BIC and AIC for mixtures of Mallows models with Spearman distance
bicMSmix and aicMSmix compute, respectively, the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) for a mixture of Mallows models with Spearman distance fitted on partial rankings.
bicMSmix(rho, theta, weights, rankings) aicMSmix(rho, theta, weights, rankings)
rho: Integer G$$x$$n matrix with the component-specific consensus rankings in each row.theta: Numeric vector of non-negative component-specific precision parameters.weights: Numeric vector of positive mixture weights (normalization is not necessary).rankings: Integer N$$x$$n matrix or data frame with partial rankings in each row. Missing positions must be coded as NA.The BIC or AIC value.
The (log-)likelihood evaluation is performed by augmenting the partial rankings with the set of all compatible full rankings (see data_augmentation), and then the marginal likelihood is computed.
When , the (log-)likelihood is exactly computed, otherwise it is approximated with the method introduced by Crispino et al. (2023). If , the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.
## Example 1. Simulate rankings from a 2-component mixture of Mallows models ## with Spearman distance. set.seed(12345) rank_sim <- rMSmix(sample_size = 50, n_items = 12, n_clust = 2) str(rank_sim) rankings <- rank_sim$samples # Fit the true model. set.seed(12345) fit <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10) # Comparing the BIC at the true parameter values and at the MLE. bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights, rankings = rank_sim$samples) bicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights, rankings = rank_sim$samples) aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights, rankings = rank_sim$samples) aicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights, rankings = rank_sim$samples) ## Example 2. Simulate rankings from a basic Mallows model with Spearman distance. set.seed(54321) rank_sim <- rMSmix(sample_size = 50, n_items = 8, n_clust = 1) str(rank_sim) # Let us censor the observations to be top-5 rankings. rank_sim$samples[rank_sim$samples > 5] <- NA rankings <- rank_sim$samples # Fit the true model with the two EM algorithms. set.seed(54321) fit_em <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10) set.seed(54321) fit_mcem <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10, mc_em = TRUE) # Compare the BIC at the true parameter values and at the MLEs. bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights, rankings = rank_sim$samples) bicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights, rankings = rank_sim$samples) bicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights, rankings = rank_sim$samples) aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights, rankings = rank_sim$samples) aicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights, rankings = rank_sim$samples) aicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights, rankings = rank_sim$samples)
Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)
Crispino M, Mollica C, Astuti V and Tardella L (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33 (98), DOI: 10.1007/s11222-023-10266-8.
Schwarz G (1978). Estimating the dimension of a model. The Annals of Statistics, 6 (2), pages 461–464, DOI: 10.1002/sim.6224.
Sakamoto Y, Ishiguro M, and Kitagawa G (1986). Akaike Information Criterion Statistics. Dordrecht, The Netherlands: D. Reidel Publishing Company.
likMSmix, data_augmentation
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