# Can the ratio importance sampling estimate by made to be unbiased with resampling?

Consider approximating the following integral: $$\mathcal{Z} = \int h(x) \pi(x) dx$$ Where $\pi$ is known only up to a normalizing constant, that is, $\pi(x) = \hat{\pi}(x)/\mathcal{Z}_\pi$. We can simulate $\{x^{(i)}\}_{i=1}^m$ from an appropriate proposal distribution $q$ and use the ratio importance sampling estimator: $$\mathcal{Z} \approx \sum_{i=1}^m w(x^{(i)})h(x^{(i)}) / \sum_{i=1}^m w(x^{(i)})$$ Where $w(x^{(i)}) = \hat{\pi}(x^{(i)})/q(x^{(i)})$. This is a consistent estimator and actually even has lower asymptotic variance than the standard version (which can be used when $\pi$ is normalized) but it cannot be said to be unbiased. In the event that one needs provably unbiased samples could this be achieved through the use of the resampling bootstrap? That is, we collect our weighted samples as usual: $\{w^{(i)},x^{(i)}\}_{i=1}^m$ then proceed to a resampling stage to generate a set of unweighted samples: $\{\hat{x}^{(i)}\}_{i=1}^m$ which are distributed according to $\pi$. At this point we can use the standard empirical average to yield an unbiased estimate: $$\mathcal{Z} \approx m^{-1} \sum_{i=1}^m h(\hat{x}^{(i)})$$ Of course, this is not likely to be a practical approach since the bias in the original estimate is not typically a problem, my question is my attempt to settle an argument over the fact that the bias doesn't matter, and even if it does, it can be worked around in this way.

This is not possible because the weights are not normalised: in the resampling stage, you need first to normalise the $\omega_i$'s into $$\bar \omega_i = \omega_i \bigg/ \sum_{j=1}^m \omega_j$$ to be able to sample the $(x_i,\omega_i)$...