mapPLMIX_multistart function

MAP estimation for a Bayesian mixture of Plackett-Luce models with multiple starting values

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

mapPLMIX_multistart(pi_inv, K, G, n_start = 1, init = rep(list(list(p =
  NULL, omega = NULL)), times = n_start), n_iter = 200,
  hyper = list(shape0 = matrix(1, nrow = G, ncol = K), rate0 = rep(0, G),
  alpha0 = rep(1, G)), eps = 10^(-6), plot_objective = FALSE,
  init_index = 1:n_start, parallel = FALSE, centered_start = FALSE)

Arguments

• pi_inv: An object of class top_ordering, collecting the numeric N$$x$$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.
• n_start: Number of starting values.
• init: List of n_start lists of named objects with initialization values: p is a numeric G$$x$$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$$x$$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.
• plot_objective: Logical: whether the objective function (that is the kernel of the log-posterior distribution) should be plotted. Default is FALSE.
• init_index: Numeric vector indicating the positions of the starting values in the init list to be actually launched. Useful to launch the most promising starting values identified after a preliminary run. Default is run all the starting points in the init list.
• parallel: Logical: whether parallelization should be used. Default is FALSE.
• 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.

Returns

A list of S3 class mpPLMIX with named elements:

• mod: List of named objects describing the best model in terms of maximized posterior distribution. See output values of the single-run mapPLMIX function for a detailed explanation of the list elements.

• max_objective: Numeric vector of the maximized objective function values for each initialization.

• convergence: Binary vector with length(init_index) convergence indicators for each initialization: 1 = convergence has been achieved, 0 = otherwise.

• call: The matched call.

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. The best model in terms of maximized posterior distribution is returned.

Examples

data(d_carconf)
MAP_mult <- mapPLMIX_multistart(pi_inv=d_carconf, K=ncol(d_carconf), G=3, 
                                            n_start=2, n_iter=400*3)
str(MAP_mult)
MAP_mult$mod$P_map
MAP_mult$mod$W_map

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

Author(s)

Cristina Mollica and Luca Tardella