Note: The proposed parameterisation of the Gaussian mixture distribution is given by $$f(x| \mu, \sigma , {\bf p}, \varphi, {\bf \varpi, \xi})=\sum_{i=1}^k p_i f\left(x| \mu + \sigma \gamma_i/\sqrt{p_i}, \sigma \eta_i/\sqrt{p_i}\right)$$ under the non-informative prior $\pi(\mu, \sigma)=1/\sigma$. Here, the vector of the $\gamma_i=\varphi \Psi_i\Big({\bf \varpi}, {\bf p}\Big)_i$'s belongs to an hypersphere of radius $\varphi$ intersecting with an hyperplane. It is thus expressed in terms of spherical coordinates within that hyperplane that depend on $k-2$ angular coordinates $\varpi_i$. Similarly, the vector of $\eta_i=\sqrt{1-\varphi^2}\Psi_i\Big({\bf \xi}\Big)_i$'s can be turned into a spherical coordinate in a k-dimensional Euclidean space, involving a radial coordinate $\sqrt{1-\varphi^2}$ and $k-1$ angular coordinates $\xi_i$. A natural prior for $\varpi$ is made of uniforms, $\varpi_1, \ldots, \varpi_{k-3}\sim U[0, \pi]$ and $\varpi_{k-2} \sim U[0, 2\pi]$, and for $\varphi$, we consider a beta prior $Beta(\alpha, \alpha)$. A reference prior on the angles $\xi$ is $(\xi_1, \ldots, \xi_{k-1})\sim U[0, \pi/2]^{k-1}$ and a Dirichlet prior $Dir(\alpha_0, \ldots, \alpha_0)$ is assigned to the weights $p_1, \ldots, p_k$.
Kamary, K., Lee, J.Y., and Robert, C.P. (2015) Non-informative reparameterisation of location-scale mixtures. arXiv.
Ultimixtdata(faithful)
xobs=faithful[,1]
#estimate=K.MixReparametrized(xobs, k=2, alpha0=.5, alpha=.5, Nsim=1e4)Run the code above in your browser using DataLab