I am looking for an MCMC algorithm that leaves a target $\pi$ invariant but that overestimates the mode.
Basically I am looking for an algorithm that whose transition kernel leaves $\pi$ stationary (can be either reversible or non-reversible) but that mixes poorly in the sense that it tends to spend too much time on the mode rather than the tails.
Do you know of any such method? Intuitively I was thinking that some gradient-based algorithm could do the job. However HMC mixes very well of course. I know this is an odd question!