samples closest to the real data to keep and reject the rest, resulting in $M=\left \lfloor{\alpha N}\right \rfloor $ samples. \[step3\] Sample $\theta^{*}$ from previous population ${\theta_{t-1}^{i}}$ with weights $v_{t-1}$.\
Peturb $\theta^*$ to give $\theta^{**} \sim K_t(\theta|\theta^*)$. If $\pi(\theta^{**}) = 0$, return to step 11. \[step4\] 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 $\theta^i_t$ via $$v^i_t = \frac{\pi (\theta_t^i)}{\sum_{j=1}^M v_{t-1}^j K_t (\theta_{t-1}^j, \theta_t^i)}.$$ Normalize the particle weights ${v_t^i}$.\
Select the proportion $\alpha$ of samples closest to the real data to keep and reject the rest, resulting in $M=\left \lfloor{\alpha N}\right \rfloor$ samples. **return** $\{\theta_T^i\}_{i=1}^M$, $\{v_T^i\}_{i=1}^M$

Choice of summary statistics
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Suppose we are interested in inferring multi-dimensional parameters for a model that we can simulate, but cannot evaluate the likelihood directly. In many practical circumstances, the data (either collected experimentally or simulated from the *in silico* model) will be very high dimensional. High-dimensional data poses difficulties within the ABC framework, as it is difficult to sensibly estimate when the output of a particular simulation is ‘close’ to the data. Even taking account of domain expertise, it can be hard to determine which features of the data are 