Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

On predictive distributions and Bayesian networks

In this paper we are interested in discrete prediction problems for a decision-theoretic setting, where the task is to compute the predictive distribution for a finite set of possible alternatives. This question is first addressed in a general Bayesian framework, where we consider a set of probability distributions defined by some parametric model class. Given a prior distribution on the model parameters and a set of sample data, one possible approach for determining a predictive distribution is to fix the parameters to the instantiation with the maximum a posteriori probability. A more accurate predictive distribution can be obtained by computing the evidence (marginal likelihood), i.e., the integral over all the individual parameter instantiations. As an alternative to these two approaches, we demonstrate how to use Rissanen's new definition of stochastic complexity for determining predictive distributions, and show how the evidence predictive distribution with Jeffrey's prior approaches the new stochastic complexity predictive distribution in the limit with increasing amount of sample data. To compare the alternative approaches in practice, each of the predictive distributions discussed is instantiated in the Bayesian network model family case. In particular, to determine Jeffrey's prior for this model family, we show how to compute the (expected) Fisher information matrix for a fixed but arbitrary Bayesian network structure. In the empirical part of the paper the predictive distributions are compared by using the simple tree-structured Naive Bayes model, which is used in the experiments for computational reasons. The experimentation with several public domain classification datasets suggest that the evidence approach produces the most accurate predictions in the log-score sense. The evidence-based methods are also quite robust in the sense that they predict surprisingly well even when only a small fraction of the full training set is used.     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

Integrated nested Laplace approximation for the analysis of count data via the combined model: A simulation study

The combined model accounts for different forms of extra-variability and has traditionally been applied in the likelihood framework, or in the Bayesian setting via Markov chain Monte Carlo. In this article, integrated nested Laplace approximation is investigated as an alternative estimation method for the combined model for count data, and compared with the former estimation techniques. Longitudinal, spatial, and multi-hierarchical data scenarios are investigated in three case studies as well as a simulation study. As a conclusion, integrated nested Laplace approximation provides fast and precise estimation, while avoiding convergence problems often seen when using Markov chain Monte Carlo.     

Toward an integrated Bayesian network approach to measurement error detection and correction

In this article, the quality of data produced by national statistical institutes and by governmental institutions is considered. In particular, the problem of measurement error is analyzed and an integrated Bayesian network decision support system based on non-parametric Bayesian networks is proposed for its detection and correction. Non-parametric Bayesian networks are graphical models expressing dependence structure via bivariate copulas associated to the edges of the graph. The network structure and the misreport probability are estimated using a validation sample. The Bayesian network model is proposed to decide: (i) which records have to be corrected; (ii) the kind and amount of correction to be adopted. The proposed correction procedure is applied to the Banca d’Italia Survey on Household Income and Wealth and, specifically, the bond amounts are analyzed. Finally, the sensitivity of the conditional distribution of the true value random variable given the observed one to different evidence configurations is studied.     

Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.     

Bayesian estimation of the efficient frontier

In this paper, we consider the estimation of the three determining parameters of the efficient frontier, the expected return, and the variance of the global minimum variance portfolio and the slope parameter, from a Bayesian perspective. Their posterior distribution is derived by assigning the diffuse and the conjugate priors to the mean vector and the covariance matrix of the asset returns and is presented in terms of a stochastic representation. Furthermore, Bayesian estimates together with the standard uncertainties for all three parameters are provided, and their asymptotic distributions are established. All obtained findings are applied to real data, consisting of the returns on assets included into the S&P 500. The empirical properties of the efficient frontier are then examined in detail.     

部分状态可见的隐马尔可夫模型状态序列的估计方法

本文考虑了部分状态可见的隐马尔可夫模型的状态序列估计问题,在分析了现有算法无法合理估计状态路径之后,以状态转移概率、观测概率和可见状态作为先验信息,通过贝叶斯分析计算可见状态前后向状态的后验概率,并给出初始条件和递推公式,运用动态规划递推得到每个观测值对应的最可能状态以及最可能的状态路径。最后,本文给出一个系统故障识别的应用例子,验证了所设计算法的可行性。     

Model selection for factor analysis: Some new criteria and performance comparisons

This paper derives Akaike information criterion (AIC), corrected AIC, the Bayesian information criterion (BIC) and Hannan and Quinn’s information criterion for approximate factor models assuming a large number of cross-sectional observations and studies the consistency properties of these information criteria. It also reports extensive simulation results comparing the performance of the extant and new procedures for the selection of the number of factors. The simulation results show the di?culty of determining which criterion performs best. In practice, it is advisable to consider several criteria at the same time, especially Hannan and Quinn’s information criterion, Bai and Ng’s IC_p2 and BIC₃, and Onatski’s and Ahn and Horenstein’s eigenvalue-based criteria. The model-selection criteria considered in this paper are also applied to Stock and Watson’s two macroeconomic data sets. The results differ considerably depending on the model-selection criterion in use, but evidence suggesting five factors for the first data and five to seven factors for the second data is obtainable.     

Accurate estimation for extra-Poisson variability assuming random effect models

In this study, the components of extra-Poisson variability are estimated assuming random effect models under a Bayesian approach. A standard existing methodology to estimate extra-Poisson variability assumes a negative binomial distribution. The obtained results show that using the proposed random effect model it is possible to get more accurate estimates for the extra-Poisson variability components when compared to the use of a negative binomial distribution where it is possible to estimate only one component of extra-Poisson variability. Some illustrative examples are introduced considering real data sets.