Thank you! This is indeed food for thought.
Exactly. Even in a simple standard normal in N dimensions, a draw is just as likely to be in any quadrant of the 2^N available. In twenty dimensions, there are more than a million quadrants (i.e., choices of signs of components of 20-vectors). So any thought of getting reasonable “coverage” of the posterior goes out the window. The fact that integrals can converge in the face of the sample of draws being so sparse never ceases to amaze me.