Efficient Learning of Continuous-Time Hidden Markov Models for Disease Progression
The Continuous-Time Hidden Markov Model (CT-HMM) is an attractive approach to modeling disease progression due to its ability to describe noisy observations arriving irregularly in time. However, the lack of an efficient parameter learning algorithm for CT-HMM restricts its use to very small models or requires unrealistic constraints on the state transitions. In this paper, we present the first complete characterization of efficient EM-based learning methods for CT-HMM models. We demonstrate that the learning problem consists of two challenges: the estimation of posterior state probabilities and the computation of end-state conditioned statistics. We solve the first challenge by reformulating the estimation problem in terms of an equivalent discrete time-inhomogeneous hidden Markov model. The second challenge is addressed by adapting three approaches from the continuous time Markov chain literature to the CT-HMM domain. We demonstrate the use of CT-HMMs with more than 100 states to visualize and predict disease progression using a glaucoma dataset and an Alzheimer's disease dataset.
The paper presents new methods to learn parameters and then perform inference (either Viterbi or forward/backward) for continous time HMM models (CT-HMM) with non uniformly distributed time points. They solve the inference problem for the transition rate matrix Q by constructing a discrete time inhomogeneous HMM then applying two recently developed methods to get sufficient statistics for their EM (Expfm, Unif - from Hobolth & Jensen 2011). They follow with a detailed time complexity analysis for the different flavors of the their algorithm (Table 1) before proceeding to experimental evaluation. Experiments include a 5 state synthetic dataset and two disease progression datasets for Glaucoma and AD.Pros: The paper is mostly well written and the problem addressed is interesting in terms of the computational challenge, the modeling aspects, and the biomedical domain. The authors provide mathematical foundation for their approach building on previous work byHobolth & Jensen 2011 for the specific task at hand. The authors do a good job explaining the biomedical datasets/questions and relating their results to what is known about the disease.Cons: 1. The synthetic data evaluation is not sufficient. Why 5 states? 10^6 samples seems an awful lot of samples to train on an not representative of real life problems. The difference between states emission (line 319-320) seems too easy? That may explain why all their algorithm's variants seem to preform similarly. Harder settings and varying different parameters may flush out differences better and give a bette sense of how the algorithm can perform on real life data. 2. Sec. 5.2 is rather cryptic, especially the description of the competing methods. Consider adding details about those in the appendix to make this section more clear.Minor comments: 1. Fig2 is not self contained. Referring to it first (line 40) is premature. 2. Line 91 "the rate the process" 3. Line 161 "in only at" 4. Sec 5.2: the claim about binary search (lines 348-350) is not clear. 5. Line 399 "to shown"