In particle filters when one is doing sequential importance sampling, the quantity of interest that is being approximated is usually a weighted sum:

$$\hat x_t = \sum_{i=1}^M \Bigl [f(v^{(i)}_{t}) \times w(v^{(i)}_t) \Bigr] \tag 1$$

where $t$ denotes a time or step index and $i$ denotes a particle/sample index.

and when you're doing resampling it's:

$$\hat x_t = {1\over M} \sum_{i=1}^M \Bigl [f(v^{(i)}_{t}) \times w(v^{(i)}_t)\Bigr] \tag 2$$

where $w(v)$ is the importance weight and $f(v)$ is the function estimate, suppose you don't resample on every step but only when the effective sample size goes below some threshold, does that mean that you perform $(1)$ on the time steps you didn't resample and $(2)$ on the time steps you did resample on?

In particle filters when one is doing sequential importance sampling, the quantity of interest that is being approximated is usually a weighted sum:

$$\hat x_t = \sum_{i=1}^M \Bigl [f(v^{(i)}_{t}) \times w(v^{(i)}_t) \Bigr] \tag 1$$

where $t$ denotes a time or step index and $i$ denotes a particle/sample index.

and when you're doing resampling it's:

$$\hat x_t = {1\over M} \sum_{i=1}^M f(v^{(i)}_{t}) \tag 2$$

where $w(v)$ is the importance weight and $f(v)$ is the function estimate, suppose you don't resample on every step but only when the effective sample size goes below some threshold, does that mean that you perform $(1)$ on the time steps you didn't resample and $(2)$ on the time steps you did resample on?