I don’t think I can follow your thoughts here - maybe you have in mind some more complex version of the model than the simple one you presented? In HMMs transition probabilities are always state dependent and switching states resets the transition rates by definition (that’s the Markov property), so I don’t really see a problem here.
Yes, but in the end it would likely be most easy to implement as a single-variable HMM with 2K states.
I however think the bigger problem here is likely that the model as given is unlikely to have its parameters informed by data in several ways:
- as a gets small relative to the distance between thresholds for switching, changing \omega and \alpha will IMHO not change the dynamics in a noticeable way
- in the picture you’ve shown, I think doubling \omega would lead to almost the same dynamics. More generally, there will always be many sinusoids that cross a given finite set of points, hence multimodality in posterior is highly likely unles C is given or can be very tightly constrained a priori.
- it looks like at least for some configuration the dynamics of H would be well approximated (and thus indistinguishable from) a linear increase/decrease. But whenver that’s the case, there will likely be many vaslty different combinations of your parameters that lead to almost the same approximately linear increase/decrase.
In fact even fitting a simple noisy sinusoid is a tricky problem (see Ideas for modelling a periodic timeseries for some fun discussion on the topic - it is possible we all missed something basic, but all the “solutions” were quite involved. It is possible the marginalization implied by HMM would help, but I am not completely sure.