A weakly informative prior for Bayesian dynamic model selection with applications in fMRI

In recent years, Bayesian statistics methods in neuroscience have been showing important advances. In particular, detection of brain signals for studying the complexity of the brain is an active area of research. Functional magnetic resonance imagining (fMRI) is an important tool to determine which parts of the brain are activated by different types of physical behavior. According to recent results, there is evidence that the values of the connectivity brain signal parameters are close to zero and due to the nature of time series fMRI data with high-frequency behavior, Bayesian dynamic models for identifying sparsity are indeed far-reaching. We propose a multivariate Bayesian dynamic approach for model selection and shrinkage estimation of the connectivity parameters. We describe the coupling or lead-lag between any pair of regions by using mixture priors for the connectivity parameters and propose a new weakly informative default prior for the state variances. This framework produces one-step-ahead proper posterior predictive results and induces shrinkage and robustness suitable for fMRI data in the presence of sparsity. To explore the performance of the proposed methodology, we present simulation studies and an application to functional magnetic resonance imaging data.     

A Stochastic Geometry Model for Functional Magnetic Resonance Images

In functional magnetic resonance imaging, spatial activation patterns are commonly estimated using a non-parametric smoothing approach. Significant peaks or clusters in the smoothed image are subsequently identified by testing the null hypothesis of lack of activation in every volume element of the scans. A weakness of this approach is the lack of a model for the activation pattern; this makes it difficult to determine the variance of estimates, to test specific neuroscientific hypotheses or to incorporate prior information about the brain area under study in the analysis. These issues may be addressed by formulating explicit spatial models for the activation and using simulation methods for inference. We present one such approach, based on a marked point process prior. Informally, one may think of the points as centres of activation, and the marks as parameters describing the shape and area of the surrounding cluster. We present an MCMC algorithm for making inference in the model and compare the approach with a traditional non-parametric method, using both simulated and visual stimulation data. Finally we discuss extensions of the model and the inferential framework to account for non-stationary responses and spatio-temporal correlation.     

Sensitivity analysis for incomplete contingency tables: the Slovenian plebiscite case

Classical inferential procedures induce conclusions from a set of data to a population of interest, accounting for the imprecision resulting from the stochastic component of the model. Less attention is devoted to the uncertainty arising from (unplanned) incompleteness in the data. Through the choice of an identifiable model for non-ignorable non-response, one narrows the possible data-generating mechanisms to the point where inference only suffers from imprecision. Some proposals have been made for assessing the sensitivity to these modelling assumptions; many are based on fitting several plausible but competing models. For example, we could assume that the missing data are missing at random in one model, and then fit an additional model where non-random missingness is assumed. On the basis of data from a Slovenian plebiscite, conducted in 1991, to prepare for independence, it is shown that such an ad hoc procedure may be misleading. We propose an approach which identifies and incorporates both sources of uncertainty in inference: imprecision due to finite sampling and ignorance due to incompleteness. A simple sensitivity analysis considers a finite set of plausible models. We take this idea one step further by considering more degrees of freedom than the data support. This produces sets of estimates (regions of ignorance) and sets of confidence regions (combined into regions of uncertainty).     

Trajectory inference and parameter estimation in stochastic models with temporally aggregated data

Stochastic models are of fundamental importance in many scientific and engineering applications. For example, stochastic models provide valuable insights into the causes and consequences of intra-cellular fluctuations and inter-cellular heterogeneity in molecular biology. The chemical master equation can be used to model intra-cellular stochasticity in living cells, but analytical solutions are rare and numerical simulations are computationally expensive. Inference of system trajectories and estimation of model parameters from observed data are important tasks and are even more challenging. Here, we consider the case where the observed data are aggregated over time. Aggregation of data over time is required in studies of single cell gene expression using a luciferase reporter, where the emitted light can be very faint and is therefore collected for several minutes for each observation. We show how an existing approach to inference based on the linear noise approximation (LNA) can be generalised to the case of temporally aggregated data. We provide a Kalman filter (KF) algorithm which can be combined with the LNA to carry out inference of system variable trajectories and estimation of model parameters. We apply and evaluate our method on both synthetic and real data scenarios and show that it is able to accurately infer the posterior distribution of model parameters in these examples. We demonstrate how applying standard KF inference to aggregated data without accounting for aggregation will tend to underestimate the process noise and can lead to biased parameter estimates.     

Functional Autoregression for Sparsely Sampled Data

We develop a hierarchical Gaussian process model for forecasting and inference of functional time series data. Unlike existing methods, our approach is especially suited for sparsely or irregularly sampled curves and for curves sampled with nonnegligible measurement error. The latent process is dynamically modeled as a functional autoregression (FAR) with Gaussian process innovations. We propose a fully nonparametric dynamic functional factor model for the dynamic innovation process, with broader applicability and improved computational efficiency over standard Gaussian process models. We prove finite-sample forecasting and interpolation optimality properties of the proposed model, which remain valid with the Gaussian assumption relaxed. An efficient Gibbs sampling algorithm is developed for estimation, inference, and forecasting, with extensions for FAR(p) models with model averaging over the lag p. Extensive simulations demonstrate substantial improvements in forecasting performance and recovery of the autoregressive surface over competing methods, especially under sparse designs. We apply the proposed methods to forecast nominal and real yield curves using daily U.S. data. Real yields are observed more sparsely than nominal yields, yet the proposed methods are highly competitive in both settings. Supplementary materials, including R code and the yield curve data, are available online.     

Statistical Inference Based on Pooled Data: A Moment-Based Estimating Equation Approach

We consider statistical inference on parameters of a distribution when only pooled data are observed. A moment-based estimating equation approach is proposed to deal with situations where likelihood functions based on pooled data are difficult to work with. We outline the method to obtain estimates and test statistics of the parameters of interest in the general setting. We demonstrate the approach on the family of distributions generated by the Box-Cox transformation model, and, in the process, construct tests for goodness of fit based on the pooled data.     

Information analysis of local suppression scheme based on a spatial-temporal model

In a wireless sensor network, data collection is relatively cheap whereas data transmission is relatively expensive. Thus, preserving battery life is critical. If the process of interest is sufficiently predictable, the suppression in transmission can be adopted to improve efficiency of sensor networks because the loss of information is not great. The prime interest lies in finding an inference-efficient way to support suppressed data collection application. In this paper, we present a suppression scheme for a multiple nodes setting with spatio-temporal processes, especially when process knowledge is insufficient. We also explore the impact of suppression schemes on the inference of the regional processes under various suppression levels. Finally, we formalize the hierarchical Bayesian model for these schemes.     

Likelihood-free inference via classification

Increasingly complex generative models are being used across disciplines as they allow for realistic characterization of data, but a common difficulty with them is the prohibitively large computational cost to evaluate the likelihood function and thus to perform likelihood-based statistical inference. A likelihood-free inference framework has emerged where the parameters are identified by finding values that yield simulated data resembling the observed data. While widely applicable, a major difficulty in this framework is how to measure the discrepancy between the simulated and observed data. Transforming the original problem into a problem of classifying the data into simulated versus observed, we find that classification accuracy can be used to assess the discrepancy. The complete arsenal of classification methods becomes thereby available for inference of intractable generative models. We validate our approach using theory and simulations for both point estimation and Bayesian inference, and demonstrate its use on real data by inferring an individual-based epidemiological model for bacterial infections in child care centers.     

Bayesian semiparametric modeling and inference with mixtures of?symmetric distributions

We propose a semiparametric modeling approach for mixtures of symmetric distributions. The mixture model is built from a common symmetric density with different components arising through different location parameters. This structure ensures identifiability for mixture components, which is a key feature of the model as it allows applications to settings where primary interest is inference for the subpopulations comprising the mixture. We focus on the two-component mixture setting and develop a Bayesian model using parametric priors for the location parameters and for the mixture proportion, and a nonparametric prior probability model, based on Dirichlet process mixtures, for the random symmetric density. We present an approach to inference using Markov chain Monte Carlo posterior simulation. The performance of the model is studied with a simulation experiment and through analysis of a rainfall precipitation data set as well as with data on eruptions of the Old Faithful geyser.     

Markov chain Monte Carlo exact inference for binomial and multinomial logistic regression models

We develop Metropolis-Hastings algorithms for exact conditional inference, including goodness-of-fit tests, confidence intervals and residual analysis, for binomial and multinomial logistic regression models. We present examples where the exact results, obtained by enumeration, are available for comparison. We also present examples where Monte Carlo methods provide the only feasible approach for exact inference.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

Accuracy of edge detection methods with local information in speckled imagery

We compare the accuracy of five approaches for contour detection in speckled imagery. Some of these methods take advantage of the statistical properties of speckled data, and all of them employ active contours using B-spline curves. Images obtained with coherent illumination are affected by a noise called speckle, which is inherent to the imaging process. These data have been statistically modeled by a multiplicative model using the G0 distribution, under which regions with different degrees of roughness can be characterized by the value of a parameter. We use this information to find boundaries between regions with different textures. We propose and compare five strategies for boundary detection: three based on the data (maximum discontinuity on raw data, fractal dimension and maximum likelihood) and two based on estimates of the roughness parameter (maximum discontinuity and anisotropic smoothed roughness estimates). In order to compare these strategies, a Monte Carlo experience was performed to assess the accuracy of fitting a curve to a region. The probability of finding the correct edge with less than a specified error is estimated and used to compare the techniques. The two best procedures are then compared in terms of their computational cost and, finally, we show that the maximum likelihood approach on the raw data using the G0 law is the best technique.     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes

Summary.  We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis–Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein–Uhlenbeck processes. We apply our methodology to the US dollar–Deutschmark exchange rate.     

Likelihood Inference for Unions of Interacting Discs

Abstract. This is probably the first paper which discusses likelihood inference for a random set using a germ‐grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation‐based maximum likelihood inference and the effect of specifying different reference Poisson models.     

Separability tests for high-dimensional,low-sample size multivariate repeated measures data

Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within-subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our approach provides useful guidance for high dimension, low-sample size data that preclude using standard likelihood-based tests. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrate the approaches appeal.     

Inferactive data analysis

We describe inferactive data analysis, so-named to denote an interactive approach to data analysis with an emphasis on inference after data analysis. Our approach is a compromise between Tukey's exploratory and confirmatory data analysis allowing also for Bayesian data analysis. We see this as a useful step in concrete providing tools (with statistical guarantees) for current data scientists. The basis of inference we use is (a conditional approach to) selective inference, in particular its randomized form. The relevant reference distributions are constructed from what we call a DAG-DAG—a Data Analysis Generative DAG, and a selective change of variables formula is crucial to any practical implementation of inferactive data analysis via sampling these distributions. We discuss a canonical example of an incomplete cross-validation test statistic to discriminate between black box models, and a real HIV dataset example to illustrate inference after making multiple queries on data.     

Parallel inference for massive distributed spatial data using low-rank models

Due to rapid data growth, statistical analysis of massive datasets often has to be carried out in a distributed fashion, either because several datasets stored in separate physical locations are all relevant to a given problem, or simply to achieve faster (parallel) computation through a divide-and-conquer scheme. In both cases, the challenge is to obtain valid inference that does not require processing all data at a single central computing node. We show that for a very widely used class of spatial low-rank models, which can be written as a linear combination of spatial basis functions plus a fine-scale-variation component, parallel spatial inference and prediction for massive distributed data can be carried out exactly, meaning that the results are the same as for a traditional, non-distributed analysis. The communication cost of our distributed algorithms does not depend on the number of data points. After extending our results to the spatio-temporal case, we illustrate our methodology by carrying out distributed spatio-temporal particle filtering inference on total precipitable water measured by three different satellite sensor systems.     

Bayesian P-splines and advanced computing in R for a changepoint analysis on spatio-temporal point processes

ABSTRACT

This work presents advanced computational aspects of a new method for changepoint detection on spatio-temporal point process data. We summarize the methodology, based on building a Bayesian hierarchical model for the data and declaring prior conjectures on the number and positions of the changepoints, and show how to take decisions regarding the acceptance of potential changepoints. The focus of this work is about choosing an approach that detects the correct changepoint and delivers smooth reliable estimates in a feasible computational time; we propose Bayesian P-splines as a suitable tool for managing spatial variation, both under a computational and a model fitting performance perspective. The main computational challenges are outlined and a solution involving parallel computing in R is proposed and tested on a simulation study. An application is also presented on a data set of seismic events in Italy over the last 20 years.