How can you model a finite mixture model where it is over a dirac point mass? [duplicate]

I am trying to model the finite mixture model case where:

$$\theta \sim \sum_{h=1}^{k}\pi_h \delta_{\theta^{*}_h}$$

where $\theta^{*}_h \sim Gamma(\alpha, \beta)$, $(\pi_1, \ldots ,\pi_k) \sim Dirichlet(1/k, \ldots, 1/k)$, and $\delta$ is the dirac point mass.

I am doing this in R, but I do not even know how to start. Would anyone have any idea how I can implement this? Thanks!

• (1) Does your notation mean that you draw $k$ independent values from a $\Gamma(\alpha,\beta)$ distribution and then independently weight them with $k$ values from a Dirichlet distribution? (2) Regardless, what exactly do you mean by "model ... a model"?? What procedure are you contemplating implementing?
– whuber
Nov 10, 2015 at 20:46
• Hi, that is exactly what I am trying to do in the first park you described. I am trying to find a way to obtain draws from this weighted mixture of dirichlet "weights" and the dirac point mass at the Gamma draws. Would you know how I can proceed with this? Nov 10, 2015 at 21:25
• I'm still trying to figure out what you want to do. If you draw independent $\pi_h$ and $\theta^{*}_h$ for each $\theta$, then what you are doing is no different than drawing a single value from a $\Gamma(\alpha,\beta)$ distribution. If you draw $k$ values once and for all from the Gamma distribution and draw a new $(\pi_i)$ for each $\theta$, then you are choosing each of those $k$ Gamma values with equal probability. Shall we presume, then, that you are defining a discrete distribution by drawing exactly one set of $k$ Gamma values and one set of $\pi_i$ and sampling repeatedly from it?
– whuber
Nov 10, 2015 at 23:06
• Hi, that is exactly what I'm trying to do, sorry if I didn't make it more clear. Nov 12, 2015 at 2:35