likMSmix {MSmix}

(Log-)likelihood for mixtures of Mallows models with Spearman distance

Description

Compute the (log-)likelihood for the parameters of a mixture of Mallow models with Spearman distance on partial rankings. Partial rankings with arbitrary missing positions are supported.

Usage

likMSmix(
  rho,
  theta,
  weights = (if (length(theta) == 1) NULL),
  rankings,
  log = TRUE
)

Arguments

rho

Integer G\timesn matrix with the component-specific consensus rankings in each row.

theta

Numeric vector of G non-negative component-specific precision parameters.

weights

Numeric vector of G positive mixture weights (normalization is not necessary).

rankings

Integer N\timesn matrix with partial rankings in each row. Missing positions must be coded as NA.

log

Logical: whether the log-likelihood must be returned. Defaults to TRUE.

Details

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 n\leq 20, the (log-)likelihood is exactly computed. When n>20, the model normalizing constant is not available and is approximated with the method introduced by Crispino et al. (2023). If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

The (log)-likelihood value.

References

Crispino M, Mollica C, Modugno L, Casadio Tarabusi E, and Tardella L (2024+). 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.

See Also

bicMSmix, aicMSmix, data_augmentation

Examples

## Example 1. Likelihood of a full ranking of n=5 items under the uniform (null) model.
likMSmix(rho = 1:5, theta = 0, weights = 1, rankings = c(3,5,2,1,4), log = FALSE)
# corresponds to...
1/factorial(5)

## Example 2. Simulate rankings from a 2-component mixture of Mallow models
## with Spearman distance.
set.seed(12345)
d_sim <- rMSmix(sample_size = 75, n_items = 8, n_clust = 2)
str(d_sim)
# Fit the true model.
rankings <- d_sim$samples
fit <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10)
# Compare log-likelihood values of the true parameter values and the MLE.
likMSmix(rho = d_sim$rho, theta = d_sim$theta, weights = d_sim$weights,
       rankings = d_sim$samples)
likMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = d_sim$samples)

## Example 3. Simulate rankings from a basic Mallow model with Spearman distance.
set.seed(12345)
d_sim <- rMSmix(sample_size = 25, n_items = 6)
str(d_sim)
# Censor data to be partial top-3 rankings.
rankings <- d_sim$samples
rankings[rankings>3] <- NA
# Fit the true model with data augmentation.
set.seed(12345)
fit <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10)
# Compare log-likelihood values of the true parameter values and the MLEs.
likMSmix(rho = d_sim$rho, theta = d_sim$theta, weights = d_sim$weights,
       rankings = d_sim$samples)
likMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = d_sim$samples)