We present approximate algorithms for performing smoothing in a class ofhigh-dimensional state-space models via sequential Monte Carlo methods("particle filters"). In high dimensions, a prohibitively large number of MonteCarlo samples ("particles") -- growing exponentially in the dimension of thestate space -- is usually required to obtain a useful smoother. Using blockingstrategies as in Rebeschini and Van Handel (2015) (and earlier pioneering workon blocking), we exploit the spatial ergodicity properties of the model tocircumvent this curse of dimensionality. We thus obtain approximate smoothersthat can be computed recursively in time and in parallel in space. First, weshow that the bias of our blocked smoother is bounded uniformly in the timehorizon and in the model dimension. We then approximate the blocked smootherwith particles and derive the asymptotic variance of idealised versions of ourblocked particle smoother to show that variance is no longer adversely effectedby the dimension of the model. Finally, we employ our method to successfullyperform maximum-likelihood estimation via stochastic gradient-ascent andstochastic expectation--maximisation algorithms in a 100-dimensionalstate-space model.
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