enclosing smaller and smaller probability mass (assuming the existence of multiple modes). Therefore we believe it is important not to let $K$ fixed to a large value from the start of the algorithm, but instead start with a small value for $K$ and then increase it progressively. Enabling a smooth and not too rapid increase of $K$ should help the chain from being stuck in low-probability regions. However in the examples discussed in sections \[sec:g-and-k\]–\[sec:2GBM\] a rapid increase in $K$ is possible.

Approximate inference using ABC with data-cloning {#sec:ABC-all}
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Acceptance of proposals in MCMC algorithms is particularly challenging when the modelled process is highly erratic, for example when the unobserved state is a diffusion process, that is a solution to a stochastic differential equation (SDE, e.g. [@fuchs2013inference]). For such class of models, trajectories for $\{X_t\}$ may result quite distant from the observed data $y$, even for values of the parameters in the bulk of their posterior distributions. In such circumstance $q^{\#}$ will often be small compared to $q^{*}$, and the proposal will rarely be accepted. For example, when using an approach as the one described above, where trajectories are simulated “blindly” from $p(X|\cdot)$, that is unconditionally to data, then 