General Prior Distribution Research Articles

Multivariate Lévy-type drift change detection and mortality modeling

In this paper we give a solution to the quickest drift change detection problem for a multivariate Lévy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point as well as for a random post-change drift parameter. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two examples, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly.

Detecting renewal states in chains of variable length via intrinsic Bayes factors

Markov chains with variable length are useful parsimonious stochastic models able to generate most stationary sequence of discrete symbols. The idea is to identify the suffixes of the past, called contexts, that are relevant to predict the future symbol. Sometimes a single state is a context, and looking at the past and finding this specific state makes the further past irrelevant. States with such property are called renewal states and they can be used to split the chain into independent and identically distributed blocks. In order to identify renewal states for chains with variable length, we propose the use of Intrinsic Bayes Factor to evaluate the hypothesis that some particular state is a renewal state. In this case, the difficulty lies in integrating the marginal posterior distribution for the random context trees for general prior distribution on the space of context trees, with Dirichlet prior for the transition probabilities, and Monte Carlo methods are applied. To show the strength of our method, we analyzed artificial datasets generated from different models and one example coming from the field of Linguistics.

-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

Statistical solution of inverse problems using a state reduction

PurposeThe application of statistical inversion theory provides a powerful approach for solving estimation problems including the ability for uncertainty quantification (UQ) by means of Markov chain Monte Carlo (MCMC) methods and Monte Carlo integration. This paper aims to analyze the application of a state reduction technique within different MCMC techniques to improve the computational efficiency and the tuning process of these algorithms.Design/methodology/approachA reduced state representation is constructed from a general prior distribution. For sampling the Metropolis Hastings (MH) Algorithm and the Gibbs sampler are used. Efficient proposal generation techniques and techniques for conditional sampling are proposed and evaluated for an exemplary inverse problem.FindingsFor the MH-algorithm, high acceptance rates can be obtained with a simple proposal kernel. For the Gibbs sampler, an efficient technique for conditional sampling was found. The state reduction scheme stabilizes the ill-posed inverse problem, allowing a solution without a dedicated prior distribution. The state reduction is suitable to represent general material distributions.Practical implicationsThe state reduction scheme and the MCMC techniques can be applied in different imaging problems. The stabilizing nature of the state reduction improves the solution of ill-posed problems. The tuning of the MCMC methods is simplified.Originality/valueThe paper presents a method to improve the solution process of inverse problems within the Bayesian framework. The stabilization of the inverse problem due to the state reduction improves the solution. The approach simplifies the tuning of MCMC methods.

Analysis of Weibull Step-Stress Model In Presence of Competing Risk

In this paper, we mainly consider the inference of a simple step-stress model based on a complete sample, when the stress changes after a prefixed number of failures. It is assumed that there is more than one cause of failure, and the lifetime of the experimental units at each stress level follows Weibull distribution with the same shape parameter and different scale parameters. The distribution function under different stress levels are connected through the generalized Khamis–Higgins model. The maximum likelihood estimates of the model parameters and the associated asymptotic confidence intervals are obtained. Further, we consider the Bayesian inference of the unknown model parameters based on fairly general prior distributions. We have also provided the results for Type-I censored data also. We assess the performances of the estimators through extensive simulation study for complete sample, and the analyses of one complete (simulated) data set and one Type-I censored solar lighting device data set have been performed for illustrative purpose. We propose different classical and Bayesian optimal criteria, and based on them we obtain the optimum stress changing time. Finally, we have indicated how the assumption on the common shape parameter of two competing causes can be relaxed.

Learning scene-aware image priors with high-order Markov random fields

Many methods have been proposed to learn image priors from natural images for the ill-posed image restoration tasks. However, many prior learning algorithms assume that a general prior distribution is suitable for over all kinds of images. Since the contents of the natural images and the corresponding low-level statistical characteristics vary from scene to scene, we argue that learning a universal generative prior for all natural images may be imperfect. Although the universal generative prior can remove artifacts and reserve a natural smoothness in image restoration, it also tends to introduce unreal flatness and clutter textures. To address this issue, in this paper, we present to learn a scene-aware image prior based on the high-order Markov random field (MRF) model (SA-MRF). With this model, we jointly learn a set of shared low-level features and different potentials for specific scene contents. In prediction, a good prior can be adapted to the given degenerated image with the scene content perception. Experimental results on the image denoising and inpainting tasks demonstrate the efficiency of the SA-MRF on both numerical evaluation and visual compression.

On the optimality of Bayesian change-point detection

By introducing suitable loss random variables of detection, we obtain optimal tests in terms of the stopping time or alarm time for Bayesian change-point detection not only for a general prior distribution of change-points but also for observations being a Markov process. Moreover, the optimal (minimal) average detection delay is proved to be equal to $1$ for any (possibly large) average run length to false alarm if the number of possible change-points is finite.

Inferential permutation tests for maximum entropy models in ecology

Maximum entropy (maxent) models assign probabilities to states that (1) agree with measured macroscopic constraints on attributes of the states and (2) are otherwise maximally uninformative and are thus as close as possible to a specified prior distribution. Such models have recently become popular in ecology, but classical inferential statistical tests require assumptions of independence during the allocation of entities to states that are rarely fulfilled in ecology. This paper describes a new permutation test for such maxent models that is appropriate for very general prior distributions and for cases in which many states have zero abundance and that can be used to test for conditional relevance of subsets of constraints. Simulations show that the test gives correct probability estimates under the null hypothesis. Power under the alternative hypothesis depends primarily on the number and strength of the constraints and on the number of states in the model; the number of empty states has only a small effect on power. The test is illustrated using two empirical data sets to test the community assembly model of B. Shipley, D. Vile, and E. Garnier and the species abundance distribution models of S. Pueyo, F. He, and T. Zillio.

INFERENTIAL PERMUTATION TESTS FOR MAXIMUM ENTROPY MODELS IN ECOLOGY

Maximum entropy (maxent) models assign probabilities to states that (1) agree with measured macroscopic constraints on attributes of the states and (2) are otherwise maximally uninformative and are thus as close as possible to a specified prior distribution. Such models have recently become popular in ecology, but classical inferential statistical tests require assumptions of independence during the allocation of entities to states that are rarely fulfilled in ecology. This paper describes a new permutation test for such maxent models that is appropriate for very general prior distributions and for cases in which many states have zero abundance and that can be used to test for conditional relevance of subsets of constraints. Simulations show that the test gives correct probability estimates under the null hypothesis. Power under the alternative hypothesis depends primarily on the number and strength of the constraints and on the number of states in the model; the number of empty states has only a small effect on power. The test is illustrated using two empirical data sets to test the community assembly model of B. Shipley, D. Vile, and E. Garnier and the species abundance distribution models of S. Pueyo, F. He, and T. Zillio.

A new approach to the credibility formula

The usual credibility formula holds whenever, (i) claim size distribution is a member of the exponential family of distributions, (ii) prior distribution conjugates with claim size distribution, and (iii) square error loss has been considered. As long as, one of these conditions is violent, the usual credibility formula no longer holds. This article, using the mean square error minimization technique, develops a simple and practical approach to the credibility theory. Namely, we approximate the Bayes estimator with respect to a general loss function and general prior distribution by a convex combination of the observation mean and mean of prior, say, approximate credibility formula. Adjustment of the approximate credibility for several situations and its form for several important losses are given.

Multichannel Speech Enhancement Based on Generalized Gamma Prior Distribution with Its Online Adaptive Estimation

We present a multichannel speech enhancement method based on MAP speech spectral magnitude estimation using a generalized gamma model of speech prior distribution, where the model parameters are adapted from actual noisy speech in a frame-by-frame manner. The utilization of a more general prior distribution with its online adaptive estimation is shown to be effective for speech spectral estimation in noisy environments. Furthermore, the multi-channel information in terms of cross-channel statistics are shown to be useful to better adapt the prior distribution parameters to the actual observation, resulting in better performance of speech enhancement algorithm. We tested the proposed algorithm in an in-car speech database and obtained significant improvements of the speech recognition performance, particularly under non-stationary noise conditions such as music, air-conditioner and open window.

Gamma Modeling of Speech Power and Its On-Line Estimation for Statistical Speech Enhancement

This study shows the effectiveness of using gamma distribution in the speech power domain as a more general prior distribution for the model-based speech enhancement approaches. This model is a super-set of the conventional Gaussian model of the complex spectrum and provides more accurate prior modeling when the optimal parameters are estimated. We develop a method to adapt the modeled distribution parameters from each actual noisy speech in a frame-by-frame manner. Next, we derive and investigate the minimum mean square error (MMSE) and maximum a posterior probability (MAP) estimations in different domains of speech spectral magnitude, generalized power and its logarithm, using the proposed gamma modeling. Finally, a comparative evaluation of the MAP and MMSE filters is conducted. As the MMSE estimations tend to more complicated using more general prior distributions, the MAP estimations are given in closed-form extractions and therefore are suitable in the implementation. The adaptive estimation of the modeled distribution parameters provides more accurate prior modeling and this is the principal merit of the proposed method and the reason for the better performance. From the experiments, the MAP estimation is recommended due to its high efficiency and low complexity. Among the MAP based systems, the estimation in log-magnitude domain is shown to be the best for the speech recognition as the estimation in power domain is superior for the noise reduction.

Bayes estimation of the binomial parameter n

Bayes estimation of the binomial parameter n based on a general prior distribution for n is studied. As special cases improper prior and Poisson prior (which is a natural choice) are considered, and formulae for the marginal and posterior distributions are obtained. It is shown that the assumption of improper priors in both p and n leads to implausible results.

On the two armed bandit with one probability known

The dynamic programming approach to the solution of the two armed bandit problem with one probability known is discussed, for a general prior distribution for the unknown probability. Properties of the objective function and of the stopping boundaries are obtained. The problem of costly observations is discussed.