BayesMallows (version 1.1.0)

estimate_partition_function: Estimate Partition Function

Description

Estimate the logarithm of the partition function of the Mallows rank model. Choose between the importance sampling algorithm described in vitelli2018BayesMallows and the IPFP algorithm for computing an asymptotic approximation described in mukherjee2016BayesMallows.

Usage

estimate_partition_function(
  method = "importance_sampling",
  alpha_vector,
  n_items,
  metric,
  nmc,
  degree,
  n_iterations,
  K,
  cl = NULL,
  seed = NULL
)

Arguments

method

Character string specifying the method to use in order to estimate the logarithm of the partition function. Available options are "importance_sampling" and "asymptotic".

alpha_vector

Numeric vector of \(\alpha\) values over which to compute the importance sampling estimate.

n_items

Integer specifying the number of items.

metric

Character string specifying the distance measure to use. Available options are "footrule" and "spearman" when method = "asymptotic" and in addition "cayley", "hamming", "kendall", and "ulam" when method = "importance_sampling".

nmc

Integer specifying the number of Monte Carlo samples to use in the importance sampling. Only used when method = "importance_sampling".

degree

Integer specifying the degree of the polynomial used to estimate \(\log(\alpha)\) from the grid of values provided by the importance sampling estimate.

n_iterations

Integer specifying the number of iterations to use in the asymptotic approximation of the partition function. Only used when method = "asymptotic".

K

Integer specifying the parameter \(K\) in the asymptotic approximation of the partition function. Only used when method = "asymptotic".

cl

Optional computing cluster used for parallelization, returned from parallel::makeCluster. Defaults to NULL. Only used when method = "importance_sampling".

seed

Optional random number seed.

Value

A vector of length degree which can be supplied to the logz_estimate argument of compute_mallows.

References

Examples

# NOT RUN {
  # IMPORTANCE SAMPLING
  # Let us estimate logZ(alpha) for 20 items with Spearman distance
  # We create a grid of alpha values from 0 to 10
  alpha_vector <- seq(from = 0, to = 10, by = 0.5)
  n_items <- 20
  metric <- "spearman"
  degree <- 10

  # We start with 1e3 Monte Carlo samples
  fit1 <- estimate_partition_function(method = "importance_sampling",
                                        alpha_vector = alpha_vector,
                                        n_items = n_items, metric = metric,
                                        nmc = 1e3, degree = degree)
  # A vector of polynomial regression coefficients is returned
  fit1

  # Now let us recompute with 1e4 Monte Carlo samples
  fit2 <- estimate_partition_function(method = "importance_sampling",
                                      alpha_vector = alpha_vector,
                                      n_items = n_items, metric = metric,
                                      nmc = 1e4, degree = degree)

  # ASYMPTOTIC APPROXIMATION
  # We can also compute an estimate using the asymptotic approximation
  K <- 20
  n_iterations <- 50

  fit3 <- estimate_partition_function(method = "asymptotic",
                                      alpha_vector = alpha_vector,
                                      n_items = n_items, metric = metric,
                                      n_iterations = n_iterations,
                                      K = K, degree = degree)

  # We write a little function for storing the estimates in a dataframe
  library(dplyr)
  powers <- seq(from = 0, to = degree, by = 1)

  compute_fit <- function(fit){
    tibble(alpha = alpha_vector) %>%
      rowwise() %>%
      mutate(logz_estimate = sum(alpha^powers * fit))
  }

  estimates <- bind_rows(
    "Importance Sampling 1e3" = compute_fit(fit1),
    "Importance Sampling 1e4" = compute_fit(fit2),
    "Asymptotic" = compute_fit(fit3),
    .id = "type")

  # We can now plot the two estimates side-by-side
  library(ggplot2)
  ggplot(estimates, aes(x = alpha, y = logz_estimate, color = type)) +
    geom_line()
  # We see that the two importance sampling estimates, which are unbiased,
  # overlap. The asymptotic approximation seems a bit off. It can be worthwhile
  # to try different values of n_iterations and K.

  # When we are happy, we can provide the coefficient vector in the
  # logz_estimate argument to compute_mallows
  # Say we choose to use the importance sampling estimate with 1e4 Monte Carlo samples:
  model_fit <- compute_mallows(potato_visual, metric = "spearman",
                               logz_estimate = fit2)

# }