> This is the essence of MCMC: when it's expensive to take independent draws of a space, we can instead ... I disagree. The essence of MCMC is that it's a clev

> This is the essence of MCMC: when it's expensive to take independent draws of a space, we can instead ... I disagree. The essence of MCMC is that it's a clever MC scheme that defeats the curse of dimensionality by using an adaptive importance sampling method. In this method, you use information from the current sample to help draw a better new sample, or at least not a much worse one. This way, even if you start form a place far away from the region of interest, you end up in that region; the trade-off is that the samples are obviously not independent anymore. In very high dimensional spaces the region of interest is always very tiny compared to the vastness of the space (that's the "curse of dimensionality"), so even when it's very cheap to draw samples, you would be hopeless without an importance sampling scheme.