## Approximate Bayesian Computation in state space models

**W**hile it took quite a while (!), with several visits by three of us to our respective antipodes, incl. my exciting trip to Melbourne and Monash University two years ago, our paper on ABC for state space models was arXived yesterday! Thanks to my coauthors, Gael Martin, Brendan McCabe, and Worapree Maneesoonthorn, I am very glad of this outcome and of the new perspective on ABC it produces. For one thing, it concentrates on the selection of summary statistics from a more econometrics than usual point of view, defining asymptotic sufficiency in this context and demonstrated that both asymptotic sufficiency and Bayes consistency can be achieved when using maximum likelihood estimators of the parameters of an auxiliary model as summary statistics. In addition, the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score vector. Using the score instead of the MLE as a summary statistics allows for huge gains in terms of speed. The method is then applied to a continuous time state space model, using as auxiliary model an augmented unscented Kalman filter. We also found in the various state space models tested therein that the ABC approach based on the marginal [likelihood] score was performing quite well, including wrt Fearnhead’s and Prangle’s (2012) approach… I like the idea of using such a generic object as the unscented Kalman filter for state space models, even when it is not a particularly accurate representation of the true model. Another appealing feature of the paper is in the connections made with indirect inference.

October 2, 2014 at 6:53 pm

Looks like a very interesting paper! One question I had while skimming the paper is concerning state estimation. In the conclusion you mention that one could use existing smoothing algorithms after obtaining the posterior for the parameters. Why could I not use the simulated states from the accepted draws in the parameter estimation stage to estimate the path of states?

October 2, 2014 at 9:19 pm

Interesting question! My quick gut-feeling answer would be to consider that the states are more poorly estimated than the parameters in the ABC process. Hence to trust the ABC approximation to the posterior on the parameters more than the ABC approximation to the posterior on the states… If only for the “dimensionality curse”. Approximating the whole distribution of the missing states is out-of-reach.