Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

Modelling the relationship between directional variables is a nearly unexplored field. The bivariate wrapped Cauchy distribution has recently emerged as the first closed family of bivariate directional distributions (marginals and conditionals belong to the same family). In this paper, we introduce a tree-structured Bayesian network suitable for modelling directional data with bivariate wrapped Cauchy distributions. We describe the structure learning algorithm used to learn the Bayesian network. We also report some simulation studies to illustrate the algorithms including a comparison with the Gaussian structure learning algorithm and an empirical experiment on real morphological data from juvenile rat somatosensory cortex cells.

[1]  P. Spirtes,et al.  Causation, prediction, and search , 1993 .

[2]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[3]  Nir Friedman,et al.  Probabilistic Graphical Models , 2009, Data-Driven Computational Neuroscience.

[4]  Aurel Wintner,et al.  On the Shape of the Angular Case of Cauchy's Distribution Curves , 1947 .

[5]  Shogo Kato,et al.  A Möbius transformation-induced distribution on the torus , 2015 .

[6]  Shogo Kato,et al.  A distribution for a pair of unit vectors generated by Brownian motion , 2009, 0909.1221.

[7]  Concha Bielza,et al.  Dendritic branching angles of pyramidal cells across layers of the juvenile rat somatosensory cortex , 2016, The Journal of comparative neurology.

[8]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[9]  Terje O. Espelid,et al.  An adaptive algorithm for the approximate calculation of multiple integrals , 1991, TOMS.

[10]  C. N. Liu,et al.  Approximating discrete probability distributions with dependence trees , 1968, IEEE Trans. Inf. Theory.

[11]  Paul Van Dooren,et al.  An adaptive algorithm for numerical integration over the n-cube , 1976 .

[12]  Hetunandan Kamisetty,et al.  The von Mises Graphical Model: Regularized Structure and Parameter Learning (CMU-CS-11-129/CMU-CB-11-101) , 2011 .

[13]  Nir Friedman,et al.  Probabilistic Graphical Models - Principles and Techniques , 2009 .

[14]  S. R. Jammalamadaka,et al.  Topics in Circular Statistics , 2001 .

[15]  Kanti V. Mardia,et al.  Bayesian analysis for bivariate von Mises distributions , 2010 .

[16]  Paul Levy,et al.  L'addition des variables aléatoires définies sur une circonférence , 1939 .

[17]  P. McCullagh.  Möbius transformation and Cauchy parameter estimation , 1996 .

[18]  Christophe Ley,et al.  Modern Directional Statistics , 2017 .

[19]  J. Kent.  The Fisher‐Bingham Distribution on the Sphere , 1982 .

[20]  Kanti V. Mardia,et al.  Statistics of Directional Data , 1972 .

[21]  E. Batschelet.  Circular statistics in biology , 1981 .

[22]  David Heckerman,et al.  Learning Gaussian Networks , 1994, UAI.

[23]  Kanti V. Mardia,et al.  A multivariate von mises distribution with applications to bioinformatics , 2008 .

[24]  Kiheung Kim.  Protein , 2005, The Lancet.

[25]  S. R. Jammalamadaka,et al.  Directional Statistics, I , 2011 .