bfnormmix {mombf}

Number of Normal mixture components under Normal-IW and Non-local priors

Description

Posterior sampling and Bayesian model selection to choose the number of components k in multivariate Normal mixtures.

bfnormmix computes posterior probabilities under non-local MOM-IW-Dir(q) priors, and also for local Normal-IW-Dir(q.niw) priors. It also computes posterior probabilities on cluster occupancy and posterior samples on the model parameters for several k.

Usage

bfnormmix(x, k=1:2, mu0=rep(0,ncol(x)), g, nu0, S0, q=3, q.niw=1,
B=10^4, burnin= round(B/10), logscale=TRUE, returndraws=TRUE, verbose=TRUE)

Arguments

x

n x p input data matrix

k

Number of components

mu0

Prior on mu[j] is N(mu0,g Sigma[j])

g

Prior on mu[j] is N(mu0,g Sigma[j]). This is a critical MOM-IW prior parameter that specifies the separation between components deemed practically relevant. It defaults to assigning 0.95 prior probability to any pair of mu's giving a bimodal mixture, see details

S0

Prior on Sigma[j] is IW(Sigma_j; nu0, S0)

nu0

Prior on Sigma[j] is IW(Sigma_j; nu0, S0)

q

Prior parameter in MOM-IW-Dir(q) prior

q.niw

Prior parameter in Normal-IW-Dir(q.niw) prior

B

Number of MCMC iterations

burnin

Number of burn-in iterations

logscale

If set to TRUE then log-Bayes factors are returned

returndraws

If set to TRUE the MCMC posterior draws under the Normal-IW-Dir prior are returned for all k

verbose

Set to TRUE to print iteration progress

Details

The likelihood is

p(x[i,] | mu,Sigma,eta)= sum_j eta_j N(x[i,]; mu_j,Sigma_j)

The Normal-IW-Dir prior is

Dir(eta; q.niw) prod_j N(mu_j; mu0, g Sigma) IW(Sigma_j; nu0, S0)

The MOM-IW-Dir prior is

d(\mu,A) Dir(\eta; q) \prod_j N(\mu_j; \mu0, g \Sigma_j) IW(\Sigma_j; \nu_0, S0)

where

d(\mu,A)= [\prod_{j<l} (\mu_j-\mu_l)' A (\mu_j-\mu_l)]

and A is the average of \Sigma_1^{-1},...,\Sigma_k^{-1}. Note that one must have q>1 for the MOM-IW-Dir to define a non-local prior.

By default the prior parameter g is set such that

P( (mu[j]-mu[l])' A (mu[j]-mu[l]) < 4)= 0.05.

The reasonale when Sigma[j]=Sigma[l] and eta[j]=eta[l] then (mu[j]-mu[l])' A (mu[j]-mu[l])>4 corresponds to a bimodal density. That is, the default g focuses 0.95 prior prob on a degree of separation between components giving rise to a bimodal mixture density.

bfnormmix computes posterior model probabilities under the MOM-IW-Dir and Normal-IW-Dir priors using MCMC output. As described in Fuquene, Steel and Rossell (2018) the estimate is based on the posterior probability that one cluster is empty under each possible k.

Value

A list with elements

k

Number of components

pp.momiw

Posterior probability of k components under a MOM-IW-Dir(q) prior

pp.niw

Posterior probability of k components under a Normal-IW-Dir(q.niw) prior

probempty

Posterior probability that any one cluster is empty under a MOM-IW-Dir(q.niw) prior

bf.momiw

Bayes factor comparing 1 vs k components under a MOM-IW-Dir(q) prior

logpen

log of the posterior mean of the MOM-IW-Dir(q) penalty term

logbf.niw

Bayes factor comparing 1 vs k components under a Normal-IW-Dir(q.niw) prior

Author(s)

David Rossell

References

Fuquene J., Steel M.F.J., Rossell D. On choosing mixture components via non-local priors. 2018. arXiv

Examples

x <- matrix(rnorm(100*2),ncol=2)

bfnormmix(x=x,k=1:3)