# Importance weight of conditioned particle in conditional SMC

In a generic particle filter, I understand the importance weights for each particle are calculated as $w_t^s \propto w_{t-1}^s \frac{p(y_t \mid z_t^s) p(z_t^s \mid z_{t-1}^s)}{q(z_t^s \mid z_{t-1}^s, y_t)}$.
I am confused about the weight of the conditioned particle in the conditional SMC (used in Particle Gibbs). Since the conditioned particle $z_t^N$ is set deterministically, does that mean that $q(z_t^N \mid z_{t-1}^N, y_t) = 1$?

Edit: I was mistaken. The particles in the path being conditioned on DO have weights. When you sample the $$N-1$$ ancestors at the beginning of each step, you may sample the conditioned-on particle as an ancestor. The weight formula here is roughly the same as the traditional formula you have written first.
To be more clear, the unnormalized weight for the $$k$$th particle, if $$k\neq B_n$$, at time $$n>1$$, assuming you just sampled the ancestor index $$A_{n-1}^k$$, is $$\frac{f_{\theta}(x_n^k \mid x_{n-1}^{A^k_{n-1}})g(y_n \mid x_n^{k})}{q_{\theta}(x_n^k \mid y_n, x_{n-1}^{A_{n-1}^k}) }.$$ For the special particle, where $$k=B_n$$, the ancestor is not random: it is $$B_{n-1}$$. So the above formula would change slightly to $$\frac{f_{\theta}(x_n^k \mid x_{n-1}^{B_{n-1}})g(y_n \mid x_n^{k})}{q_{\theta}(x_n^k \mid y_n, x_{n-1}^{B_{n-1}}) }.$$
I am using the notation from around page 10 of this paper, which describes the conditional SMC update. Sorry about the confusion earlier; I find it strange how they re-use the letter $$k$$ for the two different types of indices.
• @GonzaloBenegas there is an indication. The superscript is $k$, and $k$ is any index that does not equal $B_n$ at time $n$ Jun 23, 2018 at 23:31
• I'm not sure $k$ does not include $B_n$ in step (c). If the conditional particle has no weight, how does it interact with the rest? What's the point of its inclusion? Jun 24, 2018 at 20:34