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1. Graphical Gaussian process models for highly multivariate spatial data

   https://doi.org/10.1093/biomet/asab061  

   Dey, Debangan; Datta, Abhirup; Banerjee, Sudipto (December 2021, Biometrika)

   Summary For multivariate spatial Gaussian process models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence between the variables. This is undesirable, especially in highly multivariate settings, where popular cross-covariance functions, such as multivariate Matérn functions, suffer from a curse of dimensionality as the numbers of parameters and floating-point operations scale up in quadratic and cubic order, respectively, with the number of variables. We propose a class of multivariate graphical Gaussian processes using a general construction called stitching that crafts cross-covariance functions from graphs and ensures process-level conditional independence between variables. For the Matérn family of functions, stitching yields a multivariate Gaussian process whose univariate components are Matérn Gaussian processes, and which conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Matérn Gaussian process to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling. 

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2. Partial separability and functional graphical models for multivariate Gaussian processes

   https://doi.org/10.1093/biomet/asab046  

   Zapata, J; Oh, S Y; Petersen, A (October 2021, Biometrika)

   Summary The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, compared with multivariate data, a key difficulty is that the covariance operator is compact and thus not invertible. This paper addresses the general problem of covariance modelling for multivariate functional data, and functional Gaussian graphical models in particular. As a first step, a new notion of separability for the covariance operator of multivariate functional data is proposed, termed partial separability, leading to a novel Karhunen–Loève-type expansion for such data. Next, the partial separability structure is shown to be particularly useful in providing a well-defined functional Gaussian graphical model that can be identified with a sequence of finite-dimensional graphical models, each of identical fixed dimension. This motivates a simple and efficient estimation procedure through application of the joint graphical lasso. Empirical performance of the proposed method for graphical model estimation is assessed through simulation and analysis of functional brain connectivity during a motor task. 

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3. Decomposing Gaussians with unknown covariance

   https://doi.org/10.1093/biomet/asaf057  

   Dharamshi, A; Neufeld, A; Gao, L L; Bien, J; Witten, D (January 2026, Biometrika)

   Abstract Common workflows in machine learning and statistics rely on the ability to partition the information in a dataset into independent portions. Recent work has shown that this may be possible even when conventional sample splitting is not, such as when the number of samples, $ n $, is one or when observations are not independent and identically distributed. In the case of multivariate Gaussian data, these alternatives to sample splitting require knowledge of the covariance matrix. In many important problems, such as in spatial or longitudinal data analysis and in graphical modelling, the covariance matrix may be unknown and even of primary interest. Therefore, in this work we develop new approaches for decomposing multivariate Gaussians with unknown covariance. First, we present a general algorithm that encompasses all previous decomposition methods for Gaussian data as special cases and which can further handle the case of unknown covariance. It yields a new and more flexible alternative to sample splitting when $$ n \,{\gt}\, 1 $$. When $ n=1 $, we prove that it is impossible to partition the information in a multivariate Gaussian into independent portions without knowing the covariance matrix. Hence, we use the general algorithm to decompose a single multivariate Gaussian with unknown covariance into dependent parts with tractable conditional distributions and demonstrate their use for inference and validation. The proposed decomposition strategy extends naturally to Gaussian processes. In simulations and for electroencephalography data, we apply these decompositions to the tasks of model selection and post-selection inference in settings where alternative strategies are unavailable. 

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4. Graph-constrained analysis for multivariate functional data

   https://doi.org/10.1016/j.jmva.2025.105428  

   Dey, Debangan; Banerjee, Sudipto; Lindquist, Martin A; Datta, Abhirup (May 2025, Journal of Multivariate Analysis)

   The manuscript considers multivariate functional data analysis with a known graphical model among the functional variables representing their conditional relationships (e.g., brain region-level fMRI data with a prespecified connectivity graph among brain regions). Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model, and cannot preserve knowledge of a given graph. We propose a method for multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed recently for constructing optimal multivariate covariance functions that retain a given graphical model. The theoretical connection helps to design a new algorithm that leverages Dempster’s covariance selection for obtaining the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while respecting the graph and retaining computational scalability. The benefits of the proposed algorithms are illustrated using empirical experiments and a neuroimaging application. 

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5. A Self-Censoring Model for Multivariate Nonignorable Nonmonotone Missing Data

   https://doi.org/10.1111/biom.13916  

   Li, Yilin; Miao, Wang; Shpitser, Ilya; Tchetgen_Tchetgen, Eric J (December 2023, Biometrics)

   Abstract We introduce an itemwise modeling approach called “self-censoring” for multivariate nonignorable nonmonotone missing data, where the missingness process of each outcome can be affected by its own value and associated with missingness indicators of other outcomes, while conditionally independent of the other outcomes. The self-censoring model complements previous graphical approaches for the analysis of multivariate nonignorable missing data. It is identified under a completeness condition stating that any variability in one outcome can be captured by variability in the other outcomes among complete cases. For estimation, we propose a suite of semiparametric estimators including doubly robust estimators that deliver valid inferences under partial misspecification of the full-data distribution. We also provide a novel and flexible global sensitivity analysis procedure anchored at the self-censoring. We evaluate the performance of the proposed methods with simulations and apply them to analyze a study about the effect of highly active antiretroviral therapy on preterm delivery of HIV-positive mothers. 

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