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