Proposals for the parameters

For the proposal step, we have to balance the desire to move globally through the parameter space $\Omega$ with the need to make computationally feasible moves in areas of high probability. Therefore we split up the process and we apply a suitable proposition to the variables at each iteration. For the frequency parameters we adopt a Dirichlet proposal distribution centred at the current frequency vector used by Larget and Simon (1999). For the gamma distribution parameter and the substitution rate ratios, a normal proposal distribution centred at the current value is used, with reflecting boundary at zero and at the upper-limit defined above. Distant moves in $\Omega$ might result in a low acceptance rate whereas small modifications will prevent a full inspection of highly probable areas. The parameters used to move into the state space must therefore be carefully chosen for proper mixing (quick convergence and good sampling from the posterior probability density). With mcmcphase, these parameters can be adjusted during the burnin period.