sample. Choice of the tolerance $\epsilon$ can be avoided, to some extent, by simulating a large number, $N$, of parameter samples and datasets, calculating the corresponding distances for these and accepting the proportion $\alpha$ that lie closest to the real data. We will use this approach in this work.

In cases where the prior and posterior distributions are very different, the rejection sampling version of ABC described above can have very low acceptance rates. Algorithm 1 summarizes how samples from an approximate posterior can be generated via a more efficient version of ABC using sequential Monte Carlo techniques, known as ABC-SMC [@toni2009approximate; @sisson2007sequential; @del2006sequential]. Importance sampling is used iteratively so that instead of sampling repeatedly from the prior, parameters are sampled from an approximate posterior at each generation of the algorithm. A weight must be given to each sample to correct for the fact that it is not drawn from the prior.

\[abc-smc\]

Set generation index $t=1$.\
Sample from prior $\theta^{**} \sim \pi(\theta)$.\
Simulate dataset $\mathbf{x}^{*i} \sim f(\mathbf{x}|\theta^{**}) $ and calculate distance $d(s(\mathbf{y}),s(\mathbf{x}^{*i}))$. Set $\theta^i_t = \theta^{**}.$\
Calculate the weight $v^i_t$ for particle $\theta^i_t$ as $ v^i_t = 1 $ Normalize the particle weights ${v_t^i}$.\
Select the proportion $\alpha$ of 