The author's proposal to start at a high-probability-density (HPD) point reduces the bias that you get from not doing burn-in, but it still creates a bias. In

The author's proposal to start at a high-probability-density (HPD) point reduces the bias that you get from not doing burn-in, but it still creates a bias. In order to get a reasonable answer from MCMC, you need to run for many correlation times. Burn-in should only last a few correlation times, so you don't lose that large a fraction of your chain by throwing it away. And the part of the chain you're using is biased, so you're probably not gaining anything by keeping it.