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.

## Questions

- Is this how Särkka use state filters to do gaussian process regression?
- Ionides and King dominate my citations. Surely other people do this method too? But what are the keywords? This research is suspiciously concentrated in U Michigan, but the idea is not so esoteric. I think I am caught in a citation bubble.
- How does this work with non-Markov systems? Can we talk about mixing, or correlation decay? Should I then shoot for the new-wave mixing approaches of Kuznetsov and Mohri etc?

## 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…

## 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

- AnDH10
- Andrieu, C., Doucet, A., & Holenstein, R. (2010) Particle Markov chain Monte Carlo methods.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 72(3), 269–342. DOI. - BHIK09
- Bretó, C., He, D., Ionides, E. L., & King, A. A.(2009) Time series analysis via mechanistic models.
*The Annals of Applied Statistics*, 3(1), 319–348. DOI. - DeDJ06
- Del Moral, P., Doucet, A., & Jasra, A. (2006) Sequential Monte Carlo samplers.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 68(3), 411–436. DOI. - DeDJ11
- Del Moral, P., Doucet, A., & Jasra, A. (2011) An adaptive sequential Monte Carlo method for approximate Bayesian computation.
*Statistics and Computing*, 22(5), 1009–1020. DOI. - DoJR13
- Doucet, A., Jacob, P. E., & Rubenthaler, S. (2013) Derivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models.
*ArXiv:1304.5768 [Stat]*. - HeIK10
- He, D., Ionides, E. L., & King, A. A.(2010) Plug-and-play inference for disease dynamics: measles in large and small populations as a case study.
*Journal of The Royal Society Interface*, 7(43), 271–283. DOI. - IBAK11
- Ionides, E. L., Bhadra, A., Atchadé, Y., & King, A. (2011) Iterated filtering.
*The Annals of Statistics*, 39(3), 1776–1802. DOI. - IoBK06
- Ionides, E. L., Bretó, C., & King, A. A.(2006) Inference for nonlinear dynamical systems.
*Proceedings of the National Academy of Sciences*, 103(49), 18438–18443. DOI. - INAS15
- Ionides, E. L., Nguyen, D., Atchadé, Y., Stoev, S., & King, A. A.(2015) Inference for dynamic and latent variable models via iterated, perturbed Bayes maps.
*Proceedings of the National Academy of Sciences*, 112(3), 719–724. DOI. - LIFM12
- 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.
- RMAR07
- Rasmussen, J. G., Møller, J., Aukema, B. H., Raffa, K. F., & Zhu, J. (2007) Continuous time modelling of dynamical spatial lattice data observed at sparsely distributed times.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 69(4), 701–713. DOI.