Bayesian Analysis

Separable covariance arrays via the Tucker product, with applications to multivariate relational data

Peter D. Hoff

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Modern datasets are often in the form of matrices or arrays, potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we discuss an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We show how a particular array-matrix product can be used to generate the class of array normal distributions having separable covariance structure. We derive some properties of these covariance structures and the corresponding array normal distributions, and show how the array-matrix product can be used to define a semi-conjugate prior distribution and calculate the corresponding posterior distribution. We illustrate the methodology in an analysis of multivariate longitudinal network data which take the form of a four-way array.

Article information

Bayesian Anal., Volume 6, Number 2 (2011), 179-196.

First available in Project Euclid: 13 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 60E05: Distributions: general theory 62H05: Characterization and structure theory 62H12: Estimation 62J10: Analysis of variance and covariance 62P20: Applications to economics [See also 91Bxx]

Gaussian matrix normal multiway data network tensor Tucker decomposition


Hoff, Peter D. Separable covariance arrays via the Tucker product, with applications to multivariate relational data. Bayesian Anal. 6 (2011), no. 2, 179--196. doi:10.1214/11-BA606.

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See also

  • Related item: Genevera I. Allen. Comment on article by Hoff. Bayesian Anal., Vol. 6, Iss. 2 (2011), 197-201.
  • Related item: Hedibert Freitas Lopes. Comment on article by Hoff. Bayesian Anal., Vol. 6, Iss. 2 (2011), 203-204.