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I'm currently doing some work with Particle Filters, a sampling-based method for computing expectations of functions with respect to dynamic (ie: time-variant) random variables. For example, consider a Markov chain where you only get to observe $Y_t \sim P(Y_t|x_t)$ for unknown variables $X_t$. You would like to calculate $\mathbb{E}[f(X_t)|y_{1:t}]$ for all $t=1\ldots T$.

To the point, Particle Filters traditionally involve a "resampling" step, and I'm curious whether or not it will result in an unbiased estimate of $f$.

Suppose you have 3 random variables $X,Y,Z$. $Z$ is conditionally independent of $X$ given $Y$. You would like samples from $P(Y|Z)$ given a finite set of $N$ samples $x^{i}$ from $P(X)$ such that the estimate $\frac{1}{N}\sum_{i} w_{i}f(y^{i})$ is unbiased of $\mathbb{E}[f(Y)|Z=z]$ for some weights $w_i$.

I begin by sampling $y_{tmp}^{i} \sim P(y^{i} | x^i)$. I then assign each sample a weight $b_{i} = P(z|y^{i}_{tmp})$ for an observed realization for $z$. I "resample" $y^{i}$ from $y_{tmp}^i$ $N$ times according to a multinomial distribution with weights $b_i$, then assign each $y^{i}$ weights $w_{i} = \frac{1}{N}$.

Do the weighted mean of the resulting values $y^{i}$ give an unbiased estimate for $\mathbb{E}[f(Y)|Z=z]$ for some reasonable functions f (perhaps, bounded and continuous)?

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