Computational Methods in Systems Biology

142 E. Klinger and J. Hasenauer

True 0. 32 0. 71 1. 00 1. 41

Relative bandwidthb/bSilv

0. 5 1. 0

Relative bandwidth

b/bSilv

0. 00

0. 05

0. 10

Density variation

ab

Fig. 5. Influence of the kernel bandwidth on the density variation. (a)True
(unimodal normal) distribution and kernels with bandwidthsb, relative to the Silver-
man bandwidthbSilvfor population sizen=10^3 .(b)Density variation for kernels
with bandwidthsb, relative to the Silverman bandwidthbSilv. Population sizen=10^3 ,
drawn from the true (unimodal normal) distribution from (a).

inference and model selection as well as its scalability and compatibility with a
range of transition kernels (proposal densities).
The approximation quality, expressed in terms of the target variation of the
densityECV, has to be specified in our method. While selectingECVadequately
is important, the method does not simply replace manually tuning population
sizes by manually tuningECV. Instead,ECVis easier interpretable and thus eas-
ier to select. The examples with increasing, decreasing or approximately constant
population sizes indicate that the proposed method is not a mere reparameteri-
zation. Empirically, 0. 1 ≤ECV≤ 0 .2 worked reliably in many cases.
Our method can be employed together with arbitrary density estimators. The
choice of the estimator, however, affects the population sizes. Over-smoothing
estimators (e.g. Silverman) yielded smaller population sizes. This is consistent
with the lower variation of estimators with larger bandwidths (Fig. 5 ). Inappro-
priately chosen estimators can yield poor results with respect to approximation
quality and computation time. For instance, cross-validated bandwidth selection
can generate a notable computational overhead if the model simulation is fast.
We found no obvious difference between bootstrapping from the density or
directly from the particle population; we therefore decided to bootstrap from the
density. This avoids drawing the same particles of a population repeatedly as it
would likely occur in bootstrapping from the population (with replacement). The
assumed functional relation between the density variation and the population
size was motivated by the Silverman rule [ 21 ] but might be further improved.
For the considered applications, however, the approximation was sufficient and
extrapolation of larger population sizes was facilitated.
The difficulty of choosing population sizes has been discussed in the literature
before [ 16 ]. Our results suggest that probing population sizes over an order of
magnitude, as done in some studies (e.g. [ 9 ]), can be avoided. To the best of
our knowledge, this is the first attempt to adaptively and automatically select
population sizes for ABC-SMC.