 Solution

Choose a fixed Dirichlet distribution proposal $q(\mathbf{\omega })\propto {\prod }_i{\omega }_i^{{\eta }_i-1}$ with every ${\eta }_i>0$, independent of the current state, and propose the entire weight vector. This automatically respects the simplex constraint. With $D_j^{\prime }={\sum }_i{\omega }_i^{\prime }\phi (x_j;{\mu }_i,v_i)$, the Independence Metropolis–Hastings algorithm accepts withThis includes $q(\mathbf{\omega })/q({\mathbf{\omega }}^{\prime })$, not merely the posterior ratio. A particularly simple choice is ${\eta }_i={ϵ}_i$, the weight prior: its factors cancel and the ratio is just the mixture likelihood ratio. A proposal concentrated around a pilot estimate can mix better, provided it retains positive density throughout the relevant simplex. Both target and proposal densities use the same $k-1$ independent simplex coordinates, so this direct weight proposal has no extra transformation Jacobian determinant.