What is a good proposal distribution for Metropolis-Hastings for strictly positive parameters?

In Metropolis-Hastings sampling it is required to have a proposal distribution $g(\theta_{new}|\theta_{old},\lambda)$, where $\theta_{old}$ is the last accepted sample in the chain and $\theta_{new}$ is a newly proposed sample. Samples are drawn from $g$ which are then transformed into the acceptance probability (the proposal step) for subsequent use in the move step. For example, $g$ is often chosen multivariate normal with mean $\theta_{old}$ and variance-covariance matrix $\lambda$.

I have a parameter $\theta$ which is a precision (or scale or variance) so that $\theta >0$ is a constraint on the parameter space. This thread discusses various options in this situation. I find the best option to use a function $g$ which only proposes samples from the allowed parameter space.

My questions are: which are commonly used distributions in this situation? And, how should the proposal distribution be conditioned on $\theta_{old}$?

For example, I can imagine the uniform $U(0,b)$ distribution may be useful, but I do not see how to choose $b$ or how to condition on $\theta_{old}$ (i.e. the last accepted precision/variance in my case).

An alternative may be Gamma or inverse Gamma, e.g. $Gam(a,b)$, but then also the question is how to choose $a$ and $b$ and how to link them to $\theta_{old}$.

• It is not clear why you think the "best option" is to only propose samples from the allowed parameter space. There is nothing wrong (although it may not be ideal) to propose negative values in a random walk Metropolis which then get rejected because the acceptance probability is zero. – jaradniemi Oct 24 '17 at 19:37
• I second Jarad's comment in that in situations with "sharp corners", it may prove more efficient to make proposals that may fall out of the space than proposals that get stuck into such "corners". – Xi'an Oct 24 '17 at 19:41
• @jaradniemi My intuition is that if $\theta$ is close to zero, the rejection rate may then be very high. My application is one where $\theta$ (the precision) effectively plays the role of a penalization parameter with small values signifying stronger penalization. – tomka Oct 24 '17 at 19:45
• Thinking about it, I could reparameterize to the inverse of $\theta$ to avoid this problem. Still the argument stays valid I believe. – tomka Oct 24 '17 at 19:46
• Requiring visits very close to zero with a truncated proposal may be inefficient: when visits to this region are rare, they get massively weighted by the Metropolis ratio and the chain stays there for very long in order to compensate. – Xi'an Oct 24 '17 at 20:12

The most natural [and generic] resolution [imo] is to turn $\theta$ into $\eta=\log\theta$ in the original problem so that $\eta$ is unconstrained. This allows for the use of random walk proposals like Metropolis et al.'s. The only warning is that the prior must incorporate the change of variable through a Jacobian: $$\pi_\eta(\eta)=\pi_\theta(\exp\{\eta\})\times\exp\{\eta\}$$ Warning: This proposal is only equivalent to propose a log-normal new value in the original parameterisation if the proper Metropolis ratio is used [in the original parameterisation, the proposal is no longer a random walk]. (Note that an exponential change of variables turns the likelihood $p(D|θ,ν)$ into $p(D|\exp\{η\},ν)$, without a Jacobian there!)
Otherwise, a Uniform $\text{U}(\theta^\text{old}-\epsilon,\theta^\text{old}+\epsilon)$ proposal, as in Hastings (1970), can be chosen as a proposal, with the potential to propose negative [and hence surely rejected] values.
• Every where you see $\theta$, you plug in $e^\eta$, but you also need to multiple the prior by $e^\eta$ due to the Jacobian in the transformation. – jaradniemi Oct 24 '17 at 19:36
• But $\theta$ also occurs in the likelihood, for example. Is that of no consequence? Let's say I need posterior $p(\theta|D) \propto p(D|\theta,\nu)p(\nu)p(\theta)$ (with $\nu$ some nuisance parameter and $D$ data). Then $\theta$ is also substituted in the LL. – tomka Oct 24 '17 at 19:41
• One more question; it has been argued in the link I cited above and by @jaradniemi that an alternative is to reject proposals outside of the space of $\theta$. I wonder if any of the two methods, transformation or this simple rejection rule, can be expected to outperform the other. – tomka Oct 26 '17 at 10:22