> by using an adaptive importance sampling method Although not all MCMC schemes actually do this, e.g. naive Gibbs (re)sampling just sweeps through the domain

> by using an adaptive importance sampling method Although not all MCMC schemes actually do this, e.g. naive Gibbs (re)sampling just sweeps through the domain spaces iteratively. You're right though MCMC enables you to do this relatively easily. Still requires some work if to get this right, especially if your integrand has multiple 'spikes'.