The TM algorithm for maximising a conditional likelihood function

This paper describes an algorithm for maximising a conditional likelihood function when the corresponding unconditional likelihood function is more easily maximised. The algorithm is similar to the EM algorithm but different as the parameters rather than the data are augmented and the conditional rather than the marginal likelihood function is maximised. In exponential families the algorithm takes a particular simple form which is computationally very close to the EM algorithm. The algorithm alternates between a T-step which calculates a tilted version of the unconditional likelihood function and an M-step which maximises it. The algorithm applies to mixed graphical chain models (Lauritzen and Wermuth, 1989) and their generalisations (Edwards, 1990), and was developed with these in mind, but it may have applications beyond these. The algorithm has been implemented in the most recent version of the MINI software (Edwards, 2000), where it was named the ME algorithm. The name has been changed to avoid confusion with the algorithm described by Marschner (2001). © 2001 Biometrika Trust