# 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.

## 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.

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.