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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.

Let $w(\theta )=\pi (\theta )/\mu (\theta )$ be the importance weight. The two normalized densities giveSince $|F-Z|<C$, there is a finite constant $M$ such that $w(\theta )\le M$ everywhere. The Independence Metropolis–Hastings algorithm has an accepted transition density satisfyingConsequently the whole state space is a small set, with the one-step minorization condition $K(x,\cdot )\ge M^{-1}\pi (\cdot )$. Iterating this Doeblin condition gives uniform geometric convergence in total variation distance, so the chain is geometrically ergodic.