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Say I want to use Metropolis Hastings algorithm to get posterior draws of multivariate parameters.

In the one variable case, you could manipulate delta until you found something that worked (gave 40% acceptance).

However how would that work in the 2 variable case? Am I manipulating a vector of deltas that are the variance of a multivariate normal proposal distribution? How do I know if I am moving enough in both parameter spaces? Each parameter might require a different delta to move enough in their own probability space a a vector delta may not be adequate.

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This is a very natural question! Rather than trying to scale one direction at a time, a possible resolution is to first run a default algorithm, e.g. Gibbs or Metropolis-within-Gibbs or even maximum likelihood estimation to get a rough idea of the variance-covariance matrix of the targeted posterior. Using this first step estimate $\mathbf{M}$, one can then implement a random walk Metropolis with variance $\delta\mathbf{M}$ towards calibrating $\delta$. If this does not cover the acceptance rate of interest, it may be necessary to return to [block] Gibbs.

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  • $\begingroup$ So does this mean that if you have the conditional posterior available you use that to get an idea of $M$, then switch from Gibbs to MH? Why would you make that switch at all? Because of slow mixing using Gibbs? In some sense this would kind of be like a two-step version of the adaptive Metropolis-Hastings algorithm, I suppose. $\endgroup$ – hejseb Oct 27 '18 at 10:28

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