can directly be used to compute the approximation $\bar{\varphi}_{t}^N:=(t+1)^{-1}\sum_{s=1}^t\tilde{\pi}^N_s(\varphi)$ of $\bar{\pi}_t(\varphi)$ using the recursive formula $$\bar{\varphi}_{t}^N=\frac{t}{t+1}\bar{\varphi}_{t-1}^N+\frac{1}{t+1}\tilde{\pi}^N_t(\varphi),\quad t\geq 0.$$

Choosing $(h_t)_{t\geq 0}$: computational v.s. statistical efficiency\[sub:h\_t\]
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Both Proposition \[prop:continuity\] and the numerical results of Section \[subsec:example\] call for choosing a sequence $(h_t)_{t\geq 0}$ such that $h_t$ tends to $0$ quickly, while as mentioned in the previous subsection computational constraints will generally impose to take $(h_t)_{t\geq 0}$ such that the convergence of $h_t$ to $0$ is not too fast. In the special case where $h_t=\Theta(t^{-\alpha})$, we observed in our experiments that these latter usually lead to choose $\alpha=0.5$, in which case the example of Section \[subsec:example\] and those of Sections \[sec:num\_quantile\]-\[sub:latent\] suggest that the variance of $\theta$ under both $\tilde{\pi}_t$ and $\bar{\pi}_t$ converges to zero at a rate which is slower than the usual $t^{-1}$ convergence rate that we have for the posterior variance of $\theta$. Consequently, in practice the sequence $(\tilde{\pi}_t)_{t\geq 0}$ will be typically defined in such a way that the uncertainty quantification obtained using $\tilde{\pi}_t$ or $\bar{\pi}_t$ is too conservative. By contrast, the Bayesian credible sets may be over confident when the model is miss-specified [@Kleijn2012].

Monte Carlo Approximation of the sequences Lg and Lg \[sub:Monte\_Carlo\]
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Algorithm 