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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.

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    $\begingroup$ 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. $\endgroup$ Mar 28, 2021 at 21:38

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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.

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