Computational Methods in Systems Biology

130 E. Klinger and J. Hasenauer

pair of simulated datas∈S, obtained by a stochastic simulation of the model
for parameterθivia the function Simulate :Rdpar→S, and the observed data
sdata∈S.In[ 25 ], a transition kernel was used to perturb samples of the previ-
ous population to generate proposals for the subsequent population. We refor-
mulated the generation of parameter proposals using a generic (non-degenerate)
density functionK. This density is obtained from a kernel density estimator
KDE :∫ P→Kmapping a populationPto a density functionK:Rdpar→R+,

RdparK= 1. The densityKestimated on the current population serves as pro-
posal distribution for the subsequent population. For the first population, the
priorp 0 serves as proposal distribution. After each generation, the acceptance
thresholdis adapted via the function AdaptThreshold :D→and the popu-
lation sizenis adapted via the function AdaptPopulationSize : (P,KDE)→n,
which is described in Sect.2.3(Algorithm 3 ). Throughout this paper the accep-
tance threshold is adapted by setting the threshold for the subsequent population
to the median of the weighted distancesDof the previous population. The pop-
ulation size is initialized withn 0 ∈N. Sampling is stopped when either the
maximum number of allowed generationstmaxor the final acceptance threshold
min>0 is reached.

Algorithm 1.ABC-SMC

Input:tmax,min,n 0 ,sdata,p 0 , KDE, Simulate, ComputeDistance

Output:P

t← 0

K←p 0

←∞

n←n 0

whilet<tmaxand>mindo

(P, D)←SamplePopulation(K,p 0 ,,n,sdata, Simulate,

ComputeDistance)

K←KDE(P)

n←AdaptPopulationSize(P,KDE)

←AdaptThreshold(D)

t←t+1

end

The function SamplePopulation is described in Algorithm 2. There, a single can-
didate parameterθis stochastically drawn from the densityKby the function
SampleSingleParameter :K→θ. The model is then stochastically evaluated by
the function Simulate :θ→sat this parameterθ, yielding the simulated datas.
The parameterθis added to the next populationPonly if the distanceδof the
simulated datasto the observed datasdatais smaller than the current accep-
tance threshold. The two empty braces{}denote the empty sequence with
which the populationPand the corresponding distancesDare initialized. The

• operator applied to sequences denotes the concatenation of these sequences.