We address the problem of approximating the posterior probability distribution of the fixed
parameters of a state-space dynamical system using a sequential Monte Carlo method. The
proposed approach relies on a nested structure that employs two layers of particle filters to
approximate the posterior probability measure of the static parameters and the dynamic state
variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo
square” (SMC
2
) algorithm. However, unlike the SMC
2
scheme, the proposed technique operates
in a purely recursive manner. In particular, the computational complexity of the recursive steps
of the method introduced herein is constant over time. We analyse the approximation of integrals
of real bounded functions with respect to the posterior distribution of the system parameters
computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the
approximation errors vanish asymptotically in
L
p
(
p
≥
1) with convergence rate proportional to
1
√
N
+
1
√
M
, where
N
is the number of Monte Carlo samples in the parameter space and
N
×
M
is the number of samples in the state space. This result also holds for the approximation of the
joint posterior distribution of the parameters and the state variables. We discuss the relationship
between the SMC
2
algorithm and the new recursive method and present a simple example in
order to illustrate some of the theoretical findings with computer simulations.
Keywords:
particle filtering, parameter estimation, model inference, state space models, recursive
algorithms, Monte Carlo, error bounds.