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I'm looking for a way to fit a model akin to a Hidden Markov Model, with a Markov chain of hidden states. However, the observation at each time point depends on the past ~10 hidden states, instead of only on the current state.

Any idea on how this model could be fit?

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Maybe you can try to rewrite the latent Markov chain in a different form, so that your model becomes a more standard HMM. For instance suppose that the original hidden chain is denoted by $(X_t)$, then consider a new chain $(Z_t)$ such that $Z_t = (X_t, X_{t-1},...,X_{t-10})$. Then $(Z_t)$ is a Markov chain ($Z_t$ given $Z_{t-1}$ is independent of the previous states), and now your observation $Y_t$ at time $t$ has a distribution that depends on $Z_t$ only. So the model parametrized with $(Z_t)$ as the hidden chain is a standard HMM. You also need to define the initial state $Z_0$ appropriately.

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  • $\begingroup$ Thank you. I have talked to people who have tried this approach on the problem, and it has been useful, however, it did not scale well with the number of past states (we are considering using 20 instead of 10, for example). The space Z becomes too big, even if the transition matrix is sparse. Any suggestion on how to overcome this? $\endgroup$
    – Gonzalo Benegas
    Commented May 11, 2018 at 3:33
  • $\begingroup$ Well then, for exact calculation of the likelihood, it might be difficult unless you leverage model-specific information. However you can turn to simulation-based approaches. For instance, in section 6.2 of "Particle Gibbs with Ancestor Sampling" by Lindsten, Jordan and Schön (jmlr.org/papers/volume15/lindsten14a/lindsten14a.pdf), Bayesian inference is described for a model where Y_t depends on all the past states X_1, ..., X_t. In general particle filters are directly implementable in this setting and provide likelihood estimators, see e.g. the particle MCMC literature. $\endgroup$
    – Pierrot
    Commented May 12, 2018 at 0:27

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