Comparison of Numerical Solvers for Differential Equations for Holonomic Gradient Method in Statistics
This work addresses the computational challenge of evaluating normalizing constants for statisticians, but it is incremental as it compares existing methods rather than introducing new ones.
The paper tackles the problem of evaluating normalizing constants in statistics by comparing numerical solvers for linear ordinary differential equations derived from the holonomic gradient method, focusing on their application to definite integrals of holonomic functions.
Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation (ODE) with respect to a curve in the parameter space. We can evaluate the integral by solving the linear ODE numerically. This approach to evaluate numerically definite integrals is called the holonomic gradient method (HGM) and it is useful to evaluate several normalizing constants in statistics. We will discuss and compare methods to solve linear ODE's to evaluate normalizing constants.