region. Although there are several conditions on the prior distribution, $p$, most reasonable choices of prior distribution will satisfy these, and only a single condition on the approximate posterior distribution, $q$, is assumed. In the limit of having more samples from the distributions $p$ and $q^{(\mathbf{w})}$, selecting summary statistic weights, $\mathbf{w}$, based on optimizing the estimate from $\hat{D}_h$ will converge to give the true optimum, $\mathbf{w}^*$, of this distance between distributions. For fixed weights $\mathbf{w}^*$ in the ABC distance function, ABC-SMC will target the correct posterior distribution.

Examples {#Examples}
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We apply our algorithm of automatic, adaptive weighting of summary statistics to a variety of test problems, including a toy model and several problems based on different chemical reaction networks. The dynamics of these networks are simulated stochastically using Gillespie’s direct method [@gillespie1977exact], which allows us to sample trajectories directly from the model. Although for some of these models it is possible to solve for the likelihood analytically, we attempt parameter inference by simulation, since solving for the likelihood is very computationally expensive and, in general, the analytical solution is not available. The summary statistics collected for each of the chemical reaction network problems are in the form of a 