Importance Sampling for Sequential Data

Suppose now that instead of having one posterior $p(\mathit{\theta }\mid \mathbf{y})$ we actually have a sequence of posterior distributions. This could happen in a scenario in which data comes in sequentially and we need to estimate the posterior each time. In particular, suppose that at time $t$ we have the following data ${\mathbf{x}}_t=(x_1,\dots ,x_t{)}^{\mathrm{\top }}$ and the respective posterior is given by ${\gamma }_t({\mathbf{x}}_t)$. Then we can approximate the expected value of a function $h({\mathbf{x}}_t)$ in a similar way as above by rewriting it in terms of an importance distribution $q_t({\mathbf{x}}_t)$ $$\begin{array}{rl}{\mathbb{E}}_{{\gamma }_t}[h({\mathbf{x}}_t)]& =\int h({\mathbf{x}}_t){\gamma }_t({\mathbf{x}}_t)d{\mathbf{x}}_t\\ & =\frac{1}{Z_t}\int h({\mathbf{x}}_t)\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t)}{q_t({\mathbf{x}}_t)}{q}_t({\mathbf{x}}_t)d{\mathbf{x}}_t\\ & =\frac{{\mathbb{E}}_{q_t}[\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t)}{q_t({\mathbf{x}}_t)}h({\mathbf{x}}_t)]}{{\mathbb{E}}_{q_t}\left[\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t)}{q_t({\mathbf{x}}_t)}\right]}\\ & \approx \sum _{i=1}^N{w}_t({\mathbf{x}}_t^{(i)})h_t({\mathbf{x}}_t^{(i)})& & {\mathbf{x}}_t^{(i)}\stackrel{\text{iid}}{\sim }{q}_t(\cdot )\end{array}$$ where we have defined the normalized weights and the unnormalized weights as $$w_t({\mathbf{x}}_t^{(i)})=\frac{{\tilde{w}}_t({\mathbf{x}}_t^{(i)})}{\sum _{j=1}^N{\tilde{w}}_t({\mathbf{x}}_t^{(i)})}\text{and}{\tilde{w}}_t({\mathbf{x}}_t^{(i)})=\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t^{(i)})}{q_t({\mathbf{x}}^{(i)})}$$ We can also use these weights to obtain an estimate of the normalization constant of the posterior, $Z_t$, and of the normalized posterior itself. In order to do this, we need to recall the definition of the dirac delta mass. $$\begin{array}{rl}{\gamma }_t({\mathbf{x}}_t)& =\int {\gamma }_t({\mathbf{x}}_t^{\prime }){\delta }_{{\mathbf{x}}_t}({\mathbf{x}}_t^{\prime })d{\mathbf{x}}_t^{\prime }\\ & \approx \frac{\sum _{i=1}^N\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t^{(i)})}{q_t({\mathbf{x}}_t^{(i)})}{\delta }_{{\mathbf{x}}_t^{(i)}}({\mathbf{x}}_t)}{\sum _{j=1}^N\frac{\widetilde{{\gamma }_t}({\mathbf{x}}_t^{(j)})}{q_t({\mathbf{x}}_t^{(j)})}}\\ & =\sum _{i=1}^N{w}_t({\mathbf{x}}_t^{(i)}){\delta }_{{\mathbf{x}}_t^{(i)}}({\mathbf{x}}_t)\end{array}$$

And, similarly, to estimate the normalization constant we do the following $$Z_t=\int \widetilde{{\gamma }_t}({\mathbf{x}}_t)d{\mathbf{x}}_t\approx \sum _{i=1}^N{\tilde{w}}_t({\mathbf{x}}_t^{(i)})$$