(see the example of Section \[sub:linear\_model\]). Some computations in @MR1929161 indeed suggest that a random running time implementation of IBIS only requires ${\mathcal{O}}_{\mathbb{P}}(t)$ operations to estimate $\{\pi_s\}_{s=1}^t$. When it becomes computationally too expensive to process $Y_t$ using IBIS, @balakrishnan propose to approximate the Monte Carlo update of IBIS using the kernel smoothing idea introduced by @liu2001combined. The resulting algorithm (called one-pass SMC) is online but, unlike IBIS, is not theoretically justified.

The main idea of the method we introduce below for online approximate Bayesian learning is to approximate the sequence $(\pi_t)_{t\geq 1}$ by a sequence $(\tilde{\pi}_t)_{t\geq 1}$ which (i) can be approximated in an online fashion using Monte Carlo methods and (ii) is shown to converge to $\delta_{\{\theta_\star\}}$ as $t$ tends to infinity, in some sense to be defined. Notice that, by (ii), as $t$ increases the distance between $\pi_t$ and its approximation $\tilde{\pi}_t$ converges to zero. It is also worth mentioning at this stage that online variational updates involve expectations that are generally not computable, and thus in practice online variational methods also rely on Monte Carlo approximations (see the example of Section \[sub:online\]).

To introduce the construction of the sequence $(\tilde{\pi}_t)_{t\geq 1}$ with more details assume for now 