Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging

In this article we present a Bayesian prediction of multiplicative seasonal autoregressive moving average (SARMA) processes using the Gibbs sampling algorithm. First, we estimate the unobserved errors using the nonlinear least squares (NLS) method to approximate the likelihood function. Second, we employ conjugate priors on the model parameters and initial values and assume the model errors are normally distributed to derive the conditional posterior and predictive distributions. In particular, we show that the conditional posterior distribution of the model parameters and the variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Finally, we use these closed-form conditional posterior and predictive distributions to apply the Gibbs sampling algorithm to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. We evaluate our proposed Bayesian method using simulation study and real-world time series datasets.

Bayesian modeling of joint and conditional distributions

In this paper we use the Gibbs sampling algorithm to present a Bayesian analysis to multiplicative double seasonal autoregressive (DSAR) models, considering both estimation and prediction problems. Assuming the model errors are normally distributed and using natural conjugate and g priors on the initial values and model parameters, we show that the conditional posterior distributions of the model parameters and variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Using these closed-form conditional posterior and predictive distributions, we apply the Gibbs sampling to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. The proposed Bayesian method is evaluated using simulation study and real-world time series dataset.

A multivariate approach for statistical assessments of compound extremes

Some further results on the height of lattice paths

Consider the sample X1, X2, ..., XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F, and let Fu be their conditional excess distribution function F. We define the ordered sample by . Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F , and large u ,Fu is well approximated by the Generalized Pareto Distribution.The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds.The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.

Bayesian compressive sensing (BCS) helps address ill-posed signal recovery problems using the Bayesian estimation framework. Gibbs sampling is a technique used in Bayesian estimation that iteratively draws samples from conditional posterior distributions. However, Gibbs sampling is inherently sequential and existing parallel implementations focus on reducing the communication between computing units at the cost of increase in recovery error. In this work, we propose a two-stage parallel coefficient update scheme for wavelet-based Bayesian compressive sensing, where the first stage approximates the real distributions of the wavelet coefficients and the second stage computes the final estimate of the coefficients. While in the first stage the parallel computing units share information with each other, in the second stage, the parallel units work independently. We propose a new coefficient update scheme that updates coefficients in both stages based on data generated a few rounds ago. Such a scheme helps in relaxing the timing constraints for communication in the first stage and computations in the second stage. We design the corresponding parallel architecture and synthesize it in 7 nm technology node. We show that in a system with 8 computing units, our method helps reduce the execution time by 17.4× compared to a sequential implementation without any increase in the signal recovery error.

Heartbeat modeling allows to detect anomalies that reflect the functioning of the heart. Certain approaches face this problem by using Gaussian Mixture Models (GMMs) and other statistical classifiers by extracting the fiducial points provided by the MIT-BIH database. In this work, MIT-BIH database heartbeats are modeled into different heartbeat types from a single subject by using the Gibbs Sampling (GS) algorithm. Firstly, a data pre-processing step is performed; this step involves several tasks such as filtering the raw signals from the MIT-BIH database and reducing the heartbeat types to five. Secondly, the GS is applied to the resulting signals of one subject. Thirdly, the Euclidean distance between each heartbeat type is calculated, and lastly, the Bhattacharyya distance is used to classify heartbeats. The results obtained by the GS algorithm were also compared to results obtained by applying the Expectation Maximization (EM) algorithm to the same data-set. Results allow to conclude that GS is a proper solution for separating each heartbeat type; by providing a significant difference between each heartbeat type which can be used for classification.

This paper explores how to exploit shape information to perform object class recognition. We use a sparse part-based model to describe object categories defined by shape. The sparseness allows the relative spatial relationship between parts to be described simply. It is possible, with this model, to highlight potential locations of the object and its parts in novel images. Subsequently these areas are examined by a more flexible shape model that measures if the image data provides evidence of the existence of boundary/connecting curves between connected hypothesized parts. From these measurements it is possible to construct a very simple cost function which indicates the presence or absence of the object class. The part-based model is designed to decouple variations due to affine warps and other forms of shape deformations. The latter are modeled probabilistically using conditional probability distributions which describe the linear dependencies between the location of a part and a subset of the other parts. These conditional distributions can then be exploited to search efficiently for the instances of the part model in novel images. Results are reported on experiments performed on the ETHZ shape classes database that features heavily cluttered images and large variations in scale.

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S3. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

The smooth transition autoregressive (STAR)(k)–GARCH(l, m) model is a non-linear time series model that is able to account for changes in both regime and volatility respectively. The model can be widely applied to analyse the dynamic behaviour of data exhibiting these two phenomenons in areas such as finance, hydrology and climate change. The main aim of this paper is to perform a Bayesian analysis of STAR(k)–GARCH(l, m) models. The estimation procedure will include estimation of the mean and variance coefficient parameters, the parameters of the transition function, as well as the model orders (k, l, m). To achieve this aim, the joint posterior distribution of the model orders, coefficient and implicit parameters in the logistic STAR(k)–GARCH(l, m) model is presented. The conditional posterior distributions are then derived, followed by the design of a posterior simulator using a combination of MCMC algorithms which includes Metropolis–Hastings, Gibbs Sampler and Reversible Jump MCMC algorithms. Following this are extensive simulation studies and a case study presenting the methodology.

Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional probability distribution. The Gibbs sampler, however, is often justifiably criticized for its sometimes slow convergence, especially when it is used to fit highly structured complex models. The recently proposed Partially Collapsed Gibbs (PCG) sampler offers a new strategy for improving the convergence characteristics of a Gibbs sampler. A PCG sampler achieves faster convergence by reducing the conditioning in some or all of the component draws of its parent Gibbs sampler. Although this strategy can significantly improve convergence, it must be implemented with care to be sure that the desired stationary distribution is preserved. In some cases the set of conditional distributions sampled in a PCG sampler may be functionally incompatible and permuting the order of draws can change the stationary distribution of the chain. In this article, we draw an analogy between the PCG sampler and certain efficient EM-type algorithms that helps to explain the computational advantage of PCG samplers and to suggest when they might be used in practice. We go on to illustrate the PCG samplers in three substantial examples drawn from our applied work: a multilevel spectral model commonly used in high-energy astrophysics, a piecewise-constant multivariate time series model, and a joint imputation model for nonnested data. These are all useful highly structured models that involve computational challenges that can be solved using PCG samplers. The examples illustrate not only the computation advantage of PCG samplers but also how they should be constructed to maintain the desired stationary distribution. Supplemental materials for the examples given in this article are available online.

ABSTRACTMany study designs yield a variety of outcomes from each subject clustered within an experimental unit. When these outcomes are of mixed data types, it is challenging to jointly model the effects of covariates on the responses using traditional methods. In this paper, we develop a Bayesian approach for a joint regression model of the different outcome variables and show that the fully conditional posterior distributions obtained under the model assumptions allow for estimation of posterior distributions using Gibbs sampling algorithm.

There is an increasing interest in expanding the one‐parameter Lindley distribution to two‐parameter, three‐parameter, and five‐parameter. The univariate one‐parameter Lindley distribution is still one of the most applicable distributions in data analysis especially in lifetime data. Modeling dependent random quantities required bivariate parametric probability distributions. This study presents a new bivariate three‐parameter probability distribution called bivariate modified Lindley distribution. The one‐parameter modified Lindley distribution is used as a base line to construct the new model. Its statistical properties including cumulative function, density function, marginals, moments, conditional distributions, and copula are discussed. Simulation is constructed to declare theoretical properties, to show the flexibility of the new model and to investigate the goodness of fit. Two sets of real data, financial data and UEFA Champion’s League data, are used to show the applicability of the proposed model for different types of data.