Computation of maximum likelihood estimates in cyclic structural equation models

TL;DR

This paper introduces a block-coordinate descent algorithm for efficiently computing maximum likelihood estimates in cyclic linear structural equation models, overcoming convergence issues of traditional methods.

Contribution

It proposes a novel block-coordinate descent approach that solves closed-form updates for models with feedback cycles, with a characterization of models where the method is well-defined.

Findings

Algorithm converges for a broad class of models.

Closed-form solutions simplify computations.

Characterization criteria are polynomial-time checkable.

Abstract

Software for computation of maximum likelihood estimates in linear structural equation models typically employs general techniques from non-linear optimization, such as quasi-Newton methods. In practice, careful tuning of initial values is often required to avoid convergence issues. As an alternative approach, we propose a block-coordinate descent method that cycles through the considered variables, updating only the parameters related to a given variable in each step. We show that the resulting block update problems can be solved in closed form even when the structural equation model comprises feedback cycles. Furthermore, we give a characterization of the models for which the block-coordinate descent algorithm is well-defined, meaning that for generic data and starting values all block optimization problems admit a unique solution. For the characterization, we represent each model by its mixed graph (also known as path diagram), which leads to criteria that can be checked in time that is polynomial in the number of considered variables.