Additional file 1:

Pre-evaluation of model-prior combinations: two examples. Pre-evaluation of model-prior combinations: example 1. A single test pseudo-observed data set (10 microsatellite loci) was first simulated under a model of a single population (sample size of 30 diploid individuals) with effective size N = 10,000. Microsatellite loci were assumed to follow a generalized stepwise mutation model (GSM [37]) with a mean mutation rate (mean μ) equal to 5 × 10^-4 and a mean parameter of the geometric distribution of the length in number of repeats of mutation events (mean P) equal to 0.22. Each locus was given a possible range of 40 contiguous allelic states and was characterized by individual μ_loc and P_loc values drawn from Gamma(mean = mean μ and shape = 2) and Gamma(mean = mean P and shape = 2) distributions, respectively [12]. For ABC analysis of the test data set, we used the same population and marker models, and prior distributions of demographic parameters were as followed: Uniform[10; 1000] (figure A) or Uniform[2000; 20000] (figure B) for N, Uniform[10^-4; 10^-3] and Uniform[0.1; 0.3] for mean μ and mean P, respectively. We choose three summary statistics (s): mean number of alleles, mean expected heterozygosity [38] and mean allele size variance per population. PCA on summary statistics (A and B) and probability (s_simulated < s_observed) for each summary statistics (C) were computed from 10,000 simulations, randomly drawing parameter values from priors. Pre-evaluation of model-prior combinations: example 2. A single pseudo-observed test data set (10 microsatellite loci) was first simulated under a model of two populations (sample size of 30 diploid individuals per population) splitting at time t = 10,000 generations from an ancestral population, without subsequent migration. For all populations the effective size was N = 1,000. For ABC analysis of the test data set, we used the same population and marker models, and prior distributions of demographic parameters were as followed: Uniform[100; 1000] (figure D) or Uniform[2000; 20000] (figure E) for t, and Uniform[100; 2000] for N. The mutation model and priors for microsatellite markers are the same as in example 1. We choose eight summary statistics (s): mean number of alleles, mean expected heterozygosity [38] and mean allele size variance of each population sample, and F_ST values and genetic distances (δμ)² between pairs of populations [39,40]. PCA on summary statistics (D and E) and probability (s_simulated < s_observed) for each of the summary statistics (F) were computed from 10,000 simulaxtions, randomly drawing parameter values from priors.

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Additional file 2:

Evaluation of the variation of RMAE values expected by chance between different replicates of 500 pseudo-observed data sets. relative median absolute errors (RMAE) were computed for 10 replicates of 500 pseudo-observed data sets simulated under scenario 1. The data sets include 20 (independent) microsatellite loci and were generated under scenario 1 presented in Figure 1. Parameter values were drawn from the same distributions than the prior distributions given in the legend of Figure 1. The demographic parameters N, t1, t2, t3, t4, t5, r1 and r2 are detailed in Figure 1. Standard deviation of RMAE values were equal to 0.009, 0.019, 0.004, 0.017, 0.012, 0.013 and 0.014 for N, t1, t2, t3, t4, t5, r1 and r2, respectively. Similar levels of RMAE variation among replicates of 500 pseudo-observed data sets were obtained for other categories of genetic markers (mtDNA and nuclear sequences) and combinations of categories of markers (results not shown).

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Additional file 3:

Principal component analysis of test quantities when processing model checking for the introduction scenarios 1, 2 and 3. The scenarios 1, 2 and 3 are detailed in Figure 2. The pseudo-observed test data set analyzed here was simulated under scenario 3. PCA were processed on the test quantities corresponding to the summary statistics used to discriminate among scenarios and compute the posterior distributions of parameters (a) or on other statistics (b). The summary statistics used as test quantities are detailed in the legend of Table 1.

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