## particle efficient importance sampling

**M**arcel Scharth and Robert Kohn just arXived a new article entitled “particle efficient importance sampling“. What is—the efficiency—about?! The spectacular diminution in variance—(the authors mention a factor of 6,000 when compared with regular particle filters!—in a stochastic volatility simulation study.

**I**f I got the details right, the improvement stems from a paper by Richard and Zhang (*Journal of Econometrics*, 2007). In a state-space/hidden Markov model setting, (non-sequential) importance sampling tries to approximate the smoothing distribution one term at a time, ie p(x_{t}|x_{t-1},y_{1:n}), but Richard and Zhang (2007) modify the target by looking at

p(y_{t}|x_{t})p(x_{t}|x_{t-1})χ_{(}x_{t-1},y_{1:n}),

where the last term χ_{(}x_{t-1},y_{1:n}) is the *normalising constant* of the proposal kernel for the previous (in *t-1*) target, k(x_{t-1}|x_{t-2},y_{1:n}). This kernel is actually parameterised as k(x_{t-1}|x_{t-2},a_{t}(y_{1:n)}) and the EIS algorithm optimises those parameters, one term at a time. The current paper expands Richard and Zhang (2007) by using particles to approximate the likelihood contribution and reduce the variance once the “optimal” EIS solution is obtained. (They also reproduce Richard’s and Zhang’s tricks of relying on the same common random numbers.

**T**his approach sounds like a “miracle” to me, in the sense(s) that (a) the “normalising constant” is far from being uniquely defined (and just as far from being constant in the parameter a_{t}) and (b) it is unrelated with the target distribution (except for the optimisation step). In the extreme case when the normalising constant is also constant… in a_{t}, this step clearly is useless. (This also opens the potential for an optimisation in the choice of χ_{(}x_{t-1},y_{1:n})…)

**T**he simulation study starts from a univariate stochastic volatility model relying on two hidden correlated AR(1) models. (There may be a typo in the definition in Section 4.1, i.e. a Φ_{i} missing.) In those simulations, EIS brings a significant variance reduction when compared with standard particle filters and particle EIS further improves upon EIS by a factor of 2 to 20 (in the variance). I could not spot in the paper which choice had been made for χ()… which is annoying as I gathered from my reading that it must have a strong impact on the efficiency attached to the name of the method!

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