Description Usage Arguments Details Value Author(s) References Examples
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.
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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 |
verbose |
Set to |
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,Sigma) Dir(eta; q) prod_j N(mu_j; mu0, g Sigma) IW(Sigma_j; nu0, S0)
where
d(mu,Sigma)= [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.
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 |
David Rossell
Fuquene J., Steel M.F.J., Rossell D. On choosing mixture components via non-local priors. 2018. arXiv
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