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It's often quoted that for an N-dimensional Gaussian-like target distribution, the ideal acceptance ratio is 0.23 (if your proposal distribution is also Gaussian).

Assuming we use a gaussian proposal distribution with a diagonal covariance matrix, what would the above correspond to in $\sigma_i$ (the tuning length or scale) for each parameter, in terms of the standard deviation of the target distribution for that parameter?

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For a Gaussian target distribution of unit variance in each dimension, the optimal $\sigma$ of the proposal distribution is given as $\sigma_d \approx 2.38 d^{-1/2}$ by Gelman, Roberts, and Gilks 1996. Here $d$ is the dimension of the parameter space. If the target distribution is some other variety of Gaussian, you just have to scale it appropriately. To be exact, multiply its covariance matrix by the square of $\sigma_d$ given above.

Of course, if you already know what the target distribution is there's not much point in sampling via MCMC. That's probably why people are more apt to quote the ideal acceptance rate, in the hope that it's more generally applicable.

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