Bias in Particle Filters

Hi everyone,

I have a question about the bias (or lack of?) occurring in particle filters and the relation to the lack of bias in approximating data likelihood. More concretely, it seems to me that particle filters will give biased estimates of P(x_t|y_{1:t}), but will not give biased estimates for P(y_t|y_{1:t-1}). I hope to understand where I am mistaken.

For simplicity, I'll assume that I'm using the typical Bootstrap particle filter. The general algorithm is

input: particles (w_{t}^{i}, x_{t}^{i}) for i = 1...N sampled approximately according to Pr(x_t | y_{1:t})
output: particles (w_{t+1}^{i}, x_{t+1}^{i}) sampled approximately according to Pr(x_{t+1}|y_{1:t+1}) and estimates for Pr(y_{t+1} | y_{1:t})
a) propagate particles forward sampling from x'^{i} ~ Pr(x_{t+1}|x_{t}^{i}) for i = 1...N
b) weight samples w'^{i} = Pr(y_{t} | x'^{i})
c) Output sum_{i} w'^{i} as an estimate of Pr(y_{t+1} | y_{1:t})
d) resample particles x_{t+1}^{i} ~ Multinomial(w'^{i}, x'^{i}) and give them all weight w_{t+1}^{i} = 1/N
e) output samples (w_{t+1}^{i}, x_{t+1}^{i})

My concern lies in resampling -- it seems evident to me that sampling particles according to an approximation of Pr(x_{t+1} | y_{1:t+1}) instead of the true distribution would introduce bias into the estimate. If in turn Pr(x_{t} | y_{1:t}) is biased, then I don't see how Pr(y_{t} | y_{1:t-1}) would be unbiased either. If no resampling were performed, we would have unbiased (though extremely noise) estimates. Am I missing something here?