Ultimixt-package {Ultimixt}

set of R functions for estimating the parameters of mixture distribution with a Bayesian non-informative prior

Description

Despite a comprehensive literature on estimating mixtures of Gaussian distributions, there does not exist a well-accepted reference Bayesian approach to such models. One reason for the difficulty is the general prohibition against using improper priors (Fruhwirth-Schnatter, 2006) due to the ill-posed nature of such statistical objects. Kamary, Lee and Robert (2017) took advantage of a mean-variance reparametrisation of a Gaussian mixture model to propose improper but valid reference priors in this setting. This R package implements the proposal and computes posterior estimates of the parameters of a Gaussian mixture distribution. The approach applies with an arbitrary number of components. The Ultimixt R package contains an MCMC algorithm function and further functions for summarizing and plotting posterior estimates of the model parameters for any number of components.

Details

Package:	Ultimixt
Type:	Package
Version:	2.1
Date:	2017-03-07
License:	GPL (>=2.0)

Beyond simulating MCMC samples from the posterior distribution of the Gaussian mixture model, this package also produces summaries of the MCMC outputs through numerical and graphical methods.

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.

For a Poisson mixture, we consider

f(x|\lambda_1, \ldots, \lambda_k)=\frac{1}{x!}\sum_{i=1}^k p_i \lambda_i^x e^{-\lambda_i}

with a reparameterisation as \lambda=\bf{E}[X] and \lambda_i=\lambda \gamma_i/p_i. In this case, we can use the equivalent to the Jeffreys prior for the Poisson distribution, namely, \pi(\lambda)=1/\lambda, since it leads to a well-defined posterior with a single positive observation.

Author(s)

Kaniav Kamary

Maintainer: [e-mail]

References

Fruhwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer-Verlag, New York, New York.

Kamary, K., Lee, J.Y., and Robert, C.P. (2017) Weakly informative reparameterisation for location-scale mixtures. arXiv.

See Also

Ultimixt

Examples

	#K.MixReparametrized(faithful[,2], k=2, alpha0=.5, alpha=.5, Nsim=10000)