MSmix (version 1.0.2)

likMSmix: (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
)

Value

The (log)-likelihood value.

Arguments

rho

Integer \(G\)\(\times\)\(n\) 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 \(N\)\(\times\)\(n\) matrix or data frame 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.

References

Crispino M, Mollica C and Modugno 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)