Random Samples from the Mallows Rank Model
Generate random samples from the Mallows Rank Model (Mallows 1957) with consensus ranking ρ and scale parameter α. The samples are obtained by running the Metropolis-Hastings algorithm described in Appendix C of Vitelli et al. (2018).
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 )
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 |
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 |
diagnostic |
Logical specifying whether to output convergence
diagnostics. If |
burnin |
Integer specifying the number of iterations to discard as
burn-in. Defaults to 1000 when |
thinning |
Integer specifying the number of MCMC iterations to perform
between each time a random rank vector is sampled. Defaults to 1000 when
|
items_to_plot |
Integer vector used if |
max_lag |
Integer specifying the maximum lag to use in the computation
of autocorrelation. Defaults to 1000L. This argument is passed to
|
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, https://doi.org/10.18637/jss.v071.i12.
Mallows CL (1957).
“Non-Null Ranking Models. I.”
Biometrika, 44(1/2), 114–130.
Vitelli V, Sørensen Ø, 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.
# 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)
# 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)
# 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(samples, nmc = 10000)
# The highest posterior density interval covers alpha0 = 10.
compute_posterior_intervals(model_fit, burnin = 2000, parameter = "alpha")
# The PerMallows package has a Gibbs sampler for sampling from the Mallows
# distribution with Cayley, Kendall, Hamming, and Ulam distances. For these
# distances, using the PerMallows package is typically faster.
# Let us sample 100 rankings from the Mallows model with Cayley distance,
# with the same consensus ranking and scale parameter as above.
library(PerMallows)
# Set the scale parameter of the PerMallows package corresponding to
# alpha0 in BayesMallows
theta0 = alpha0 / n_items
# Sample with PerMallows::rmm
sample1 <- rmm(n = 100, sigma0 = rho0, theta = theta0, dist.name = "cayley")
# Generate the same sample with sample_mallows
sample2 <- sample_mallows(rho0 = rho0, alpha0 = alpha0, n_samples = 100,
burnin = 1000, thinning = 1000, metric = "cayley")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.