Suppose you are interested in sampling some parameter $x$. We sample proposals of $x$ (called $x^{*}$) from some normal distribution $q \sim N(\mu_{x},\sigma^{2}_{x})$. Denote $x'$ as all other parameters of interest in our model, except for $x$; and the currently accepted value of $x$ as $x^{(m)}$.

Now, we accept or reject this based on a Metropolis-Hastings scheme, such that it is accepted with probability $\alpha$, whereby: $ \alpha = \min \left\{1, \frac{p(x^{*}|x')q(x^{(m)})}{p(x^{(m)}|x')q(x^{*})} \right\}. $

Now, further suppose that I am not satisfied with the Acceptance Rate of this (or for any other reason), so I am interested in tuning the proposal. My question is, what is the correct way to "tune the proposal". Let the tuning parameter be denoted as $c_{x}$.

Option 1: Do the same as written above, except now $q \sim N(\mu_{x},\sigma^{2}_{x}c_{x})$.

Option 2: Sample $x^{*} \sim N(\mu_{x},\sigma^{2}_{x}c_{x})$, then accept-reject with $\alpha$.

Many thanks in advance.

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    $\begingroup$ It seems you are confusing $q$ [the posterior or any other target] and $p$ [the proposal]. Tuning stands within the proposal distribution, e.g., in changing the scale parameter. $\endgroup$ – Xi'an Feb 27 '17 at 14:17
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    $\begingroup$ Furthermore, the remark about "tuning your posterior" sounds like you are confused about something, as tuning the MCMC algorithm (such as targeting a certain acceptance rate) would not change the posterior distribution. $\endgroup$ – Juho Kokkala Feb 27 '17 at 19:02
  • $\begingroup$ @Xi'an thank you for your comments. I have made these changes. Are you suggesting it is option 1? $\endgroup$ – akkp Feb 27 '17 at 21:36
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    $\begingroup$ You changed the prior instead of the proposal: MCMC is not concerned with evaluating the prior. $\endgroup$ – Xi'an Feb 27 '17 at 21:46

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