# Parsimonious graphical dependence models

## News

• Author: Harry Joe
• Date: 03 July 2019

Multivariate models with parsimonious dependence have been used for a large number of variables, and have mainly been developed for multivariate Gaussian. Graphical dependence model representations include Bayesian networks, conditional independence graphs, and truncated vines. The class of Gaussian truncated vines is a subset of Gaussian Bayesian networks and Gaussian conditional independence graphs, but has an extension to non‐Gaussian dependence with (i) combinations of continuous and discrete random variables with arbitrary univariate margins, and (ii) accommodation of latent variables. To illustrate the importance of graphical models with latent variables that do not rely on the Gaussian assumption, a paper published in The Canadian Journal of Statistics explains the combined factor‐vine structure is presented and applied to a data set of stock returns.

The paper is available via this link and the author explains his findings in further detail below:

Parsimonious graphical dependence models constructed from vines

Harry Joe

The Canadian Journal of Statistics, Volume 46, Issue 4, December 2018, pages 532-555

The aim of graphical dependence models is to show relations of many variables such as log returns of many stocks or items in a psychological test to measure an abstract concept. In several methods for graphical models, edges are drawn only for pairs of variables with the strongest dependence or conditional dependence. Parsimonious graphs with the number of edges being of the order of the number of variables have many edges representing conditional dependence.

The methods of Bayesian networks, conditional independence graphs, and truncated vines are compared on whether they can (a) readily extend from Gaussian (linear) dependence to non-Gaussian dependence and (b) accommodate unobserved (latent) variables. Criterion (a) is important because with large data sets that are now common, the classical Gaussian assumption is seldom valid. Criterion (b) is important because dependence in many observed related variables can commonly be explained by latent variables. Based on comparisons of computational feasibility and fit to multivariate data, the truncated vine approach is best to satisfy (a) and (b). Vines have been developed much later than Bayesian networks and conditional independence graphs but are now widely applied for non-Gaussian dependence. Truncated vines consists of a nested sequence of trees; the parsimony comes from tree 1 linking d-1 pairs among d variables, and a few higher order trees connecting pairs of variables to explaining conditional dependencies. The main illustrative application is for log returns for 30 stocks in the Dow Jones index; the dependence can be well explained through one latent variable combined with residual conditional dependence given the latent variable.

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