the Gaussian kernel with bandwidth parameter equals to one, in which case $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$ is explicitly computable and $d_{\mathrm{MMD}}$ is a metric on $\mathcal{P}({\mathbb{R}})$ [see e.g. @muandet2016kernel for more details on the MMD distance]. As expected, we see in Figure \[fig:toy\_mmd\] that $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$ is smaller for $\alpha=1$ than for $\alpha=0.5$. In addition, we observe that the distance between $\tilde{\pi}_t$ and $\pi_t$ decreases as $t$ increases, which is due to the fact that the two distributions converge to $\delta_{{\mathbb{E}}[Y_1]}$ as $t\rightarrow\infty$. The right plot of Figure \[fig:toy\_mmd\] shows the corresponding results for the MMD distance between $\bar{\pi}_t$ and $\pi_t$ and for $\alpha\in\{0.5,0.7,1\}$. Computing $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ explicitly is feasible but computationally expensive. For this reason the right plot of Figure \[fig:toy\_mmd\] in fact presents results for $d_{\mathrm{MMD}}(\bar{\pi}^N_t,\pi_t)$, with $\bar{\pi}^N_t$ the empirical distribution associated to an i.i.d. sample of size $N=5\,000$ from $\bar{\pi}_t$. Unsurprisingly, $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ decreases with $\alpha$ and $t$ while, as argued in Section \[sub:averaging\], we observe that the temporal evolution of $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ is more regular than that of $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$. Lastly, it is interesting to see that, for this example, $d_{\mathrm{MMD}}(\bar{\pi}_t,\pi_t)$ tends to be larger than $d_{\mathrm{MMD}}(\tilde{\pi}_t,\pi_t)$ despite of the fact that $\bar{\theta}_t$ is (much) closer to the posterior mean of $\theta$ than $\tilde{\theta}_t$ 