Convergence and initialization of the MAPH EM iteration

Establish convergence of the iterates of the exact-M-step expectation-maximization algorithm for Multi-Absorbing Phase-Type (MAPH) distributions under the usual EM regularity conditions, determine whether the iteration converges to a global maximum despite the nonconcavity of the MAPH likelihood, and develop a principled starting rule to mitigate sensitivity to initialization.

Background

The paper proves that its exact M-step makes the observed-data likelihood non-decreasing along the EM iteration, but explicitly distinguishes this ascent property from convergence of the parameter iterates or attainment of a global maximum. Because the MAPH likelihood surface is nonconcave, the intensive-care application exhibits substantial dependence on the starting point, with single-start runs sometimes stopping well below the best fit found through multiple starts. The authors therefore identify stationary-point convergence under standard EM assumptions, global-optimum behavior, and principled initialization as unresolved problems.

References

Several directions remain open. The M-step being exact, the observed-data likelihood is non-decreasing along the iteration (\Cref{cor:monotone}), so the ascent guarantee that motivated much of the earlier discussion now holds. What monotone ascent does not deliver is convergence of the iterates themselves, nor convergence to a global maximum: the likelihood surface of a $\text{MAPH}_{m,n}$ law is not concave and the ICU fits show a marked sensitivity to the starting point, single-start runs stalling tens of nats below the best fit found by a multi-start panel. Stationary-point convergence under the usual EM regularity conditions, and a principled starting rule, are the natural next questions.

— Multi-Absorbing Phase-Type Distributions for Right-Censored Competing Risks Data  (2609.19921 - Qiao et al., 17 Sep 2026) in Section 6, Conclusion