Computational Methods in Systems Biology

A Scheme for Adaptive Selection of Population Sizes 137

05

Generationt

0. 0

0. 1

0. 2

Density variation

ECV

0.05

0.1

0.2

010

Generationt

101

102

103

104

Population size

n 0

101

102

103

104

08

Generationt

10 −^1

100

101

102

Acceptancethreshold



08

Generationt

0

500

Population size

n neff

010

Generationt

101

102

103

Population size

n

ECV

0.05

0.07

0.1

0.15

0.2

0. 10. 2

Density

variationECV

− 10

0

10

θ^2

t=1 t=2

− 10

0

10

θ^2

t=3 t=4

−10 0 10

θ 1

− 10

0

10

θ^2

t=5

−10 0 10

θ 1

t=6

0.00 0.08

Probability density

Generationt b 1

e 1

d

c

12 4

No. modesnmodes

101

102

103

Population size

n

KDE

Global

Cross-val.

Local

f

a

b 2

e 2

Fig. 2. Adaptive population size selection for multimodal posteriors. (a)
Probability density of the first six generations of an ABC-SMC run with variation
ECV =0.1, initial population sizen 0 = 500, observed datasdata =(1,1), model
parametersσ^2 =0.5andnmodes= 4. The bandwidth was selected according to the
Silverman rule (global bandwidth, Sect.2.2).(b)(b 1 ) Acceptance threshold;(b 2 ) pop-
ulation sizenand effective population sizenefffor (a). At generationt,is set to the
median of the observation–particle distances of generationt−1.(c)Mean population
size for initial population sizesn 0 =10^1 , 102 , 103 , 104 ,ECV=0.1,nmodes=4,σ^2 =2
andsdata=(1,1) averaged over 10 ABC-SMC runs.(d)TargetECV (dashed) and
actual density variation (solid) forECV =0. 05 , 0. 1 , 0 .2,σ^2 =0.5,sdata=(1,1) and
nmodes=4.(e)VariationECV and population sizenfornmodes=4,σ^2 =0.5and
sdata=(1,1). (e 1 ) Population sizenover generationtfor different variationsECV.
(e 2 ) Median population sizenas function ofECV.(f)Population sizenas function of
the number of posterior modesnmodesfor global (Silverman), cross-validated and local
bandwidth selection withECV=0.1,σ^2 = 2 andsdata=(1,1).

selection. We then examined the dependency of the population sizenon the vari-
ationECV. First, the population sizes remained approximately constant over the
generations for each fixedECV(Fig. 2 e 1 ). Second, the median population sizes
increased with decreasingECV (Fig. 2 e 2 ), as expected.
As quantitative changes of the distribution related to changes of the accep-
tance threshold did not influence the selected population size, we assessed its