I'm new to MCMC, so this might be obvious.
Let's say we're using MCMC to estimate a posterior distribution. We run MCMC, and it returns a representative sample from that posterior distribution.
The sample is just a collection of data points. To actually use it, we just plot a (normalized) histogram of those data points to draw our estimated posterior.
However, we also know the posterior probability of each data point in the sample! This is even explicitly calculated in the Metropolis MCMC algorithm. Why do we ignore this?
Edit: Here's a similar question.