time series, to imitate data that could be collected from a biological experiment.

To demonstrate the effectiveness of taking a flexible choice of distance weights, we make two comparisons. Firstly, we compare results obtained using Algorithm 2 to those generated using a uniform choice of weights: $w_i = 1 \hspace{2mm} \forall i$. Secondly, we compare to results generated using weights that scale with each summary statistic. Here we use $w_i = 1/\sigma_i \hspace{2mm} \forall i$, where $\sigma_i$ is the standard deviation of the given summary statistic. This is a frequently used choice of weight for summary statistics [@beaumont2002approximate]. We note that a table summarising the parameters used in the implementation of all the test problems can be found in Appendix A.

Toy model {#toy_model}
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We consider a tractable toy problem with a sufficient statistic to illustrate our method. We observe $$x_i \sim \text{Unif}([0,\theta]),$$ for $i=1,\dots, r$. In this case, the maximum of the observed values is a sufficient statistic, $s(\mathbf{x}) = \max_i x_i$. We can sample the true posterior distribution $p(\theta | \mathbf{y})$ directly via MCMC and compare to the approximations obtained via ABC.

![image](Fig2.eps){width="45.00000%"}

The results in Figure \[fig:toy\_model\] indicate that the method in Algorithm 2 is able 