\left( \frac{(n-1)\rho_k(i)}{n\nu_k(i)} \right)^{1-\alpha} B_{k,\alpha},$$ where $$B_{k,\alpha} = \frac{\Gamma(k)^2}{\Gamma(k-\alpha+1)\Gamma(k+\alpha-1)}.$$

At each generation of ABC-SMC we seek to find weights $\mathbf{w} \in E \subset \mathbb{R}^{\kappa}$ such that $$\mathbf{w}^* = \operatorname*{\arg\!\max}_{\mathbf{w} \in E} \left(1 - \hat{D}_{\alpha} \left( \{\xi_i\}_{i=1}^M || \{\theta_i\}_{i=1}^M \right) \right),$$ where $\{\xi_i\}_{i=1}^M$ are samples from the prior distribution, and $\{\theta_i\}_{i=1}^M$ are samples from the approximate posterior distribution, which depends on the summary statistic weights, $\mathbf{w}$.

To perform the optimization in weight space in our implementation of Algorithm 2, we use a constrained nonlinear optimizer, implemented via $\tt{fmincon}$ in MATLAB [@MATLAB:2016].

Set generation index $t=1$.\
Sample from prior $\theta^{**} \sim \pi(\theta)$.\
Simulate dataset $\mathbf{x}^{*i} \sim f(\mathbf{x}|\theta^{**}) $. Set $\theta^i_t = \theta^{**}.$\
Calculate the weight $v^i_t$ for particle $\theta^i_t$ as $ v^i_t = 1 $. Let $$L(\mathbf{w}) = 1 - \hat{D}_{\alpha} \left( \left\{ \xi^i \right\}_{i=1}^M || \left\{ \theta_t^i \right\}_{i=1}^M \right),$$ where $\left\{ \theta_t^i \right\}_{i=1}^M$ are the closest $M=\left \lfloor{\alpha N} \right \rfloor$ samples when ranked according to ABC distance from the pseudo dataset, $d_{\mathbf{w}}(s(\mathbf{y}),s(\mathbf{x}^{i*}))$. Maxmize $L(\mathbf{w})$ as a function of summary statistic weights, $\mathbf{w}$. Keep the samples $\left\{ \theta_t^i \right\}_{i=1}^M$ corresponding to the maxmimum of $L(\mathbf{w})$ (i.e. the maximum distance between prior and approximate posterior).\
Normalize the particle weights $\left\{ v_t^i \right\}_{i=1}^M$.\
\[step3\] 