I have a HMM where forward-backward probability increases in Iteration 1, then it decreases and then increases (as well as converges). Probability values after iterations 0,1,2,3,4 are:
1.1E-15, 1.5E-11, 9.7E-17, 1.19E-16, 1.21E-16
I think we can consider 'i'th iteration as model output only if probability reaches local maxima at that point as well as it converges. Can the output of this model not be trusted?
Please refer to the excel file available at: https://www.cs.jhu.edu/~jason/465/PowerPoint/lect24-hmm.xls I'm referring to the 'p' values in cells HB3-HL3. In my problem, I get 1.1E-15 1.5E-11 9.7E-17 1.19E-16 1.21E-16 after iterations 0,1,2,3,4.
In order to identify if your HMM has converged you should check not only the likelihood function probability, Pr(O|theta), but also the observation matrix estimations and the parameters which characterize the sojourn time per hidden state. In your case, you assume geometric state duration distribution for the sojourn time per hidden state because you use the HMM. In my case, I used the Non-Homogeneous Hidden Semi Markov Model (NHHSMM) which means that I can relax the aforementioned assumption. In this direction I choose the Weibull probability density function in order to describe the sojourn time per hidden state. This means that I checked the estimations of the scale (wa) and shape (wb) Weibull parameters from iteration to iteration. As you can see in the attached file it is possible to have convergence regarding the Pr(O|theta) but the estimations of the observation or scale/shape parameters don’t be converged. Therefore, you should be sure that all of your model’s parameters are converged after some iterations. More information regarding the NHHSMM you can find in my paper with title 'Fatigue damage diagnostics and prognostics of composites utilizing structural health monitoring data and stochastic processes'.
Article Fatigue damage diagnostics and prognostics of composites uti...
Hello. In addition to the informative and interesting answer of Nick Eleftheroglou, I would suggest that you consider numerical errors since you are dealing with very small probabilities on the order of 10^-16. The expectation maximisation algorithm guarantees local convergence of the maximum likelihood parameter estimates only. Since HMM is based on the EM algorithm, you will only get local optimality. What you might consider is running multiple HMMs with different numbers of parameters (and maybe different "structures") and trying to find the "most parsimonious" model for your data based on something like Akaike's information theoretic criterion or similar. You will always get a better fit by modelling the noise, but this is probably not what you want. The reference below gives an example.
Article Evaluation and Estimation of Various Markov Models with Appl...
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