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Random Samples from the Mallows Rank Model

Source: R/sample_mallows.R
sample_mallows.Rd

Generate random samples from the Mallows Rank Model (Mallows 1957) with consensus ranking \(\rho\) and scale parameter \(\alpha\). The samples are obtained by running the Metropolis-Hastings algorithm described in Appendix C of Vitelli et al. (2018) .

Usage

sample_mallows(
  rho0,
  alpha0,
  n_samples,
  leap_size = max(1L, floor(n_items/5)),
  metric = "footrule",
  diagnostic = FALSE,
  burnin = ifelse(diagnostic, 0, 1000),
  thinning = ifelse(diagnostic, 1, 1000),
  items_to_plot = NULL,
  max_lag = 1000L
)

Arguments

rho0

Vector specifying the latent consensus ranking in the Mallows rank model.

alpha0

Scalar specifying the scale parameter in the Mallows rank model.

n_samples

Integer specifying the number of random samples to generate. When diagnostic = TRUE, this number must be larger than 1.

leap_size

Integer specifying the step size of the leap-and-shift proposal distribution.

metric

Character string specifying the distance measure to use. Available options are "footrule" (default), "spearman", "cayley", "hamming", "kendall", and "ulam". See also the rmm function in the PerMallows package (Irurozki et al. 2016) for sampling from the Mallows model with Cayley, Hamming, Kendall, and Ulam distances.

diagnostic

Logical specifying whether to output convergence diagnostics. If TRUE, a diagnostic plot is printed, together with the returned samples.

burnin

Integer specifying the number of iterations to discard as burn-in. Defaults to 1000 when diagnostic = FALSE, else to 0.

thinning

Integer specifying the number of MCMC iterations to perform between each time a random rank vector is sampled. Defaults to 1000 when diagnostic = FALSE, else to 1.

items_to_plot

Integer vector used if diagnostic = TRUE, in order to specify the items to plot in the diagnostic output. If not provided, 5 items are picked at random.

max_lag

Integer specifying the maximum lag to use in the computation of autocorrelation. Defaults to 1000L. This argument is passed to stats::acf. Only used when diagnostic = TRUE.

References

Irurozki E, Calvo B, Lozano JA (2016). “PerMallows: An R Package for Mallows and Generalized Mallows Models.” Journal of Statistical Software, 71(12), 1–30. doi:10.18637/jss.v071.i12 .

Mallows CL (1957). “Non-Null Ranking Models. I.” Biometrika, 44(1/2), 114–130.

Vitelli V, Sørensen, Crispino M, Arjas E, Frigessi A (2018). “Probabilistic Preference Learning with the Mallows Rank Model.” Journal of Machine Learning Research, 18(1), 1–49. https://jmlr.org/papers/v18/15-481.html.

See also

Other rank functions: compute_expected_distance(), compute_observation_frequency(), compute_rank_distance(), create_ranking(), get_mallows_loglik()

Examples

# Sample 100 random rankings from a Mallows distribution with footrule distance
set.seed(1)
# Number of items
n_items <- 15
# Set the consensus ranking
rho0 <- seq(from = 1, to = n_items, by = 1)
# Set the scale
alpha0 <- 10
# Number of samples
n_samples <- 100
# We first do a diagnostic run, to find the thinning and burnin to use
# We set n_samples to 1000, in order to run 1000 diagnostic iterations.
test <- sample_mallows(rho0 = rho0, alpha0 = alpha0, diagnostic = TRUE,
                       n_samples = 1000, burnin = 1, thinning = 1)
#> Items not provided by user. Picking 5 at random.

#> [1] "Press [enter] to see the next plot"

# When items_to_plot is not set, 5 items are picked at random. We can change this.
# We can also reduce the number of lags computed in the autocorrelation plots
test <- sample_mallows(rho0 = rho0, alpha0 = alpha0, diagnostic = TRUE,
                       n_samples = 1000, burnin = 1, thinning = 1,
                       items_to_plot = c(1:3, 10, 15), max_lag = 500)

#> [1] "Press [enter] to see the next plot"

# From the autocorrelation plot, it looks like we should use
# a thinning of at least 200. We set thinning = 1000 to be safe,
# since the algorithm in any case is fast. The Markov Chain
# seems to mix quickly, but we set the burnin to 1000 to be safe.
# We now run sample_mallows again, to get the 100 samples we want:
samples <- sample_mallows(rho0 = rho0, alpha0 = alpha0, n_samples = 100,
                          burnin = 1000, thinning = 1000)
# The samples matrix now contains 100 rows with rankings of 15 items.
# A good diagnostic, in order to confirm that burnin and thinning are set high
# enough, is to run compute_mallows on the samples
model_fit <- compute_mallows(
  setup_rank_data(samples),
  compute_options = set_compute_options(nmc = 10000))
# The highest posterior density interval covers alpha0 = 10.
burnin(model_fit) <- 2000
compute_posterior_intervals(model_fit, parameter = "alpha")
#>   parameter  mean median           hpdi central_interval
#> 1     alpha 9.840  9.838 [9.330,10.360]   [9.338,10.373]