A Legacy of EM Algorithms

Area of Science:

• Computational Statistics
• Statistical Algorithms
• Mathematical Optimization

Background:

• Nan Laird's significant contributions to computational statistics, particularly the expectation-maximisation (EM) algorithm and longitudinal modeling.
• The widespread citation and impact of Laird's work in statistical research.
• The need for more generalized and efficient statistical algorithms.

Purpose of the Study:

• To revisit the derivation of Laird's key algorithms using the minorisation-maximisation (MM) principle.
• To demonstrate how the MM principle generalizes the EM principle.
• To explore new algorithmic possibilities and improved convergence rates.

Main Methods:

• Application of the minorisation-maximisation (MM) principle.
• Generalization of the expectation-maximisation (EM) principle.
• Construction of surrogate functions using mathematical inequalities.

Main Results:

• The MM principle provides a unified and simplified derivation for existing algorithms.
• The MM principle can yield novel algorithms with potentially faster convergence rates than classical methods.
• The MM principle offers a flexible framework applicable beyond missing data problems.

Conclusions:

• The MM principle is a powerful generalization of the EM principle in computational statistics.
• MM algorithms offer advantages in efficiency and applicability, especially for high-dimensional data.
• This framework enhances the understanding and development of statistical algorithms.