This gives the overall gradient computation: where we sample t k n and update ω k n while moving from left-to-right, and sample s ,k n at each n. Note that although the sequential importance sampler moves left-to-right like a filter, we use only the final weights ω N to compute the expectation.
the resulting expectation is based on the distribution P (s 1:N |t 1:N ), so that no backwards "smoothing" pass (Godsill et al., 2004) is needed to eliminate bias.