Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM

Round 1

Reviewer 1 Report

This well-written paper provides new useful information on deterministic approximations (in particular, the so-called Riemann approximations) concerning the well-known Expectation Maximization (EM) algorithm (see Theorems 1--4 on pages 5, 8, 10 and 15, respectively). I like the idea of sketching and explaining the proofs of these significant theorems in the main body of the paper while relegating their detailed proofs to the Appendix. Since I believe this paper will be of interest to theorists and practitioners alike, my recommendation is that (a slightly revised version of) it be accepted for publication in the journal "Algorithms".  When the authors prepare the revised version of their paper they should eliminate all the mathematical and linguistic inaccuracies in it. For example, on line 202, the word "results" should be replaced with the word "result", and on page 15, line -10, the word "hence" should be replaced with the word "is". 

Author Response

Point 1: When the authors prepare the revised version of their paper they should eliminate all the mathematical and linguistic inaccuracies in it. For example, on line 202, the word "results" should be replaced with the word "result", and on page 15, line -10, the word "hence" should be replaced with the word "is". 

Response 1: We have corrected the aforementioned typos, and proofread the papers to eliminate all the other we could find.

Reviewer 2 Report

In this article, the authors propose a class of expectation–maximization (EM) algorithms and give the proof of correctness for models of the exponential family. They introduce a method of deterministic approximation of the conditional probability density function by a Riemann sums at the expectation (E) step and prove that it satisfies the assumptions of the main result which guarantees convergence. They also study the tempered EM algorithm with the Riemann approximation added. The presented methods are illustrated through experiments.

The article is organized in a reader-friendly manner. The presented methods and their proofs are clearly written. The experiments provided explore the algorithms presented in this article.

Author Response

We thank the Reviewer for underlining the strength and contributions of our study.