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State filtering for parameter inference

Simultaneous inference and estimation by filtering

State filters are cool for estimating time-varying hidden states. How about learning the parameters of the model generating your states? Classic ways that you can do this in dynamical systems include basic linear system identification, and general system identification. But can you identify the fixed parameters (not just hidden states) with a state filter? Yes, modulo certain convergence conditions.

I will explain how to do that here, as soon as I understand myself. But you could always see wikipedia.

Related: indirect inference. Precise relation will have to wait, since I currently do not care enough about indirect inference.

Contents

Questions

Basic Construction

There are a few variations.

But we start with the basic continuous time state space model.

Here we have an unobserved Markov state process \(x(t)\) on \(\mathcal{X}\) and an observation process \(y(t)\) on \(\mathcal{Y}\). For now they will be assumed to be finite dimensional vectors over \(\mathbb{R}.\) They will additionally depend upon a vector of parameters \(\theta\) We observe the process at discrete times \(t(1:T)=(t_1, t_2,\dots, t_T),\) and we will write the observations \(y(1:T)=(y(t_1), y(t_2),\dots, y(1_T)).\)

We presume our processes are completely specified by the following conditional densities (which might not have closed-form expression)

The transition density ..math:

f(x(t_i)|x(t_{i-1}), \theta)

The observation density (which seems overgeneral TBH…)

To be continued…

Convergence

Awaiting filing

Recently enjoyed: Sahani Pathiraja’s state filter does something cool, in attempting to identify process model noise - a conditional nonparametric density of process errors, that may be used to come up with some neat process models. I’m not convinced about her use of kernel density estimators, since these scale badly precisely when you need them most, in high dimension; but any nonparametric density estimator would, I assume, work.

Implementations

pomp does state filtering inference in R.

For some example of doing this in Stan see Sinhrks’ statn-statespace.

Refs

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Lindström, E., Ionides, E., Frydendall, J., & Madsen, H. (2012) Efficient Iterated Filtering. In IFAC-PapersOnLine (System Identification, Volume 16) (Vol. 45, pp. 1785–1790). IFAC & Elsevier Ltd. DOI.
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