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

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
)

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 \(G\)\(\times\)\(K\) matrix with the MAP estimates of the component-specific support parameters.
z_hat: Numeric \(N\)\(\times\)\(G\) 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.

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.
K: Number of possible items.
G: Number of mixture components.
init: List of named objects with initialization values: p is a numeric \(G\)\(\times\)\(K\) 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 \(G\)\(\times\)\(K\) 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.

Author

Cristina Mollica and Luca Tardella

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.

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>.

Examples

Run this code


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

Run the code above in your browser using DataLab