of the approximated posterior (optionally with the help of the regression adjustment in section \[sec:regression-adjustment\]), as well as its highest mode, and therefore we use such information to propose parameters in the next stage corresponding to a larger $K$. Not following random walk dynamics helps in avoiding getting trapped in local modes that might emerge when powering-up the posterior surface for increasing $K$. This is why, in order to accommodate the reduced support of the current targeted distribution for increasing $K$, we recompute the covariance matrix $\hat{\Sigma}_{k}$ for the independence sampler at iterations $j\in\{s_1,...,s_q\}$.

Notice that at the end of step 9 when we reach an iteration $j\in\{s_1,...,s_q\}$ and $K$ has to be enlarged, we need to “balance” the information contained in the numerator of $\alpha$ with the one in the denominator, and this is why we go back to step 7 instead of 8 and recompute $q^*$. Basically after increasing $K$ the number of clones in the numerator would be larger than the one in the denominator, hence the need to first go back to 7, where we recompute the denominator using the same value of $K$ employed for the numerator. This re-evaluation is performed only when $j\in \{s_1,...,s_q\}$ 