A Scheme for Adaptive Selection of Population Sizes 139

0

20

40

Modelm 1

X

Y

0. 0 0. 1

Timet

0

20

40

Modelm 2

Concentrations

m 1 m 2

Model

0

1

Model probability

n: 60000

ECV:

0.05

0.1

0.2

0. (^0000) m 2


1. 0025
   n=60000ECV
   =0.05
   ECV
   =0.1
   ECV
   =0.2


2. 0


3. 1


4. 2


5. 3
   Kolmogorov-
   Smirnovdistance
   (^005)
   1
   Model
   probability
   m 1 m 2
   0 5
   Generationt
   0
   2000
   Population
   size
   n neff


6. 
   

   0 0. 2 0. 4 0. 6
   Rate log 10 k 1
   0
   2
   4
   p(log
   10
   k^1
   |m

   
   

   m
   ) 1
   True rate
   Generation
   0
   2
   4
   6
   8
   abc
   d
   0.05 0.1 0.2
   Density
   variationECV
   101
   102
   103
   Population
   size
   n
   efg
   Fig. 3. Model comparison of two chemical reaction kinetics models. (a)
   Species concentrationsXandY over timetfor a single realization of each of the
   two modelsm 1 andm 2 with ratesk 1 =2.1andk 2 = 30.(b–d)Results of an ABC-
   SMC run for data generated from modelm 1 ,k 1 =2.1(log 10 k 1 =0.32) forECV=0.05:
   (b) Model posterior distribution; (c) Parameter posterior distribution and true para-
   meter (dashed line) used to generate the data; and (d) Population sizenand effective
   population sizenefffor different generationst.(e)Model posterior distribution for
   adaptively selected population sizes and large, constant population size. In all cases
   sampling of new populations was stopped as soon as the acceptance threshold reached
   1 .5. Inset shows modelm 2 only.(f)Kolmogorov-Smirnov distances of adaptive popu-
   lation size posteriors and large, constant population size posteriors for log 10 k 1 , relative
   to the reference posterior. For each scenario, four independent runs were performed.
   (g)Mean population sizenover all generations for different values ofECV.
   (Fig. 3 c). The adaptation forECV yielded population sizes between 1458 and
   2699 and effective population sizes between 1101 and 2284 particles (Fig. 3 d).
   Unexpectedly, the population sizes decayed in the last generations (Fig. 3 d).
   We then examined the quality of the posterior approximation as a function
   ofECV. As the posterior was not analytically accessible we used as reference an
   average ABC-SMC estimate obtained from four repetitions with the large and
   constant populations size of 60000 particles. We found that non-zero mass was
   attributed to both models by the reference posterior although the mass at model
   m 2 was small (Fig. 3 e). ForECV =0. 2 , 0 .1 modelm 2 was completely extinct
   (Fig. 3 e), only for smallerECV =0.05 we obtainedp(m 2 )>0. We quantified
   the mismatch between posterior approximation and reference posterior for the
   log-rate log 10 k 1 in terms of the Kolmogorov-Smirnov (KS) distance (Fig. 3 f)
   between the posteriors obtained using adaptation of the population sizes and the