(see Table \[table:ex\]). A possible explanation for this phenomenon is that, in this particular example, $\tilde{\pi}_t$ belongs to the same family of distributions than $\pi_t$, for all $t\geq 0$.

$N\in\mathbb{N}$, $c_{\mathrm{ESS}}\in(0,1]$ and a resampling algorithm $\mathcal{R}(\cdot, \cdot)$ (see Section \[sub:resampling\])

\[line:for\]Let $\theta_1^n\sim \tilde{\pi}_0({\mathrm{d}}\theta_0)$ and set $w_1^n=  \tilde{f}_{\theta_1^n}(Y_1)$ and $W_{1}^n=w_1^n/\sum_{m=1}^N w_1^m$

Set $\mathrm{ESS}_{t-1}= 1/\sum_{m=1}^N (W_{t-1}^m)^2$

\[line:R\] Let $(\hat{\theta}^{1}_{t-1},\dots,\hat{\theta}^{N}_{t-1})\sim \mathcal{R}(\{\theta_{t-1}^n,W_{t-1}^{n}\}_{n=1}^N,N)$ and set $w_{t-1}^n=1$ Let $\hat{\theta}^{n}_{t-1}=\theta^{n}_{t-1}$

Set $\theta_t^n=\hat{\theta}^n_{t-1} + c\,t^{-\alpha} \epsilon^n_{t-1}$ where $\epsilon^n_{t-1}\sim t_{d,\nu}(0, I_d)$ Set $\theta_t^n=\hat{\theta}^n_{t-1} + c\,t^{-\alpha} \epsilon^n_{t-1}$ where $\epsilon^n_{t-1}\sim \mathcal{N}_d(0,I_d)$ \[line:last\] Set $w_t^n= w_{t-1}^n\tilde{f}_{\theta_t^n}(Y_t)$ and $W_{t}^n=w_t^n/\sum_{m=1}^N w_t^m$

**Output:** An approximation $\tilde{\pi}_0^N:=\frac{1}{N}\sum_{n=1}^N  \delta_{\{\theta_1^n\}}$ of $\tilde{\pi}_0$ and, for all $t\geq 1$, an approximation $\tilde{\pi}_t^N:=\sum_{n=1}^N W_t^n\delta_{\{\theta_t^n\}}$ of $\tilde{\pi}_t$ .

A simple online approximate Bayesian learning algorithm \[sub:Monte\_Carlo1\]
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As mentioned in the introductory section, computing an efficient Monte Carlo approximation of $(\tilde{\pi}_t)_{t\geq 0}$ is particularly easy when $h_t>0$ for all $t\geq 1$ since, in this case, a standard particle filter (PF) can be used.

In Algorithm \[algo:pi\_tilde\_online\] we provide a simple PF algorithm that can be used to approximate a sequence $(\tilde{\pi}_t)_{t\geq 0}$ such that $h_0=0$ and, for all $t\geq 1$, $\Sigma_t=I_d$ and $h_t=c\, t^{-\alpha}$ for some constants $c,\alpha>0$. For every $\alpha>0$, examples of sequences $(t_p)_{p\geq 0}$ that verify the conditions of our 