How can the (Ensemble) Kalman filter be viewed as a MCMC algorithm?

I am struggling with quotes like

The EnKF applies a Markov chain Monte Carlo (MCMC) method ... (1, p. 6)

or

In fact, the Kalman filter is a MCMC algorithm in the case of a linear and Gaussian state space model with known parameters. (2, p. 5)

I cannot find the construction of a Markov chain in the description of neither linear nor ensemble Kalman filter. Could anyone please clarify how this is meant?

[1] Evensen, Geir. "The ensemble Kalman filter for combined state and parameter estimation." Control Systems, IEEE 29.3 (2009): 83-104.

[2] Johannes, Michael, and Nicholas Polson. "MCMC methods for financial econometrics." The Handbook of Financial Econometrics 65 (2002).

• IMHO, the second quote is incorrect. The Kalman filter with backward sampling, usually referred to as the forward filtering backward sampling algorithm, provides a Monte Carlo (but not a Markov chain) approach to jointly sampling the states in a linear Gaussian state space model with known parameters. – jaradniemi Apr 7 '16 at 16:47
• Could you add references to the sources you cite rather than just linking to them as "source"? This would help future-proof your question in case external links change address. – Silverfish Apr 7 '16 at 18:37
• For what it's worth, the 2nd source seems to have been "softened" a bit in what looks like an updated/final version: www0.gsb.columbia.edu/faculty/mjohannes/PDFpapers/JP_2006.pdf – Matt Krause Apr 7 '16 at 22:16
• @jaradniemi If it does not provide a Markov chain, how is it then a Markov Chain Monte Carlo (MCMC) algorithm? – arney Apr 8 '16 at 13:14

1 Answer

To my knowledge, neither the WEnKF or the KF are MCMC algorithms. I think that most of the ambiguity comes from the fact these algorithms use the markovian structure of the state space to estimate sequentially the filtering distribution $p(x_t|y_{1:t})$ using particle-based methods (so there are indeed Markov chain and Monte-Carlo inside). But in any case, the WEnKF generates any Markov chain having as stationary distribution the filtering distribution (which is the general purpose of a MCMC). If it must be categorized, an appropriate term for it could be a "particle system with mean–field interaction" but definitively not a MCMC.