# 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

• 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

Essentially the analytic solution is to do the eigendecomposition of Q into $VLV^T$. The pmf of the process at $t$ is, then $p_i(t)=\sum_{i=1}^{4}c_i v_i e^{\lambda_i}$ where the constants, $c_i$ are obtained by solving for some initial probability vector $b$ the equation $Ec=b$.