EM algorithm for HMM-GLM models in Julia?

Specific Domains Probabilistic programming
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Hi all,

Is there anyone here who tried to use Julia for HMM-GLM models using Expectation Maximization?
I’m looking for good references to try it out. I looked at HiddenMarkovModels.jl but I’m not sure how to use it to learn the GLM parameters.

Would be great to hear if someone tried it already.

Thank you!

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Hi, I’m the developer of HiddenMarkovModels.jl. Did you see the tutorial on Markov-switching regression? There’s a lot of different terminology but it seems to be close to what you need.
There’s also a dedicated package called MarSwitching.jl, whose interface might be easier to work with for you (but perhaps less flexible).

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Wow! Interesting! Thank you for the reply!
I have some questions just to make sure I’m following:

  1. Can I use the control vector as a GLM of the inputs (features)? Is this the idea? Is it possible to use logistic regression GLM too?
  2. Can I decide if the features affect both the transition matrix and the weights of the control vector or just one of them?

I like your package as it is very flexible and intuitive, so it would be great to use it for HMM-GLM.

Thank you!

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Yes, if you have a regression y_t = f(x_t, \theta[i_t]) whose parameters \theta depend on the hidden state i_t of a Markov model, then in my package docs the “controls” are the regression inputs x_t and the “observations” are the regression outputs y_t.

Yes, but unfortunately for every new controlled model you have to code the estimation procedure yourself. The “learning” section of the tutorial demonstrates this:

  • the first part of the fit! function is standard transition estimation for HMMs,
  • the second part of the fit! function estimates \theta[i] for each possible value of i.

The key here is that to recover \theta[i], you use every pair (x_t, y_t) but you weigh them individually by the posterior probability of being in state i, which is stored in \gamma_{i,t}. In the case of linear regression there is an explicit formula (which I hope I got right), for logistic regression you have to use numerical optimization with a slightly adjusted loss function. Does that make sense?

Yes, this is alluded to in the “model” section of the tutorial. By overloading HMMs.transition_matrix(hmm, control), you can impose this additional dependency. But then the transition estimation in the fit! function must also be adapted manually.

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Perfect! This is a great answer that got me the info to get started!
I will read how to optimize the logistic parameters and will try to use your package as it seems like a great ecosystem for it. Thank you for the informative answer!

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Awesome! Ping me again if you struggle.
The logistic regression alone should be straightforward to code, you just need to find a package that allows you to estimate it with weighted samples (you can do it in MJL.jl for example, see Weights · MLJ).
However, if you let your transition matrix depend on the controls, you need to make sure that the resulting model allows easy estimation, because gradient descent is much harder for stochastic matrices.

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