained by spatial interpolation using the information provided by the sites where measurements are available. The prediction procedure allows for estimation of the behaviour of the mean function of the non-homogeneous Poisson process associated with those sites. The models considered here are applied to ozone data obtained from the monitoring network of Mexico City.