A Scheme for Adaptive Selection of Population Sizes 129parameters. However, it is often difficult to select the model and its parameters
which are likely to explain a given experimental finding [ 1 ].
The Bayesian paradigm provides a natural framework to treat parameter
estimation and model selection. Unfortunately, for stochastic models it is often
impossible to calculate the likelihood efficiently, prohibiting a range of meth-
ods, such as e.g. variational inference [ 7 ], to approximate the Bayesian posterior.
This has lead to the development of Approximate Bayesian Computation (ABC)
schemes [ 2 , 14 ]. Amongst the different ABC schemes (see e.g. [ 1 , 24 ] and refer-
ences therein) one particularly popular scheme uses a Sequential Monte Carlo
(SMC) technique and is therefore called the Approximate Bayesian Computa-
tion - Sequential Monte Carlo (ABC-SMC) scheme [ 22 , 25 , 26 ]. In ABC-SMC,
the posterior distribution of a parameter is approximated through a particle
population. This population is sequentially refined from generation to genera-
tion improving the approximation. However, “it appears unfortunately difficult
to give useful general guidelines how to select the population size as it is highly
case depending” [ 16 ]. This is problematic as too small population sizes yield
large approximation errors and might even hamper convergence, while too large
population sizes result in an unnecessary computational burden.
We therefore investigated a method to select population sizes for ABC-SMC
adaptively and automatically and describe it in this paper. The method is
applied to examples with multimodal posteriors, model selection for Markov
jump process models and multiscale, agent-based models. An implementation
of the proposed method is provided as part of the pyABC framework (http://
pyabc.readthedocs.io/en/latest).
2 Methods
In the following, we introduce the ABC-SMC method and provide the corre-
sponding algorithms. Transition kernels are discussed and related to kernel den-
sity estimation. Based on this relation, we suggest a scheme for the adaptive
selection of population sizes.
2.1 ABC-SMC Algorithm
In ABC-SMC, populations of weighted parameter samples are sequentially con-
structed to approximate the posterior distribution of the parameter of interest.
The ABC-SMC scheme considered in this study, and provided in Algorithm 1 ,
is similar to the one from [ 25 ]. ByP={(wi,θi)}ni=1we denote a population^1 of
sizenof weighted parameter samples with weightswi>0,
∑
iwi= 1 and para-
metersθi∈Rdparof dimensiondpar. We denote the sequence of corresponding
weighted distancesδi∈R+byD={(wi,δi)}ni=1. The distanceδiis determined
by evaluating the distance functiond= ComputeDistance :S×S→R+for a
(^1) A population is a sequence of pairs. Pairs (w, θ) can occur multiple times in a pop-
ulation. The curly braces{}denote a finite sequence (not a set).