Importance Sampling for Sequential Data

Suppose now that instead of having one posterior \(p(\boldsymbol{\mathbf{\theta}}\mid \boldsymbol{\mathbf{y}})\) 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 \(\boldsymbol{\mathbf{x}}_{t} = (x_1, \ldots, x_t)^\top\) and the respective posterior is given by \(\gamma_t(\boldsymbol{\mathbf{x}}_t)\). Then we can approximate the expected value of a function \(h(\boldsymbol{\mathbf{x}}_t)\) in a similar way as above by rewriting it in terms of an importance distribution \(q_t(\boldsymbol{\mathbf{x}}_t)\) \[\begin{align} \mathbb{E}_{\gamma_t}\left[h(\boldsymbol{\mathbf{x}}_t)\right] &= \int h(\boldsymbol{\mathbf{x}}_t)\gamma_t(\boldsymbol{\mathbf{x}}_t) d\boldsymbol{\mathbf{x}}_t \\ &= \frac{1}{Z_t}\int h(\boldsymbol{\mathbf{x}}_t)\frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t)}{q_t(\boldsymbol{\mathbf{x}}_t)} q_t(\boldsymbol{\mathbf{x}}_t) d\boldsymbol{\mathbf{x}}_t \\ &= \frac{\mathbb{E}_{q_t}\left[\frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t)}{q_t(\boldsymbol{\mathbf{x}}_t)}h(\boldsymbol{\mathbf{x}}_t)\right]}{\mathbb{E}_{q_t}\left[\frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t)}{q_t(\boldsymbol{\mathbf{x}}_t)}\right]} \\ &\approx \sum_{i=1}^N w_t(\boldsymbol{\mathbf{x}}_t^{(i)}) h_t(\boldsymbol{\mathbf{x}}_t^{(i)}) && \boldsymbol{\mathbf{x}}_t^{(i)} \overset{\text{iid}}{\sim} q_t(\, \cdot\,) \end{align}\] where we have defined the normalized weights and the unnormalized weights as \[ w_t(\boldsymbol{\mathbf{x}}_t^{(i)}) = \frac{\widetilde{w}_t(\boldsymbol{\mathbf{x}}_t^{(i)})}{\sum_{j=1}^N \widetilde{w}_t(\boldsymbol{\mathbf{x}}_t^{(i)})} \qquad \text{and} \qquad \widetilde{w}_t(\boldsymbol{\mathbf{x}}_t^{(i)}) = \frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t^{(i)})}{q_t(\boldsymbol{\mathbf{x}}^{(i)})} \] 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. \[\begin{align} \gamma_t(\boldsymbol{\mathbf{x}}_t) &= \int \gamma_t(\boldsymbol{\mathbf{x}}_t')\delta_{\boldsymbol{\mathbf{x}}_t}(\boldsymbol{\mathbf{x}}_t') d\boldsymbol{\mathbf{x}}_t' \\ &\approx \frac{\sum_{i=1}^N \frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t^{(i)})}{q_t(\boldsymbol{\mathbf{x}}_t^{(i)})}\delta_{\boldsymbol{\mathbf{x}}_t^{(i)}}(\boldsymbol{\mathbf{x}}_t)}{\sum_{j=1}^N \frac{\widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t^{(j)})}{q_t(\boldsymbol{\mathbf{x}}_t^{(j)})}} \\ &= \sum_{i=1}^N w_t(\boldsymbol{\mathbf{x}}_t^{(i)}) \delta_{\boldsymbol{\mathbf{x}}_t^{(i)}}(\boldsymbol{\mathbf{x}}_t) \end{align}\]

And, similarly, to estimate the normalization constant we do the following \[ Z_t = \int \widetilde{\gamma_t}(\boldsymbol{\mathbf{x}}_t) d \boldsymbol{\mathbf{x}}_t \approx \sum_{i=1}^N \widetilde{w}_t(\boldsymbol{\mathbf{x}}_t^{(i)}) \]