Posterior Distribution of a Dirichlet Process from Quantal Response Data

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators.The Canadian Journal of Statistics40: 328–344; 2012 © 2012 Statistical Society of Canada

Generalized mixed linear, threshold, and logistic sire models and Markov chain, Monte Carlo simulation procedures were used to estimate genetic parameters for calving rate and calf survival in a multibreed beef cattle population. Data were obtained from a 5-generation rotational crossbreeding study involving Angus, Brahman, Charolais, and Hereford (1969 to 1995). Gelbvieh and Simmental bulls sired terminal-cross calves from a sample of generation 5 cows. A total of 1,458 cows sired by 158 bulls had a mean calving rate of 78% based on 4,808 calving records. Ninety-one percent of 5,015 calves sired by 260 bulls survived to weaning. Mean heritability estimates and standard deviations for daughter calving rate from posterior distributions were 0.063 +/- 0.024, 0.150 +/- 0.049, and 0.130 +/- 0.047 for linear, threshold, and logistic models, respectively. For calf survival, mean heritability estimates and standard deviations from posterior distributions were 0.049 +/- 0.022, 0.160 +/- 0.058, and 0.190 +/- 0.078 from linear, threshold, and logistic models, respectively. When transformed to an underlying normal scale, linear sire, mixed model, heritability estimates were similar to threshold and logistic sire mixed model estimates. Posterior density distributions of estimated heritabilities from all models were normal. Spearman rank correlations between sire EPD across statistical models were greater than 0.97 for daughter calving rate and for calf survival. Sire EPD had similar ranges across statistical models for daughter calving rate and for calf survival.

Real and simulated experimental data and theoretical data from quantal response experiments were used to make a comparison between the analysis of data from a quantal response experiment and data from a direct enumeration experiment. The method of analysis for each is differentiated, thereby enhancing the utility of the quantal response experiment in sterilization studies. From this comparison it appears that the Stumbo estimate of the D value is biased. Furthermore, the Stumbo estimate depends upon the spore load per replicate in quantal response type experiments, which makes experimental comparisons difficult. Another estimate of D is demonstrated which overcomes some of these shortcomings.

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.

본 논문에서는 겉보기 무관 회귀모형을 고려하고 디리크레 프로세스 혼합모형을 오차항의 분포로 하는 비모수 베이지안 방법을 제안한다. 제안된 모형을 바탕으로 사후분포를 유도하고 디리크레 프로세스 혼합모형의 붕괴깁스표집 방법을 통해 마코프 체인 몬테 칼로 알고리듬을 구성하고 사후추론을 실시한다. 모형의 성능을 비교하기 위해 모의실험을 실시하고, 더 나아가 한국지역의 강수량 예측에 대한 실제 자료에 적용해 본다. In this paper, we consider a seemingly unrelated regression (SUR) model and propose a nonparametric Bayesian approach to SUR with a Dirichlet process mixture of normals for modeling an unknown error distribution. Posterior distributions are derived based on the proposed model, and the posterior inference is performed via Markov chain Monte Carlo methods based on the collapsed Gibbs sampler of a Dirichlet process mixture model. We present a simulation study to assess the performance of the model. We also apply the model to precipitation data over South Korea.

In modeling HIV/AIDS progression, we carried out a comprehensive investigation into the risk factors for statespecific-failure rates to identify the influential co-variates using Bayesian Model averaging method (BMA).BMA provides a posterior probability via Markov Chain Monte Carlo (MCMC) for each variable that belongs to the model.It accounts for model uncertainty by averaging all plausible models using their posterior probabilities as the weights for model-averaged predictions and estimates of the required parameters.Patients' age, and gender, among other co-variates, have been found to influence the state-specific-failure rates highly.However, the impact of each of the factors on the state specific-failure was not quantified.This paper seeks to evaluate and quantify the contribution of the patient's age and gender, CD4 cell count during any two consecutive visits, and state movement on the state-specific-failure rates for patients transiting either to the same, better or worse state.We used R Studio statistical Programming software to implement the method by applying BMS and BMA packages.State movement had a comparatively large coefficient with a posterior inclusion probability (PIP) of 0.8788 (87.88%).Hence, the most critical variable followed by observation-two-CD4-cell-count with a PIP of 0.1416 (14.16%), age and gender were the last with a PIP of 0.0556 (5.56%) and 0.0510 (5.10%) respectively for patients transiting to the same state.For patients transiting to a better state, the patients' age group dominated with a PIP of 0.9969 (99.69%), followed by patients' gender with a PIP of 0.0608 (6.08%).Patients' CD4 cell count during the second observation had the least PIP of 0.0399 (3.99%).For patients transiting to a worse disease state, patients CD4 cell count during the second observation proved to be the most important, with a PIP of 0.6179(61.79%)followed by state movement with a PIP of 0.2599 (25.99%), patients gender tailed with a PIP of 0.0467 (4.67%).

Hidden Markov random field (HMRF) models are widely used for image segmentation, as they appear naturally in problems where a spatially constrained clustering scheme is asked for. A major limitation of HMRF models concerns the automatic selection of the proper number of their states, i.e., the number of region clusters derived by the image segmentation procedure. Existing methods, including likelihood- or entropy-based criteria, and reversible Markov chain Monte Carlo methods, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (DP, infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori; infinite mixture models based on the original DP or spatially constrained variants of it have been applied in unsupervised image segmentation applications showing promising results. Under this motivation, to resolve the aforementioned issues of HMRF models, in this paper, we introduce a nonparametric Bayesian formulation for the HMRF model, the infinite HMRF model, formulated on the basis of a joint Dirichlet process mixture (DPM) and Markov random field (MRF) construction. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally demonstrate its advantages over competing methodologies.

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena. Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation. Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions. Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process. Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

The Khasminskii theorem have come to play an important role in the nonexplosion solutions of stochastic differential equations (SDEs) without the linear growth condition. In this paper, by using Peng's G-expectation theory, we establish an even more general Khasminskii-type test for stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) that cover a wide class of highly nonlinear G-SDDEs.

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time tau and provide a novel asymptotic estimate of the cumulative distribution function of tau . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

SummaryDirichlet process mixtures are flexible nonparametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are typically not consistent for the number of clusters if the concentration parameter is held fixed and data come from a finite mixture. Here we show that consistency for the number of clusters can be achieved if the concentration parameter is adapted in a fully Bayesian way, as commonly done in practice. Our results are derived for data coming from a class of finite mixtures, with mild assumptions on the prior for the concentration parameter and for a variety of choices of likelihood kernels for the mixture.