  Approximate Bayesian computation (ABC) methods are standard tools for
inferring parameters of complex models when the likelihood function is
analytically intractable. A popular approach to improving the poor acceptance
rate of the basic rejection sampling ABC algorithm is to use sequential Monte
Carlo (ABC SMC) to produce a sequence of proposal distributions adapting
towards the posterior, instead of generating values from the prior distribution
of the model parameters. Proposal distribution for the subsequent iteration is
typically obtained from a weighted set of samples, often called particles, of
the current iteration of this sequence. Current methods for constructing these
proposal distributions treat all the particles equivalently, regardless of the
corresponding value generated by the sampler, which may lead to inefficiency
when propagating the information across iterations of the algorithm. To improve
sampler efficiency, we introduce a modified approach called stratified distance
ABC SMC. Our algorithm stratifies particles based on their distance between the
corresponding synthetic and observed data, and then constructs distinct
proposal distributions for all the strata. Taking into account the distribution
of distances across the particle space leads to substantially improved
acceptance rate of the rejection sampling. We further show that efficiency can
be gained by introducing a novel stopping rule for the sequential process based
on the stratified posterior samples and demonstrate these advances by several
examples.
