Say the observations are $x_i$ and the states are $y_i$ in a sequential model.

I understand that particle filtering works by generating "particles" from $p(y_i | x_1,\ldots,x_i)$ for approximating $p(y_{i+1} | x_1,\ldots,x_{i+1})$.

How do we decide on how many "particles" to use as we go along on the chain? Do we choose a fixed number in the beginning, and stick to it (one that works well experimentally), or do we change the number of particles used as the particle filtering algorithm proceeds?