Computational Methods in Systems Biology

A Scheme for Adaptive Selection of Population Sizes 131

For model selection, the parameterθ, the priorp 0 , as well as the proposal den-
sityKcan be decomposed into a component over the models and one over the
model specific parameters. AssumingMmodels, the parameterθis a sequence
θ=(θm)Mm=1withθm∈Rdpar,m anddpar,mthe dimension of the parameter
space of modelm. Moreover, the prior factorizes asp 0 (θ)=p 0 (θm|m)p 0 (m)and
similarly the proposal density asK(θ)=Kparameter(θm|m)Kmodel(m).

Algorithm 2.SamplePopulation

Input:K,p 0 ,,n,sdata, Simulate, ComputeDistance

Output:(P, D)

P←{},D←{},Z← 0

while|P|<ndo

repeat

repeat

θ←SampleSingleParameter(K)

untilp 0 (θ)>0;

s←Simulate(θ)

δ←ComputeDistance(s,sdata)

untilδ<;

w←p 0 (θ)/K(θ)

Z←Z+w

P←P+{(w, θ)},D←D+{(w, δ)}

end

P←{(w/Z, θ)|(w, θ)∈P},D←{(w/Z, δ)|(w, δ)∈D}

The result of the ABC-SMC Algorithm 1 detailed above is an approximation of
the posterior density, represented by a particle population (or by a KDE of it).
The algorithm is implemented as part of the pyABC framework (http://pyabc.
readthedocs.io/en/latest).

2.2 Kernel Density Estimation

In ABC-SMC the populations of weighted parameters are sequentially refined
by decreasing the acceptance threshold from generation to generation. Of cru-
cial importance in this process is the sampling of parameter proposals based on
parameters accepted in the previous generation. This is commonly achieved by
selecting an accepted parameter and perturbing it to generate a proposal [ 25 ].
This method is equivalent to sampling proposals from a non-parametric distrib-
ution approximationK, i.e. a kernel density estimate. In this study, the proposal
densitiesKin Algorithms 1 and 2 were determined by kernel density estimators
sharing the same general form.