136 E. Klinger and J. Hasenauer
3.1 Appropriateness of the Functional Approximation ofECV
for Normal Distributions
We first asked if the chosen functional formECV(n;α, β)=αn−βdid indeed
capture the relation between population size and density variation reasonably
well. A perfect approximation was not necessary, only an approximation which
was good enough to ensure that the population size evolved towards the desired
ECVwas required. For a first assessment we considered the case of a unimodal
normal distribution. Indeed, the chosen functional form matched the relation
betweenECVandnon the bootstrapped populations (Fig. 1 b). Control samples
from the true density revealed that in the extrapolated regime, the curve seemed
to slightly overestimateECV but still captured the scaling behavior (Fig. 1 b).
We therefore continued with the first example.
3.2 Stability of the Population Size Adaptation for an Analytical
Model
We considered a model, with a multimodal posterior (similar to [ 10 ]) to investi-
gate the stability of the population sizes over the course of the generations, as
well as a possible dependency on the number of posterior modes or the employed
KDE. In this model, the simulated datas∈R^2 were obtained by sampling from
s∼N(sq(θ, nmodes),σ^2 I), in whichIdenotes the identity matrix inR^2 ,σ^2 >0,
nmodes∈{ 1 , 2 , 4 }denotes the number of posterior modes, and sq a squaring-like
function squaringθ=(θ 1 ,θ 2 ) elementwise according to
sq(θ, nmodes)=⎧
⎪⎨
⎪⎩
(θ 1 ,θ 2 )ifnmodes=1,
(θ^21 ,θ 2 )ifnmodes=2,
(θ^21 ,θ^22 )ifnmodes=4.The form of the squaring function sq ensured that the number of posterior modes
equalednmodes. The parameterθ∈[− 10 ,10]^2 was subject to posterior inference,
with uniform priorθ∼U([− 10 ,10]^2 ) over the square [− 10 ,10]^2. The distance
functiondwasd(s, sdata)=|s 1 −sdata, 1 |+|s 2 −sdata, 2 |. For this model,B=10
bootstrapped populations were used to estimate the density variation.
We performed ABC-SMC runs fornmodes=1, 2 ,4 modes, with observed
datasdata=(1,1),σ^2 =0.5andECV =0.1. The modes were correctly cap-
tured after a few generations for all scenarios (see Fig. 2 afornmodes= 4). We
then investigated how the population size evolved. To our surprise, even though
the acceptance threshold decreased substantially (Fig. 2 b 1 ), the population size
and effective population size decayed only slightly (Fig. 2 b 2 ). Runs with ini-
tial population sizesn 0 =10^1 , 102 , 103 , 104 converged within 3 generations to
approximately the same population size (Fig. 2 c). There was no further sys-
tematic dependency of the population sizes of later generations on the initial
population sizen 0 (Fig. 2 c).
Furthermore, we found that the actually achieved variation matched the
targetECV well on average (Fig. 2 d), confirming the adaptive population size