Perturbed Bayesian Inference for Online Parameter Estimation

In this paper we introduce perturbed Bayesian inference, a new Bayesian based approach for online parameter inference. Given a sequence of stationary observations $(Y_t)_{t\geq 1}$ and a parametric model $\{f_\theta,\theta\in \mathbb{R}^d\}$, the sequence $(\tilde{\pi}_t^N)_{t\geq 1}$ of \textit{perturbed posterior distributions} has the following properties: (i) $\tilde{\pi}_t^N$ does not depend on $(Y_s)_{s>t}$, (ii) the time and space complexity of computing $\tilde{\pi}_t^N$ from $\tilde{\pi}^N_{t-1}$ and $Y_{t}$ is at most $cN$, where $c<+\infty$ is independent of $t$, and (iii) for $N$ large enough and all $\alpha\in(0,1/2)$ the sequence $(\tilde{\pi}^N_t)_{t\geq 1}$ converges almost surely as $t\rightarrow+\infty$ to $\theta_\star:=\text{argmax}_{\theta\in \mathbb{R}^d}\mathbb{E}[\log f_\theta(Y_1)]$ at rate $t^{-\alpha}$. This convergence result is obtained under classical conditions that can be found in the literature on maximum likelihood estimation and on Bayesian asymptotics, and is illustrated on several examples.