Bayesian Point Of View Research Articles

Sequential Multiple Model Filtering with Interrupted Measurements

This paper considers the joint detection and filtering problem of discrete-time stochastic systems when the measurements are interrupted in a random fashion. By formulating the measurement interruptions into two-state Markov chains, a sequential multiple model filter is developed from the Bayesian point of view. With a soft switching mechanism, the proposed filter automatically abandons the useless measurements in the interrupted time intervals, and captures the correct measurements for recursive estimation. Compared with the widely used interacting multiple model filter, the new filter has a more simple structure and requires less time for computation. A numerical example shows that the proposed multiple model filter can effectively solve the target tracking problem with interrupted range measurements.

Inverted exponential distribution as a life distribution model from a Bayesian viewpoint

The CODATA Data Science Journal is a peer-reviewed, open access, electronic journal, publishing papers on the management, dissemination, use and reuse of research data and databases across all research domains, including science, technology, the humanities and the arts. The scope of the journal includes descriptions of data systems, their implementations and their publication, applications, infrastructures, software, legal, reproducibility and transparency issues, the availability and usability of complex datasets, and with a particular focus on the principles, policies and practices for open data.All data is in scope, whether born digital or converted from other sources.

Decision theoretic Bayesian hypothesis testing with the selection goal

Consider a probability model P θ,α , where θ=(θ 1 ,θ 2 ,…,θ k ) T is a parameter vector of interest, and α is some nuisance parameter. The problem of testing null hypothesis H 0 : θ 1 =θ 2 =…=θ k against selecting one of k alternative hypotheses H i :θ i =θ [ k ] > θ [1] , i =1,2,…, k , where θ [ k ] =max{θ 1 ,θ 2 ,…,θ k } and θ [1] =min{θ 1 ,θ 2 ,…,θ k }, is formulated from a Bayesian decision theoretic point of view. This problem can be viewed as selecting a component with the largest parameter value if the null hypothesis is rejected. General results are obtained for the Bayes rule under monotonic permutation invariant loss functions. Bayes rules are obtained for k one-parameter exponential families of distributions under conjugate priors. The example of normal populations is considered in more detail under the non-informative (improper) priors. It is demonstrated through this example that the classical hypothesis testing yields a poor power as compared to the Bayes rules when the alternatives are such that a small fraction of the components of θ have significantly high values while most of them have low values. Consequences of this for the high dimensional data such as microarray data are pointed out.

Some Asymptotic Results on Generalized Penalized Spline Smoothing

The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models (GLMM). We consider the asymptotic rates such that Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a full Bayesian view point by imposing a priori distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coeffocients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov Chain Monte Carlo (MCMC) results with their asymptotic approximation in a simulation study.

Two-way imputation: A Bayesian method for estimating missing scores in tests and questionnaires, and an accurate approximation

Previous research has shown that method two-way with error for multiple imputation in test and questionnaire data produces small bias in statistical analyses. This method is based on a two-way ANOVA model of persons by items but it is improper from a Bayesian point of view. Proper two-way imputations are generated using data augmentation. Simulation results show that the resulting method two-way with data augmentation produces unbiased results in Cronbach's alpha, the mean of squares in ANOVA, the item means, and small bias in the mean test score and the factor loadings from principal components analysis. The data with imputed scores result in statistics having a slightly larger standard deviation than the original complete data. Method two-way with error produces results that are only slightly more biased, especially for low percentages of missingness. Thus, it may serve as an accurate approximation to the more involved method two-way with data augmentation.

Bayesian Interpretation of a Distance Function for Navigating High-Dimensional Descriptor Spaces

A distance function to analyze molecular similarity relationships in high-dimensional descriptor spaces and focus search calculations on "active subspaces" is defined in Bayesian terms. As a measure of similarity, database compounds are ranked according to their distance from the center of a subspace formed by known active molecules. From a Bayesian point of view, distance calculations are transformed into a "log-odds" estimate. Following this approach, maximizing the likelihood of a compound to be active corresponds to minimizing the distance from the center of an active subspace. Since the methodology generates a ranking of database molecules according to decreasing similarity to template compounds, it can be conveniently compared to similarity search tools, and the Bayesian function is found to compare favorably to two standard fingerprints in multiple template-based database searching.

A Condition to Obtain the Same Decision in the Homogeneity Testing Problem from the Frequentist and Bayesian Point of View

We develop a Bayesian procedure for the homogeneity testing problem of r populations using r × s contingency tables. The posterior probability of the homogeneity null hypothesis is calculated using a mixed prior distribution. The methodology consists of choosing an appropriate value of π0 for the mass assigned to the null and spreading the remainder, 1 − π0, over the alternative according to a density function. With this method, a theorem which shows when the same conclusion is reached from both frequentist and Bayesian points of view is obtained. A sufficient condition under which the p-value is less than a value α and the posterior probability is also less than 0.5 is provided.

Bayesian Network Learning with Parameter Constraints

The task of learning models for many real-world problems requires incorporating domain knowledge into learning algorithms, to enable accurate learning from a realistic volume of training data. This paper considers a variety of types of domain knowledge for constraining parameter estimates when learning Bayesian networks. In particular, we consider domain knowledge that constrains the values or relationships among subsets of parameters in a Bayesian network with known structure. We incorporate a wide variety of parameter constraints into learning procedures for Bayesian networks, by formulating this task as a constrained optimization problem. The assumptions made in module networks, dynamic Bayes nets and context specific independence models can be viewed as particular cases of such parameter constraints. We present closed form solutions or fast iterative algorithms for estimating parameters subject to several specific classes of parameter constraints, including equalities and inequalities among parameters, constraints on individual parameters, and constraints on sums and ratios of parameters, for discrete and continuous variables. Our methods cover learning from both frequentist and Bayesian points of view, from both complete and incomplete data. We present formal guarantees for our estimators, as well as methods for automatically learning useful parameter constraints from data. To validate our approach, we apply it to the domain of fMRI brain image analysis. Here we demonstrate the ability of our system to first learn useful relationships among parameters, and then to use them to constrain the training of the Bayesian network, resulting in improved cross-validated accuracy of the learned model. Experiments on synthetic data are also presented.

On model selection in Bayesian regression

We discuss the problem of constructing a suitable regression model from a nonparametric Bayesian viewpoint. For this purpose, we consider the case when the error terms have symmetric and unimodal densities. By the Khintchine and Shepp theorem, the density of response variable can be written as a scale mixture of uniform densities. The mixing distribution is assumed to have a Dirichlet process prior. We further consider appropriate prior distributions for other parameters as the components of the predictive device. Among the possible submodels, we select the one which has the highest posterior probability. An example is given to illustrate the approach.

The Estimation Error in the Chain-Ladder Reserving Method: A Bayesian Approach

We derive the estimation error in a Bayesian framework and discuss the estimates of Mack and of Buchwalder, Bühlmann, Merz and Wüthrich (BBMW) from a Bayesian point of view.

The Estimation Error in the Chain-Ladder Reserving Method: A Bayesian Approach

We derive the estimation error in a Bayesian framework and discuss the estimates of Mack and of Buchwalder, Bühlmann, Merz and Wüthrich (BBMW) from a Bayesian point of view.

Probability biases as Bayesian inference

AbstractIn this article, I will show how several observed biases in human probabilistic reasoning can be partially explained as good heuristics for making inferences in an environment where probabilities have uncertainties associated to them. Previous results show that the weight functions and the observed violations of coalescing and stochastic dominance can be understood from a Bayesian point of view. We will review those results and see that Bayesian methods should also be used as part of the explanation behind other known biases. That means that, although the observed errors are still errors under the laboratory conditions in which they are demonstrated, they can be understood as adaptations to the solution of real life problems. Heuristics that allow fast evaluations and mimic a Bayesian inference would be an evolutionary advantage, since they would give us an efficient way of making decisions.

Trans-dimensional inverse problems, model comparison and the evidence

Summary In most geophysical inverse problems the properties of interest are parametrized using a fixed number of unknowns. In some cases arguments can be used to bound the maximum number of parameters that need to be considered. In others the number of unknowns is set at some arbitrary value and regularization is used to encourage simple, non-extravagant models. In recent times variable or self-adaptive parametrizations have gained in popularity. Rarely, however, is the number of unknowns itself directly treated as an unknown. This situation leads to a transdimensional inverse problem, that is, one where the dimension of the parameter space is a variable to be solved for. This paper discusses trans-dimensional inverse problems from the Bayesian viewpoint. A particular type of Markov chain Monte Carlo (MCMC) sampling algorithm is highlighted which allows probabilistic sampling in variable dimension spaces. A quantity termed the evidence or marginal likelihood plays a key role in this type of problem. It is shown that once evidence calculations are performed, the results of complex variable dimension sampling algorithms can be replicated with simple and more familiar fixed dimensional MCMC sampling techniques. Numerical examples are used to illustrate the main points. The evidence can be difficult to calculate, especially in high-dimensional non-linear inverse problems. Nevertheless some general strategies are discussed and analytical expressions given for certain linear problems.

Suboptimal estimation of signals from uncertain observations using approximations of mixtures

In this paper, we propose a suboptimal filtering algorithm for the estimation of a Gaussian signal from uncertain observations, using covariance information. The suboptimal estimators are obtained as the expectation of the signal, given the observations, when a certain approximation is considered for the conditional distribution. The approximation is carried out via successive approximations of mixtures of Gaussian distributions by Gaussian distributions. On the other hand, by assuming that the uncertainty probability is unknown, a recursive estimation algorithm is proposed for that probability. This algorithm is obtained under a Bayesian viewpoint; specifically, by considering a Beta as the a priori distribution for the unknown parameter, the proposed estimators provide approximations for the mean of the a posteriori distributions.

A Comparison of Rarefaction and Bayesian Methods for Predicting the Allelic Richness of Future Samples on the Basis of Currently Available Samples

Rarefaction methods have been introduced into population genetics (from ecology) for predicting and comparing the allelic richness of future samples (or sometimes populations) on the basis of currently available samples, possibly of different sizes. Here, we focus our attention on one such problem: Predicting which population is most likely to yield the future sample having the highest allelic richness. (This problem can arise when we want to construct a core collection from a larger germplasm collection.) We use extensive simulations to compare the performance of the Monte Carlo rarefaction (repeated random subsampling) method with a simple Bayesian approach we have developed-which is based on the Ewens sampling distribution. We found that neither this Bayesian method nor the (Monte Carlo) rarefaction method performed uniformly better than the other. We also examine briefly some of the other motivations offered for these methods and try to make sense of them from a Bayesian point of view.

Nonlinear local electrovascular coupling. II: From data to neuronal masses

In the companion article a local electrovascular coupling (LEVC) model was proposed to explain the continuous dynamics of electrical and vascular states within a cortical unit. These states produce certain mesoscopic reflections whose discrete time series can be reconstructed from electroencephalography (EEG) and functional magnetic resonance imaging (fMRI). In this article we develop a recursive optimization algorithm based on the local linearization (LL) filter and an innovation method to make statistical inferences about the LEVC model from both EEG and fMRI data, i.e., to estimate the unobserved states and the unknown parameters of the model. For a better understanding, the LL filter is described from a Bayesian point of view, providing the particulars for the case of hybrid data (e.g., EEG and fMRI), which could be sampled at different rates. The dynamics of the exogenous synaptic inputs going into the cortical unit are also estimated by introducing a set of Gaussian radial basis functions. In order to study the dynamics of the electrical and vascular states in the striate cortex of humans as well as their local interrelationships, we applied this algorithm to EEG and fMRI recordings obtained concurrently from two subjects while passively observing a radial checkerboard with a white/black pattern reversal. The EEG and fMRI data from the first subject was used to estimate the electrical/vascular states and parameters of the LEVC model in V1 for a 4.0 Hz reversion frequency. We used the EEG data from the second subject to investigate the changes in the dynamics of the electrical states when the frequency of reversion is varied from 0.5-4.0 Hz. Then we made use of the estimated electrical states to predict the effects on the vasculature that such variations produce.

Accounting for threshold uncertainty in extreme value estimation

Tail data are often modelled by fitting a generalized Pareto distribution (GPD) to the exceedances over high thresholds. In practice, a threshold \(u\) is fixed and a GPD is fitted to the data exceeding \(u\). A difficulty in this approach is the selection of the threshold above which the GPD assumption is appropriate. Moreover the estimates of the parameters of the GPD may depend significantly on the choice of the threshold selected. Sensitivity with respect to the threshold choice is normally studied but typically its effects on the properties of estimators are not accounted for. In this paper, to overcome the difficulties of the fixed-threshold approach, we propose to model extreme and non-extreme data with a distribution composed of a piecewise constant density from a low threshold up to an unknown end point \(alpha\) and a GPD with threshold \(alpha\) for the remaining tail part. Since we estimate the threshold together with the other parameters of the GPD we take naturally into account the threshold uncertainty. We will discuss this model from a Bayesian point of view and the method will be illustrated using simulated data and a real data set.

A flexible Bayesian algorithm for sample size calculations in misclassified data

The problem of obtaining a flexible and easy to implement algorithm in order to derive the optimal sample size when the data are subject to misclassification is critical to practitioners. The topic is addressed from the Bayesian point of view where a special structure of the a priori parameter information is investigated. The proposed methodology is applied in specific examples.

Bayesian estimation and prediction for some life distributions based on record values

Some statistical data are most easily accessed in terms of record values. Examples include meteorology, hydrology and athletic events. Also, there are a number of industrial situations where experimental outcomes are a sequence of record-breaking observations. In this paper, Bayesian estimation for the two parameters of some life distributions, including Exponential, Weibull, Pareto and Burr type XII, are obtained based on upper record values. Prediction, either point or interval, for future upper record values is also presented from a Bayesian view point. Some of the non-Bayesian results can be achieved as limiting cases from our results. Numerical computations are given to illustrate the results.

Two-view multibody structure-and-motion with outliers through model selection

Multibody structure-and-motion (MSaM) is the problem to establish the multiple-view geometry of several views of a 3D scene taken at different times, where the scene consists of multiple rigid objects moving relative to each other. We examine the case of two views. The setting is the following: Given are a set of corresponding image points in two images, which originate from an unknown number of moving scene objects, each giving rise to a motion model. Furthermore, the measurement noise is unknown, and there are a number of gross errors, which are outliers to all models. The task is to find an optimal set of motion models for the measurements. It is solved through Monte-Carlo sampling, careful statistical analysis of the sampled set of motion models, and simultaneous selection of multiple motion models to best explain the measurements. The framework is not restricted to any particular model selection mechanism because it is developed from a Bayesian viewpoint: Different model selection criteria are seen as different priors for the set of moving objects, which allow one to bias the selection procedure for different purposes.