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Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age.

 

Graphical models are powerful tools that are regularly used to investigate complex dependence structures in high-throughput biomedical datasets. They allow for holistic, systems-level view of the various biological processes, for intuitive and rigorous understanding and interpretations. In the context of large networks, Bayesian approaches are particularly suitable because it encourages sparsity of the graphs, incorporate prior information, and most importantly account for uncertainty in the graph structure. These features are particularly important in applications with limited sample size, including genomics and imaging studies. In this paper, we review several recently developed techniques for the analysis of large networks under non-standard settings, including but not limited to, multiple graphs for data observed from multiple related subgroups, graphical regression approaches used for the analysis of networks that change with covariates, and other complex sampling and structural settings. We also illustrate the practical utility of some of these methods using examples in cancer genomics and neuroimaging.

 

Children who experience a traumatic brain injury (TBI) are at elevated risk for a range of negative cognitive and neuropsychological outcomes. Identifying which children are at greatest risk for negative outcomes can be difficult due to the heterogeneity of TBI. To address this barrier, the current study applied a novel method of characterizing brain connectivity networks, Bayesian multi‐subject vector autoregressive modelling (BVAR‐connect), which used white matter integrity as priors to evaluate effective connectivity—the time‐dependent relationship in functional magnetic resonance imaging (fMRI) activity between two brain regions—within the default mode network (DMN). In a prospective longitudinal study, children ages 8–15 years with mild to severe TBI underwent diffusion tensor imaging and resting state fMRI 7 weeks after injury; post‐concussion and anxiety symptoms were assessed 7 months after injury. The goals of this study were to (1) characterize differences in positive effective connectivity of resting‐state DMN circuitry between healthy controls and children with TBI, (2) determine if severity of TBI was associated with differences in DMN connectivity and (3) evaluate whether patterns of DMN effective connectivity predicted persistent post‐concussion symptoms and anxiety. Healthy controls had unique positive connectivity that mostly emerged from the inferior temporal lobes. In contrast, children with TBI had unique effective connectivity among orbitofrontal and parietal regions. These positive orbitofrontal‐parietal DMN effective connectivity patterns also differed by TBI severity and were associated with persisting behavioural outcomes. Effective connectivity may be a sensitive neuroimaging marker of TBI severity as well as a predictor of chronic post‐concussion symptoms and anxiety.

 

Summary In this article, we develop a graphical modeling framework for the inference of networks across multiple sample groups and data types. In medical studies, this setting arises whenever a set of subjects, which may be heterogeneous due to differing disease stage or subtype, is profiled across multiple platforms, such as metabolomics, proteomics, or transcriptomics data. Our proposed Bayesian hierarchical model first links the network structures within each platform using a Markov random field prior to relate edge selection across sample groups, and then links the network similarity parameters across platforms. This enables joint estimation in a flexible manner, as we make no assumptions on the directionality of influence across the data types or the extent of network similarity across the sample groups and platforms. In addition, our model formulation allows the number of variables and number of subjects to differ across the data types, and only requires that we have data for the same set of groups. We illustrate the proposed approach through both simulation studies and an application to gene expression levels and metabolite abundances on subjects with varying severity levels of chronic obstructive pulmonary disease. Bayesian inference; Chronic obstructive pulmonary disease (COPD); Data integration; Gaussian graphical model; Markov random field prior; Spike and slab prior.