A grouped beta process model for multivariate resting‐state EEG microstate analysis on twins

EEG microstate analysis investigates the collection of distinct temporal blocks that characterize the electrical activity of the brain. Brain activity within each microstate is stable, but activity switches rapidly between different microstates in a nonrandom way. We propose a Bayesian nonparametric model that concurrently estimates the number of microstates and their underlying behaviour. We use a Markov switching vector autoregressive (VAR) framework, where a hidden Markov model (HMM) controls the nonrandom state switching dynamics of the EEG activity and a VAR model defines the behaviour of all time points within a given state. We analyze the resting‐state EEG data from twin pairs collected through the Minnesota Twin Family Study, consisting of 70 epochs per participant, where each epoch corresponds to 2 s of EEG data. We fit our model at the twin pair level, sharing information within epochs from the same participant and within epochs from the same twin pair. We capture within twin‐pair similarity, using an Indian buffet process, to consider an infinite library of microstates, allowing each participant to select a finite number of states from this library. The state spaces of highly similar twins may completely overlap while dissimilar twins could select distinct state spaces. In this way, our Bayesian nonparametric model defines a sparse set of states that describe the EEG data. All epochs from a single participant use the same set of states and are assumed to adhere to the same state switching dynamics in the HMM model, enforcing within‐participant similarity.     

Nonparametric Bayesian analysis for multi-site hidden Markov model

The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in a variety of applications including pattern recognition. Unlike typical mixture models, hidden Markov states can represent the heterogeneity in data and it can be extended to a multivariate case using a hierarchical Bayesian approach. This article provides a nonparametric Bayesian modeling approach to the multi-site HMM by considering stick-breaking priors for each row of an infinite state transition matrix. This extension has many advantages over a parametric HMM. For example, it can provide more flexible information for identifying the structure of the HMM than parametric HMM analysis, such as the number of states in HMM. We exploit a simulation example and a real dataset to evaluate the proposed approach.     

A Bayesian HMM with random effects and an unknown number of states for DNA copy number analysis

Hidden Markov models (HMMs) have been shown to be a flexible tool for modelling complex biological processes. However, choosing the number of hidden states remains an open question and the inclusion of random effects also deserves more research, as it is a recent addition to the fixed-effect HMM in many application fields. We present a Bayesian mixed HMM with an unknown number of hidden states and fixed covariates. The model is fitted using reversible-jump Markov chain Monte Carlo, avoiding the need to select the number of hidden states. We show through simulations that the estimations produced are more precise than those from a fixed-effect HMM and illustrate its practical application to the analysis of DNA copy number data, a field where HMMs are widely used.     

Have I seen you before? Principles of Bayesian predictive classification revisited

A general inductive Bayesian classification framework is considered using a simultaneous predictive distribution for test items. We introduce a principle of generative supervised and semi-supervised classification based on marginalizing the joint posterior distribution of labels for all test items. The simultaneous and marginalized classifiers arise under different loss functions, while both acknowledge jointly all uncertainty about the labels of test items and the generating probability measures of the classes. We illustrate for data from multiple finite alphabets that such classifiers achieve higher correct classification rates than a standard marginal predictive classifier which labels all test items independently, when training data are sparse. In the supervised case for multiple finite alphabets the simultaneous and the marginal classifiers are proven to become equal under generalized exchangeability when the amount of training data increases. Hence, the marginal classifier can be interpreted as an asymptotic approximation to the simultaneous classifier for finite sets of training data. It is also shown that such convergence is not guaranteed in the semi-supervised setting, where the marginal classifier does not provide a consistent approximation.     

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.     

Hierarchical Bayesian Approach to a Multi-Site Hidden Markov Model

Multivariate data with a sequential or temporal structure occur in various fields of study. The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in areas of pattern recognition through the extension of independent and identically distributed mixture models. Unlike in typical mixture models, the heterogeneity of data is represented by hidden Markov states. This article extends the HMM to a multi-site or multivariate case by taking a hierarchical Bayesian approach. This extension has many advantages over a single-site HMM. For example, it can provide more information for identifying the structure of the HMM than a single-site analysis. We evaluate the proposed approach by exploiting a spatial correlation that depends on the distance between sites.     

An EM-Based Viterbi Approximation Algorithm for Mixed-State Latent Factor Models

In this article, a state-space model based on an underlying hidden Markov chain model (HMM) with factor analysis observation process is introduced. The HMM generates a piece-wise constant state evolution process and the observations are produced from the state vectors by a conditionally heteroscedastic factor analysis observation process. More specifically, we concentrate on situations where the factor variances are modeled by univariate Generalized Quadratic Autoregressive Conditionally Heteroscedastic processes (GQARCH). An expectation maximization (EM) algorithm combined with a mixed-state version of the Viterbi algorithm is derived for maximum likelihood estimation. The various regimes, common factors, and their volatilities are supposed unobservable and the inference must be carried out from the observable process. Extensive Monte Carlo simulations show promising results of the algorithms, especially for segmentation and tracking tasks.     

Some applications of nonlinear and non-Gaussian state–space modelling by means of hidden Markov models

Nonlinear and non-Gaussian state–space models (SSMs) are fitted to different types of time series. The applications include homogeneous and seasonal time series, in particular earthquake counts, polio counts, rainfall occurrence data, glacial varve data and daily returns on a share. The considered SSMs comprise Poisson, Bernoulli, gamma and Student-t distributions at the observation level. Parameter estimations for the SSMs are carried out using a likelihood approximation that is obtained after discretization of the state space. The approximation can be made arbitrarily accurate, and the approximated likelihood is precisely that of a finite-state hidden Markov model (HMM). The proposed method enables us to apply standard HMM techniques. It is easy to implement and can be extended to all kinds of SSMs in a straightforward manner.     

Parallel Markov chain Monte Carlo for Bayesian hierarchical models with big data,in two stages

Due to the escalating growth of big data sets in recent years, new Bayesian Markov chain Monte Carlo (MCMC) parallel computing methods have been developed. These methods partition large data sets by observations into subsets. However, for Bayesian nested hierarchical models, typically only a few parameters are common for the full data set, with most parameters being group specific. Thus, parallel Bayesian MCMC methods that take into account the structure of the model and split the full data set by groups rather than by observations are a more natural approach for analysis. Here, we adapt and extend a recently introduced two-stage Bayesian hierarchical modeling approach, and we partition complete data sets by groups. In stage 1, the group-specific parameters are estimated independently in parallel. The stage 1 posteriors are used as proposal distributions in stage 2, where the target distribution is the full model. Using three-level and four-level models, we show in both simulation and real data studies that results of our method agree closely with the full data analysis, with greatly increased MCMC efficiency and greatly reduced computation times. The advantages of our method versus existing parallel MCMC computing methods are also described.     

Semiparametric hidden Markov model with non-parametric regression

The hidden Markov model regression (HMMR) has been popularly used in many fields such as gene expression and activity recognition. However, the traditional HMMR requires the strong linearity assumption for the emission model. In this article, we propose a hidden Markov model with non-parametric regression (HMM-NR), where the mean and variance of emission model are unknown smooth functions. The new semiparametric model might greatly reduce the modeling bias and thus enhance the applicability of the traditional hidden Markov model regression. We propose an estimation procedure for the transition probability matrix and the non-parametric mean and variance functions by combining the ideas of the EM algorithm and the kernel regression. Simulation studies and a real data set application are used to demonstrate the effectiveness of the new estimation procedure.     

Bayesian analysis of human movement curves

Summary.  We consider the Bayesian analysis of human movement data, where the subjects perform various reaching tasks. A set of markers is placed on each subject and a system of cameras records the three-dimensional Cartesian co-ordinates of the markers during the reaching movement. It is of interest to describe the mean and variability of the curves that are traced by the markers during one reaching movement, and to identify any differences due to covariates. We propose a methodology based on a hierarchical Bayesian model for the curves. An important part of the method is to obtain identifiable features of the movement so that different curves can be compared after temporal warping. We consider four landmarks and a set of equally spaced pseudolandmarks are located in between. We demonstrate that the algorithm works well in locating the landmarks, and shape analysis techniques are used to describe the posterior distribution of the mean curve. A feature of this type of data is that some parts of the movement data may be missing—the Bayesian methodology is easily adapted to cope with this situation.     

Stochastic models for greenhouse gas emission rate estimation from hydroelectric reservoirs: a Bayesian hierarchical approach

Herein, we propose a fully Bayesian approach to the greenhouse gas emission problem. The goal of this work is to estimate the emission rate of polluting gases from the area flooded by hydroelectric reservoirs. We present models for gas concentration evolution in two ways: first, by proposing them from ordinary differential equation solutions and, second, by using stochastic differential equations with a discretization scheme. Finally, we present techniques to estimate the emission rate for the entire reservoir. In order to carry out the inference, we use the Bayesian framework with Monte Carlo via Markov Chain methods. Discretization schemes over continuous differential equations are used when necessary. These models applied to greenhouse gas emission and Bayesian inference for this purpose are completely new in statistical literature, as far as we know, and contribute to estimate the amount of polluting gases released from hydroelectric reservoirs in Brazil. The proposed models are applied in a real data set and results are presented.     

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 with missing covariates for policy considerations in fox enclosures

Foxhound training enclosures are facilities where wild-trapped foxes are placed into large fenced areas for dog training purposes. Although the purpose of these facilities is to train dogs without harming foxes, dog-related mortality has been reported to be an issue in some enclosures. Using data from a fox enclosure in Virginia, we investigate factors that influence fox survival in these dog training facilities and propose a set of policies to improve fox survival. In particular, a Bayesian hierarchical model is formulated to compute fox survival probabilities based on a fox's time in the enclosure and the number of dogs allowed in the enclosure at one time. These calculations are complicated by missing information on the number of dogs in the enclosure for many days during the study. We elicit expert knowledge for a prior on the number of dogs to account for the uncertainty in the missing data. Reversible jump Markov Chain Monte Carlo is used for model selection in the presence of missing covariates. We then use our model to examine possible changes to foxhound training enclosure policy and what effect those changes may have on fox survival.     

Bayesian semiparametric inference for the accelerated failure-time model

Bayesian semiparametric inference is considered for a loglinear model. This model consists of a parametric component for the regression coefficients and a nonparametric component for the unknown error distribution. Bayesian analysis is studied for the case of a parametric prior on the regression coefficients and a mixture-of-Dirichlet-processes prior on the unknown error distribution. A Markov-chain Monte Carlo (MCMC) method is developed to compute the features of the posterior distribution. A model selection method for obtaining a more parsimonious set of predictors is studied. The method adds indicator variables to the regression equation. The set of indicator variables represents all the possible subsets to be considered. A MCMC method is developed to search stochastically for the best subset. These procedures are applied to two examples, one with censored data.     

Unbiased Bayesian inference for population Markov jump processes via random truncations

We consider continuous time Markovian processes where populations of individual agents interact stochastically according to kinetic rules. Despite the increasing prominence of such models in fields ranging from biology to smart cities, Bayesian inference for such systems remains challenging, as these are continuous time, discrete state systems with potentially infinite state-space. Here we propose a novel efficient algorithm for joint state/parameter posterior sampling in population Markov Jump processes. We introduce a class of pseudo-marginal sampling algorithms based on a random truncation method which enables a principled treatment of infinite state spaces. Extensive evaluation on a number of benchmark models shows that this approach achieves considerable savings compared to state of the art methods, retaining accuracy and fast convergence. We also present results on a synthetic biology data set showing the potential for practical usefulness of our work.     

A Spatio-Temporal Model for Functional Magnetic Resonance Imaging Data – with a View to Resting State Networks

Abstract.  Functional magnetic resonance imaging (fMRI) is a technique for studying the active human brain. During the fMRI experiment, a sequence of MR images is obtained, where the brain is represented as a set of voxels. The data obtained are a realization of a complex spatio-temporal process with many sources of variation, both biological and technical. We present a spatio-temporal point process model approach for fMRI data where the temporal and spatial activation are modelled simultaneously. It is possible to analyse other characteristics of the data than just the locations of active brain regions, such as the interaction between the active regions. We discuss both classical statistical inference and Bayesian inference in the model. We analyse simulated data without repeated stimuli both for location of the activated regions and for interactions between the activated regions. An example of analysis of fMRI data, using this approach, is presented.     

Bayesian networks for imputation

Summary.  Bayesian networks are particularly useful for dealing with high dimensional statistical problems. They allow a reduction in the complexity of the phenomenon under study by representing joint relationships between a set of variables through conditional relationships between subsets of these variables. Following Thibaudeau and Winkler we use Bayesian networks for imputing missing values. This method is introduced to deal with the problem of the consistency of imputed values: preservation of statistical relationships between variables ( statistical consistency ) and preservation of logical constraints in data ( logical consistency ). We perform some experiments on a subset of anonymous individual records from the 1991 UK population census.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.