Effective sample size of a weighted sample I am trying to understand the meaning of the effective sample size when we have weights for the sample. I have an intuitive understanding of why effective sample size can be less than the raw sample size when correlations exist in the data, but can somebody please explain why this is the case when the sample is weighted? Can weights be regarded as introducing some kind of correlation into the data? Thanks!
 A: 
Can weights be regarded as introducing some kind of correlation into the data?

Assume you have two uncorrelated standard normal variables with lots of samples.  If all points on / closest to a positively sloped line are heavily weighted relative to other points, you would induce positive correlation. Similarly, if all points on / closest to a negatively sloped line are heavily weighted relative to other points, you would induce negative correlation.
Sometimes weighting points near a line is intentional, see for instance RANSAC (random sample consensus).

I am trying to understand the meaning of the effective sample size when we have weights for the sample.

In a sample with independent identically distributed (i.i.d.) observations, precision (inverse of variance) of parameter estimates (e.g. mean $\hat{\mu}$) is additive, accumulating over the sample observations.   If you weight your sample observations or your sample is not i.i.d, the precision of your parameter estimate and the effective size of your sample is reduced relative to an equally weighted i.i.d. sample with the same number of observations.
Intuitively effective sample size is your actual sample size scaled by the ratio of precision of a parameter estimate using your weighted sample divided by the precision of the same parameter estimate in a counterfactual i.i.d. sample.
For purposes of intuitively understanding reduced effective sample size, consider
$$N_\textrm{effective} \approx 
N_\textrm{actual} \frac{\textrm{Precision}_W(\hat{\mu})}{\textrm{Precision}_\textrm{iid}(\hat{\mu})} =
N_\textrm{actual} \frac{\textrm{Var}_\textrm{iid}(\hat{\mu})}{\textrm{Var}_W(\hat{\mu})} 
$$
See discussion and references in Effective sample size for more details.
