0}$ are given in Section \[sec:main\]. Section \[sec:num\] proposes some numerical experiments which notably aim at illustrating on challenging problems (including a latent variable model) the benefit of the method introduced in this work for approximate Bayesian parameter inference. In this section we also consider an online binary classification example, with 53 predictors, on which the proposed algorithm outperforms stochastic gradient descent and a standard online variational algorithm. Section \[sec:conclusion\] concludes and all the proofs are gathered in the Appendix.

Online Approximate Bayesian learning\[sec:approach\]
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In this section we introduce in detail the proposed approach for online approximate Bayesian learning. In Section \[sub:pi\_tilde\] we provide the precise definition of the sequence $(\tilde{\pi}_t)_{t\geq 0}$ and explain in what sense $\tilde{\pi}_t$ can be interpreted as an approximation of the posterior distribution $\pi_t$, for all $t\geq 1$. In Section \[sub:averaging\] we introduce the sequence $(\bar{\pi}_t)_{t\geq 0}$ of average measures and in Section \[subsec:example\] we study the properties of $\tilde{\pi}_t$ and of $\bar{\pi}_t$ on a toy example. In Section \[sub:Monte\_Carlo1\] we introduce a simple online Monte Carlo algorithm (Algorithm \[algo:pi\_tilde\_online\]) that can be used to estimate $(\tilde{\pi}_t)_{t\geq 0}$ and $(\bar{\pi}_t)_{t\geq 0}$ in the special case where $\tilde{\pi}_t$ can be interpreted as the filtering distribution 