selectPLMIX_single: Bayesian selection criteria for mixtures of Plackett-Luce models

Description

Compute Bayesian comparison criteria for mixtures of Plackett-Luce models with a different number of components.

Usage

selectPLMIX_single(
  pi_inv,
  G,
  MCMCsampleP = NULL,
  MCMCsampleW = NULL,
  MAPestP,
  MAPestW,
  deviance,
  post_est = "mean"
)

Value

A list of named objects:

point_estP: Numeric \(G\)\(\times\)\((K+1)\) matrix with the point estimates of the Plackett-Luce mixture parameters. The \((K+1)\)-th column contains estimates of the mixture weights.
point_estW: Numeric \(G\)\(\times\)\((K+1)\) matrix with the point estimates of the Plackett-Luce mixture parameters. The \((K+1)\)-th column contains estimates of the mixture weights.
D_bar: Posterior expected deviance.
D_hat: Deviance function evaluated at point_est.
pD: Effective number of parameters computed as D_bar-D_hat.
pV: Effective number of parameters computed as half the posterior variance of the deviance.
DIC1: Deviance Information Criterion with penalty term equal to pD.
DIC2: Deviance Information Criterion with penalty term equal to pV.
BPIC1: Bayesian Predictive Information Criterion obtained from DIC1 by doubling its penalty term.
BPIC2: Bayesian Predictive Information Criterion obtained from DIC2 by doubling its penalty term.
BICM1: Bayesian Information Criterion-Monte Carlo.
BICM2: Bayesian Information Criterion-Monte Carlo based on the actual MAP estimate given in the MAPestP and MAPestW arguments (unlike BICM1, no approximation of the MAP estimate from the MCMC sample).

Arguments

pi_inv: An object of class top_ordering, collecting the numeric \(N\)\(\times\)\(K\) data matrix of partial orderings, or an object that can be coerced with as.top_ordering.
G: Number of mixture components.
MCMCsampleP: Numeric \(L\)\(\times\)\(G*K\) matrix with the MCMC samples of the component-specific support parameters.
MCMCsampleW: Numeric \(L\)\(\times\)\(G\) matrix with the MCMC samples of the mixture weights.
MAPestP: Numeric \(G\)\(\times\)\(K\) matrix of MAP component-specific support parameter estimates.
MAPestW: Numeric vector of the \(G\) MAP estimates of the mixture weights.
deviance: Numeric vector of posterior deviance values.
post_est: Character string indicating the point estimates of the Plackett-Luce mixture parameters to be computed from the MCMC sample. This argument is ignored when MAP estimates are supplied in the MAPestP and MAPestW arguments. Default is "mean". Alternatively, one can choose "median".

Author

Cristina Mollica and Luca Tardella

Details

Two versions of DIC and BPIC are returned corresponding to two alternative ways of computing the penalty term: the former was proposed by Spiegelhalter et al. (2002) and is denoted with pD, whereas the latter was proposed by Gelman et al. (2004) and is denoted with pV. DIC2 coincides with AICM, that is, the Bayesian counterpart of AIC introduced by Raftery et al. (2007).

References

Mollica, C. and Tardella, L. (2017). Bayesian Plackett-Luce mixture models for partially ranked data. Psychometrika, 82(2), pages 442--458, ISSN: 0033-3123, <doi:10.1007/s11336-016-9530-0>.

Ando, T. (2007). Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models. Biometrika, 94(2), pages 443--458.

Raftery, A. E, Satagopan, J. M., Newton M. A. and Krivitsky, P. N. (2007). BAYESIAN STATISTICS 8. Proceedings of the eighth Valencia International Meeting 2006, pages 371--416. Oxford University Press.

Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian data analysis. Chapman & Hall/CRC, Second Edition, ISBN: 1-58488-388-X. New York.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), pages 583--639.