mapPLMIX: MAP estimation for a Bayesian mixture of Plackett-Luce models

In PLMIX: Bayesian Analysis of Finite Mixtures of Plackett-Luce Models for Partial Rankings/Orderings

Description

Perform MAP estimation via EM algorithm for a Bayesian mixture of Plackett-Luce models fitted to partial orderings.

Usage

mapPLMIX(pi_inv, K, G, init = list(p = NULL, omega = NULL),
  n_iter = 1000, hyper = list(shape0 = matrix(1, nrow = G, ncol = K),
  rate0 = rep(0, G), alpha0 = rep(1, G)), eps = 10^(-6),
  centered_start = FALSE, plot_objective = FALSE)

Arguments

pi_inv	An object of class top_ordering, collecting the numeric NxK data matrix of partial orderings, or an object that can be coerced with as.top_ordering.
K	Number of possible items.
G	Number of mixture components.
init	List of named objects with initialization values: p is a numeric GxK matrix of component-specific support parameters; omega is a numeric vector of G mixture weights. If starting values are not supplied (NULL), they are randomly generated with a uniform distribution. Default is NULL.
n_iter	Maximum number of EM iterations.
hyper	List of named objects with hyperparameter values for the conjugate prior specification: shape0 is a numeric GxK matrix of shape hyperparameters; rate0 is a numeric vector of G rate hyperparameters; alpha0 is a numeric vector of G Dirichlet hyperparameters. Default is noninformative (flat) prior setting.
eps	Tolerance value for the convergence criterion.
centered_start	Logical: whether a random start whose support parameters and weights should be centered around the observed relative frequency that each item has been ranked top. Default is FALSE. Ignored when init is not NULL.
plot_objective	Logical: whether the objective function (that is the kernel of the log-posterior distribution) should be plotted. Default is FALSE.

Details

Under noninformative (flat) prior setting, the EM algorithm for MAP estimation corresponds to the EMM algorithm described by Gormley and Murphy (2006) to perform frequentist inference. In this case, the MAP solution coincides with the MLE and the output vectors log_lik and objective coincide as well.

The mapPLMIX function performs the MAP procedure with a single starting value. To address the issue of local maxima in the posterior distribution, see the mapPLMIX_multistart function.

Value

A list of S3 class mpPLMIX with named elements:

W_map	Numeric vector with the MAP estimates of the G mixture weights.
P_map	Numeric GxK matrix with the MAP estimates of the component-specific support parameters.
z_hat	Numeric NxG matrix of estimated posterior component membership probabilities.
class_map	Numeric vector of N mixture component memberships based on MAP allocation from the z_hat matrix.
log_lik	Numeric vector of the log-likelihood values at each iteration.
objective	Numeric vector of the objective function values (that is the kernel of the log-posterior distribution) at each iteration.
max_objective	Maximized objective function value.
bic	BIC value (only for the default flat priors, otherwise NULL).
conv	Binary convergence indicator: 1 = convergence has been achieved, 0 = otherwise.
call	The matched call.

Author(s)

Cristina Mollica and Luca Tardella

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.

Gormley, I. C. and Murphy, T. B. (2006). Analysis of Irish third-level college applications data. Journal of the Royal Statistical Society: Series A, 169(2), pages 361–379, ISSN: 0964-1998, DOI: 10.1111/j.1467-985X.2006.00412.x.

See Also

mapPLMIX_multistart

Examples

data(d_carconf)
MAP <- mapPLMIX(pi_inv=d_carconf, K=ncol(d_carconf), G=3, n_iter=400*3)
str(MAP)
MAP$P_map
MAP$W_map

PLMIX documentation built on Sept. 4, 2019, 5:03 p.m.