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. 
 A: 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 $(x_i,\omega_i)$...
A: The fundamental idea behind importance sampling is to be able to estimate better the tails of the distribution.  It is a variance reduction trick in Monte Carlo.  Generally you need a large sample to get observations in the tale of the distribution.  Importance sampling gives more weight to the tails or in general any part of the distribution that needs more attention than it would get from basic (sometimes called naive) Monte Carlo.  The term is due to Hammersley and Handscombe who wrote a book on Monte Carlo and variance reduction techniques way back in the 1950s.  Because you know how you biased the sampling with the weights you chose you can determine weights to get unbiased estimates.  There is no free lunch though.  While you gain accuracy for some estimates you will lose for others.  For example if the distribution has a finite mean the adjusted unbiased estimate will have a larger variance than what you would have gotten taking an unweighted average using naive Monte Carlo.  When talking about bootstrap you are usually referring to the process of drawing inference about a population based on the mechanism of sampling with replacement from the original data set.  In theory it does not involve Monte Carlo.  In practice you cannot usually get bootstrap estimates without a Monte Carlo approximation.  I sense that your understanding of bootstrap may be fuzzy.  To learn more about the bootstrap Tim Hesterberg's chapter in David Moore's introductory statistics book can be very helpful.  Also the book by Efron and Tibshirani or one of mine explain the bootstrap well.  Hesterberg's new Mathematical Statistics book also covers bootstrap.
