Bayesian Analysis (2011)
6, Number 2, pp. 203–204
Comment on Article by Hoff Hedibert Freitas Lopes∗ I would like to start by congratulating Peter for sharing a stimulating paper on Bayesian multidimensional modeling of Gaussian data. The paper shows in a constructive and didactical manner that the array normal distribution shares most of the nice properties of the well-known multivariate normal and the matrix normal distributions. One of the nicest features of this new class of distributions is its flexibility to easily accommodate extensions for time series, spatio-temporal and other types of commonly known statistical models for multivariate and matrix-variate data. The following points are very subjective and relate to my own research interest, but I have no doubt other readers will find the paper very interesting and will formulate their own questions and make connections to their own research on multidimensional modeling of Gaussian data. Seemingly unrelated regressions. One important class of of separable covariance structures in econometrics is the class of seemingly unrelated regression (SUR) models, popularized by Arnold Zellner in the 1960s (Zellner, 1962, 1963), which are now common place, for instance, in i) various versions of vector autoregressive (VAR) modeling of macroeconomic and financial econometrics data and in ii) multinomial probit modeling of microeconomic and marketing data. Factor models. In Lopes, Salazar and Gamerman (2008) and Lopes et al. (2010) we handle space-time dependences and multi-level dependences, respectively, through the prior distributions on factor loadings and common factors themselves. Calder (2007), for instance, models multiple pollutant levels at multiple locations and multiple time periods. The discussion of the current paper suggests some basic factor model extensions to the array normal distribution inspired by Lathauwer et al.’s (2000) higher order singular value decompositions. Indeed, this is a very important extension and it would be of great value to search for a common ground between these highly “regularlized”, dynamic and hierarchical factor models and the array factor normal models. For a more concrete example, consider the international trade of Section 4. A hierarchical model with a similar flavor to the ones in Lopes et al. (2010) would be yijt |ft ∼ N (βij fijt , H) with
βij |θi ∼ N (θi , R)
for i, j = 1, . . . , 30, i 6= j, H = diag(h21 , . . . , h26 ) and, for simplicity, βij loading vectors of dimension 29. The matrix R would take into account the countries’ spatial (possibly economical, not geographical) co-dependence, while θi would take into account country specific co-variates. The next level of the hierarchy would assume that θi ∼ N (θ0 , V0 ), with θ0 and V0 taking standard multivariate normal and inverse-Wishart priors. Finally, the factor score fijt would follow, say, a standard first order autoregression, i.e. fijt ∼ 2 2 N (φij fij,t−1 , τij ) and parameters (φij , τij ) would follow standard univariate normal and ∗ The
University of Chicago Booth School of Business, Chicago, IL, mailto:
[email protected]
© 2011 International Society for Bayesian Analysis
DOI:10.1214/11-BA606B
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Comment on Article by Hoff
inverse gamma distributions. Separability. Would not the restriction of separable covariance structures prevent the class of array normal distributions to be fully appreciated by, say, those working with space-time geostatistical models? For example, the spatial dynamic factor model (SDFM) of Lopes et al. (2008) introduces conditional separability when common factors are known, but the covariance structure of the observables is not necessarily separable.
References Calder, C. A. (2007). “Dynamic factor process convolution models for multivariate space-time data with application to air quality assessment.” Environmental and Ecological Statistics, 14: 229–247. Lathauwer, L. D., Moor, B. D., and Vandewalle, J. (2000). “A multilinear singular value decomposition.” SIAM Journal on Matrix Analysis and Applications, 21(4): 1253–1278. Lopes, H. F., Salazar, E., and Gamerman, D. (2008). “Spatial dynamic factor models.” Bayesian Analysis, 3: 759–792. Lopes, H. F., Schmidt, A. M., Salazar, E., Gomez, M., and Achkar, M. (2010). “Measuring the vulnerability of the Uruguayan population to vector-borne diseases via spatially hierarchical factor models.” Technical report, University of Chicago Booth School of Business. http://faculty.chicagobooth.edu/hedibert.lopes/ research/pdf/LSSGA-2011.pdf. Zellner, A. (1962). “An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias.” Journal of the American Statistical Association, 57: 348–368. — (1963). “Estimators for seemingly unrelated regression equations: some exact finite sample results.” Journal of the American Statistical Association, 58: 977–992.
