Probabilistic Modeling Using Tree Linear Cascades

TL;DR

This paper introduces tree linear cascades, a flexible class of structural equation models with uncorrelated errors, and demonstrates their identifiability and efficient fitting methods, with applications to stock price data.

Contribution

It presents the concept of tree linear cascades, proves their identifiability despite weak assumptions, and provides an analytical solution for fitting these models.

Findings

Models are identifiable under weak assumptions.

Analytical solution for constrained regression in tree-structured models.

Application demonstrated on stock price data.

Abstract

We introduce tree linear cascades, a class of linear structural equation models for which the error variables are uncorrelated but need not be Gaussian nor independent. We show that, in spite of this weak assumption, the tree structure of this class of models is identifiable. In a similar vein, we introduce a constrained regression problem for fitting a tree-structured linear structural equation model and solve the problem analytically. We connect these results to the classical Chow-Liu approach for Gaussian graphical models. We conclude by giving an empirical-risk form of the regression and illustrating the computationally attractive implications of our theoretical results on a basic example involving stock prices.