Metropolis Hastings with Gamma Proposal Density

I am trying to use Metropolis Hastings to sample from a shifted gamma distribution. Since it is shifted, it has a domain of $$(n, \infty)$$. I tried using a Gaussian proposal density and ran into the issue of encountering values that are not in the domain of what I am trying to sample from. I am hoping to use a gamma distribution as my proposal density but am not quite sure how that would work. When using a Gaussian for this algorithm, I know you change $$\mu$$ so that the proposal density is centered at the current state, but how would this work for a Gamma density? Do you change the shape parameter? Any advice or clarifying resources would be appreciated.

• welcome to cross validated, for the shifted gamma do you know the value $n$ at the step in the MH step when you will make a draw? If so then you could use $n + Gamma(\alpha, \beta)$ as a draw, assuming I've understood correctly what you mean by shifted gamma. Mar 28, 2021 at 21:38

I cannot find the connected earlier questions on $$\mathsf X$$ validated but many addressed this issue of making a proposal $$q(\cdot;\cdot)$$ that covers more than the support of the target distribution $$\pi(\cdot)$$. This is not an issue: when $$x'\sim q(x;x_t)$$is such that$$\pi(x')=0$$the Metropolis-Hastings acceptance ratio $$1 \wedge \dfrac{q(x';x_t)\pi(x')}{q(x_t;x')\pi(x_t)}$$ is equal to zero and therefore $$x'$$ is rejected, hence $$x_{t+1}=x_t$$. If using a proposal with the same support as $$\pi$$, this occurrence obviously disappears but it does not mean that the resulting Metropolis-Hastings algorithm is more efficient.
Concerning the specific choice of using a Gamma proposal, $$\mathcal G(\alpha_t,\beta_t)$$, taking$$\alpha_t=\beta x_t\quad\beta_t=\beta$$makes for a centred proposal, with the rate $$\beta$$ free to calibrate in terms of the acceptance rate. Note that the support of $$\pi(\cdot)$$ must be at most $$\mathbb R_+$$ in this case.