I want to sample from the distribution of a mixture distribution. The hierarchical model is $x_i\sim f$, where:

$$f(x\mid \theta_1,\dots,\theta_p, w_1,\dots,\omega_p) = \sum_{j=1}w_p\varphi(x\mid\theta_p),$$ and $\theta\sim\pi(\theta)$ for some appropriate prior and $w\sim Dirichlet(w\mid \alpha)$.

Is it possible to sample from such distribution using a Metropolis within Gibbs sampler? Is there some reference I can read to implement such sampler. My main concern is to properly account for the fact that $\sum_{j=1}^p w_j = 1$.

  • $\begingroup$ Metropolis gives complete freedom in the choice of the proposal so what is exctly your question? $\endgroup$ – Xi'an Aug 12 '19 at 11:10

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