  We provide a new strategy built on the divide-and-conquer approach by
Lindsten et al. (2017) to investigate the smoothing problem in a hidden Markov
model. We employ this approach to decompose a hidden Markov model into
sub-models with intermediate target distributions based on an auxiliary tree
structure and produce independent samples from the sub-models at the leaf nodes
towards the original model of interest at the root. We review the target
distribution in the sub-models suggested by Lindsten et al. and propose two new
classes of target distributions, which are the estimates of the (joint)
filtering distributions and the (joint) smoothing distributions. The first
proposed type is straightforwardly constructible by running a filtering
algorithm in advance. The algorithm using the second type of target
distributions has an advantage of roughly retaining the marginals of all random
variables invariant at all levels of the tree at the cost of approximating the
marginal smoothing distributions in advance. We further propose the
constructions of these target distributions using pre-generated Monte Carlo
samples. We show empirically the algorithms with the proposed intermediate
target distributions give stable and comparable results as the conventional
smoothing methods in a linear Gaussian model and a non-linear model.
