# Optimal inference for nonlinear state space models

When considering a linear-Gaussian state space model, it is often referred that, optimal inference is tractable which is very rare in state space models. When considering a nonlinear state space model, it is said that, in general the posterior distribution is intractable, therefore, inference methods should be used to approximate the posterior.

What I would like to ask is that, what is the exceptions for nonlinear state space models? Which nonlinear state space model is tractable analytically? Are there any examples?

PS: For instance, Doucet says:

Given the ﬁltering distribution one can then routinely proceed to ﬁltered point estimates such as the posterior mode or mean of the state. This problem is known as the Bayesian ﬁltering problem or the optimal ﬁltering problem. [...] Except in a few special cases, including linear Gaussian state space models (Kalman ﬁlter) and hidden ﬁnite-state space Markov chains, it is impossible to evaluate these distributions analytically.

To reexpress, I would like to ask that, which nonlinear state space model is tractable?

• Is there a specific book or paper that you're referencing? It may be helpful if you mention what it is. – Phonon Oct 1 '12 at 14:04
• If you don't get any reasonable answers here, consider requesting migration to stats.stackexchange.com. – Phonon Oct 1 '12 at 14:40
• Often nonlinear models are linearized (see en.wikipedia.org/wiki/Linearization), which makes them tractable to linear methods. – Jim Clay Oct 1 '12 at 17:14
• Phonon: I couldn't figure out how to migrate questions. Can you give advise? / Jim thanks for that, but I am looking for tractable nonlinear models. Linearization is a sort of approximate inference, in particular Extended Kalman Filter is an example of it. – Deniz Oct 1 '12 at 18:42
• @DenizAkyildiz If you want it migrated, I can do it for you. – Phonon Oct 2 '12 at 14:25