I think this question has been addressed many times on Cross Validated: proposing a negative value when the target distribution is of no consequence since the target density is equal to zero at this negative value. A Metropolis-Hasting algorithm that is described in the question as a random-walk version of MH thus works in such a context: for each proposal with a negative variance, the chain repeats its current value.

I think this question has been addressed many times on Cross Validated: proposing a negative value when the target distribution is of no consequence since the target density is equal to zero at this negative value. A Metropolis-Hasting algorithm that is described in the question as a random-walk version of MH thus works in such a context: for each proposal with a negative variance, the chain repeats its current value.

See, e.g.,

• MCMC Metropolis-Hastings' jumping distribution for non-negative parameters
• Narrow distribution and Hasting-Metropolis
• Elementary MCMC pseudocode