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Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. (English) Zbl 1079.65006

A particle system is a collection (ϑ (j,H) ,w (j,H) ) jH where ϑ (j,H) Θ are “particles” and w (j,H) >0 are their “weights”. The system targets a distribution π on Θ if for any measurable ϕ with |E π (ϕ)|<,

E ^ H (ϕ)= j=1 H w (j,H) ϕ(ϑ (j,H) ) j=1 H w (j,H) E π (ϕ)·

A sequential Monte Carlo algorithm (a particle filter) produces recursively (using mutation-correction-resampling scheme) a sequence of particle systems which target a sequence of distributions π t on Θ t . In the Bayes estimation problems Θ t =Θ is the parameter space and π t is an a posteriori distribution of the parameter ϑ given the sample of size t. In the state-space filtering or smoothing Θ t is the space of states trajectories and π t is the conditional distribution of the trajectory given the data.

The author obtains conditions for the central limit theorem of the form H(E ^ H (ϕ)-E π (ϕ))N(0,V t (ϕ)) where V t (ϕ) is described using recursive formulae. These conditions hold for many of sequential Monte Carlo algorithms including the resample-move algorithm and the residual resampling scheme. Asymptotics of V t (ϕ) as t are investigated for Bayesian problems.

MSC:
65C05Monte Carlo methods
62F15Bayesian inference
60F05Central limit and other weak theorems
62L10Sequential statistical analysis