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I need to implement a program that generates a sample from a really complicated distribution $f$ of 3 variables. I need to implement it using Metropolis-Hastings Algorithm's and its variation and I was thinking to use Metropolis Random Walk with a multivariate Normal distribution. But to use Random-Walk, the proposal must be symmetric. Is the multivariate normal always symmetric or is there any condition (about the matrix of covariances or anything else) that it has to satisfy so it is symmetric?

Thanks in advance!

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The multivariate normal is symmetric insofar as it applies to being a proposal distribution. One can just examine the ratio of PDFs:

$$ \begin{eqnarray} \frac{\mathcal{N}(X \mid Y, \Sigma)}{\mathcal{N}(Y \mid X, \Sigma)} &=& \frac{(2\pi)^{-p/2} |\Sigma|^{-1/2} \exp\left( -\frac{1}{2} (X - Y)'\Sigma^{-1}(X-Y) \right)}{(2\pi)^{-p/2} |\Sigma|^{-1/2} \exp\left( -\frac{1}{2} (Y -X)'\Sigma^{-1}(Y-X) \right)}\\ &=& \exp \left( -\frac{1}{2} (X - Y)'\Sigma^{-1}(X-Y) + \frac{1}{2}(Y -X)'\Sigma^{-1}(Y-X) \right) \\ &=& \exp\left( -\frac{1}{2} (X - Y)'\Sigma^{-1}(X-Y) + \frac{1}{2}(X -Y)'\Sigma^{-1}(X-Y) \right) \\ &=& \exp(0) \\ &=& 1, \end{eqnarray} $$

where $$ (Y -X)'\Sigma^{-1}(Y-X) = (X - Y)'\Sigma^{-1}(X-Y) $$ because the double minus signs 'pass' through the vector and matrix multiplications and are ignored by the transpose.

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