# Proposal for transition matrix for Metropolis-Hastings phylogenetic inference

I am using the Metropolis-Hastings algorithm for phylogenetic inference. To do so I would like to draw the substitution matrix Q from the generalized time-reversible model.

To do so I need proposal distributions for the stationary distribution and the 6 substitution probabilities. For the stationary distributions I would use a Dirichlet, but I am not sure what to take for the substitution probabilities.

My question: Can I use a Dirichlet distribution for the substitution probabilities too?

My problem is: Dirichlet samples sum up to 1 but this is not necessarily true for the 6 parameters. The domain of the proposal distribution is only a subset of the domain of the target distribution but every single substitution probability can take all values in [0,1] (Which is everything that matters to me).

I hope you understand my question :D

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You say that the substitution probabilities are not required to sum to 1, are they independent? is the sum required to be <= 1? –  Nick May 30 '12 at 19:34
They are independent and strictly positive. Because every row in a substitution matrix should sum up to 1. There is an upper bound. So they should not be much higher then 1 . These are the only constraints. –  peri4n May 31 '12 at 12:10
If they are independent, you could draw the substitution rates from Beta distributions. –  Nick May 31 '12 at 14:44