Jumping over the scale parameter

Updating \(\alpha\) in every step of the MCMC algorithm may not be necessary, as the number of posterior samples typically is more than large enough to obtain good estimates of its posterior distribution. With the alpha_jump argument, we can tell the MCMC algorithm to update \(\alpha\) only every alpha_jump-th iteration. To update \(\alpha\) every 10th time \(\boldsymbol{\rho}\) is updated, we do

bmm_visual <- compute_mallows(
  data = complete_data, 
  compute_options = 
    set_compute_options(nmc = 21000, burnin = 1000, alpha_jump = 10)
  )

Other distance metric

By default, compute_mallows uses the footrule distance, but the user can also choose to use Cayley, Kendall, Hamming, Spearman, or Ulam distance. Running the same analysis of the potato data with Spearman distance is done with the command

bmm <- compute_mallows(
  data = complete_data, 
  model_options = set_model_options(metric = "spearman"),
  compute_options = set_compute_options(nmc = 21000, burnin = 1000)
)

For the particular case of Spearman distance, BayesMallows only has integer sequences for computing the exact partition function with 14 or fewer items. In this case a precomputed importance sampling estimate is part of the package, and used instead.

Convergence diagnostics

As with the potato data, we can do a test run to assess the convergence of the MCMC algorithm. This time we use the beach_data object that we generated above, based on the stated preferences. We also set save_aug = TRUE to save the augmented rankings in each MCMC step, hence letting us assess the convergence of the augmented rankings.

bmm_test <- compute_mallows(
  data = beach_data,
  compute_options = set_compute_options(save_aug = TRUE))

Running assess_convergence for \(\alpha\) and \(\boldsymbol{\rho}\) shows good convergence after 1000 iterations.

assess_convergence(bmm_test)

Trace plot for scale parameter.

assess_convergence(bmm_test, parameter = "rho", items = 1:6)

Trace plot for modal ranking.

To check the convergence of the data augmentation scheme, we need to set parameter = "Rtilde", and also specify which items and assessors to plot. Let us start by considering items 2, 6, and 15 for assessor 1, which we studied above.

assess_convergence(
  bmm_test, parameter = "Rtilde", items = c(2, 6, 15), assessors = 1)

Trace plot for augmented rankings.

The convergence plot illustrates how the augmented rankings vary, while also obeying their implied ordering.

By further investigation of the transitive closure, we find that no orderings are implied between beach 1 and beach 15 for assessor 2. That is, the following statement returns zero rows.

subset(tc, assessor == 2 & bottom_item %in% c(1, 15) & top_item %in% c(1, 15))
#> [1] assessor    bottom_item top_item   
#> <0 rows> (or 0-length row.names)

With the following command, we create trace plots to confirm this:

assess_convergence(
  bmm_test, parameter = "Rtilde", items = c(1, 15), assessors = 2)

Trace plot for augmented rankings where items have not been compared.

As expected, the traces of the augmented rankings for beach 1 and 15 for assessor 2 do cross each other, since no ordering is implied between them.

Ideally, we should look at trace plots for augmented ranks for more assessors to be sure that the algorithm is close to convergence. We can plot assessors 1-8 by setting assessors = 1:8. We also quite arbitrarily pick items 13-15, but the same procedure can be repeated for other items.

assess_convergence(
  bmm_test, parameter = "Rtilde", items = 13:15, assessors = 1:8)

Trace plots for items 13-15 and assessors 1-8.

The plot indicates good mixing.

Posterior distributions

Based on the convergence diagnostics, and being fairly conservative, we discard the first 2,000 MCMC iterations as burn-in, and take 20,000 additional samples.

bmm_beaches <- compute_mallows(
  data = beach_data,
  compute_options = 
    set_compute_options(nmc = 22000, burnin = 2000, save_aug = TRUE)
)

The posterior distributions of \(\alpha\) and \(\boldsymbol{\rho}\) can be studied as shown in the previous sections. Posterior intervals for the latent rankings of each beach are obtained with compute_posterior_intervals:

compute_posterior_intervals(bmm_beaches, parameter = "rho")
#>    parameter    item mean median    hpdi central_interval
#> 1        rho  Item 1    7      7     [7]              [7]
#> 2        rho  Item 2   15     15    [15]             [15]
#> 3        rho  Item 3    3      3   [3,4]            [3,4]
#> 4        rho  Item 4   12     12 [11,13]          [11,14]
#> 5        rho  Item 5    9      9  [8,10]           [8,10]
#> 6        rho  Item 6    2      2   [1,2]            [1,2]
#> 7        rho  Item 7    8      8   [8,9]           [8,10]
#> 8        rho  Item 8   12     12 [11,13]          [11,14]
#> 9        rho  Item 9    1      1   [1,2]            [1,2]
#> 10       rho Item 10    6      6   [5,6]            [5,6]
#> 11       rho Item 11    4      4   [3,4]            [3,5]
#> 12       rho Item 12   13     13 [12,14]          [12,14]
#> 13       rho Item 13   10     10  [9,10]           [9,10]
#> 14       rho Item 14   13     14 [11,14]          [11,14]
#> 15       rho Item 15    5      5   [4,5]            [4,6]

We can also rank the beaches according to their cumulative probability (CP) consensus (Vitelli et al. 2018) and their maximum posterior (MAP) rankings. This is done with the function compute_consensus, and the following call returns the CP consensus:

compute_consensus(bmm_beaches, type = "CP")
#>      cluster ranking    item cumprob
#> 1  Cluster 1       1  Item 9 0.89815
#> 2  Cluster 1       2  Item 6 1.00000
#> 3  Cluster 1       3  Item 3 0.72665
#> 4  Cluster 1       4 Item 11 0.95160
#> 5  Cluster 1       5 Item 15 0.95400
#> 6  Cluster 1       6 Item 10 0.97645
#> 7  Cluster 1       7  Item 1 1.00000
#> 8  Cluster 1       8  Item 7 0.62585
#> 9  Cluster 1       9  Item 5 0.85950
#> 10 Cluster 1      10 Item 13 1.00000
#> 11 Cluster 1      11  Item 4 0.46870
#> 12 Cluster 1      12  Item 8 0.84435
#> 13 Cluster 1      13 Item 12 0.61905
#> 14 Cluster 1      14 Item 14 0.99665
#> 15 Cluster 1      15  Item 2 1.00000

The column cumprob shows the probability of having the given rank or lower. Looking at the second row, for example, this means that beach 6 has probability 1 of having latent rank \(\rho_{6} \leq 2\). Next, beach 3 has probability 0.738 of having latent rank \(\rho_{3}\leq 3\). This is an example of how the Bayesian framework can be used to not only rank items, but also to give posterior assessments of the uncertainty of the rankings. The MAP consensus is obtained similarly, by setting type = "MAP".

compute_consensus(bmm_beaches, type = "MAP")
#>      cluster map_ranking    item probability
#> 1  Cluster 1           1  Item 9     0.04955
#> 2  Cluster 1           2  Item 6     0.04955
#> 3  Cluster 1           3  Item 3     0.04955
#> 4  Cluster 1           4 Item 11     0.04955
#> 5  Cluster 1           5 Item 15     0.04955
#> 6  Cluster 1           6 Item 10     0.04955
#> 7  Cluster 1           7  Item 1     0.04955
#> 8  Cluster 1           8  Item 7     0.04955
#> 9  Cluster 1           9  Item 5     0.04955
#> 10 Cluster 1          10 Item 13     0.04955
#> 11 Cluster 1          11  Item 4     0.04955
#> 12 Cluster 1          12  Item 8     0.04955
#> 13 Cluster 1          13 Item 14     0.04955
#> 14 Cluster 1          14 Item 12     0.04955
#> 15 Cluster 1          15  Item 2     0.04955

Keeping in mind that the ranking of beaches is based on sparse pairwise preferences, we can also ask: for beach \(i\), what is the probability of being ranked top-\(k\) by assessor \(j\), and what is the probability of having latent rank among the top-\(k\). The function plot_top_k plots these probabilities. By default, it sets k = 3, so a heatplot of the probability of being ranked top-3 is obtained with the call:

plot_top_k(bmm_beaches)

Top-3 rankings for beach preferences.

The plot shows, for each beach as indicated on the left axis, the probability that assessor \(j\) ranks the beach among top-3. For example, we see that assessor 1 has a very low probability of ranking beach 9 among her top-3, while assessor 3 has a very high probability of doing this.

The function predict_top_k returns a dataframe with all the underlying probabilities. For example, in order to find all the beaches that are among the top-3 of assessors 1-5 with more than 90 % probability, we would do:

subset(predict_top_k(bmm_beaches), prob > .9 & assessor %in% 1:5)
#>     assessor    item    prob
#> 301        1  Item 6 0.99435
#> 303        3  Item 6 0.99600
#> 305        5  Item 6 0.97605
#> 483        3  Item 9 1.00000
#> 484        4  Item 9 0.99975
#> 601        1 Item 11 0.95030

Note that assessor 2 does not appear in this table, i.e., there are no beaches for which we are at least 90 % certain that the beach is among assessor 2’s top-3.

Convergence diagnostics

The function compute_mallows_mixtures computes multiple Mallows models with different numbers of mixture components. It returns a list of models of class BayesMallowsMixtures, in which each list element contains a model with a given number of mixture components. Its arguments are n_clusters, which specifies the number of mixture components to compute, an optional parameter cl which can be set to the return value of the makeCluster function in the parallel package, and an ellipsis (...) for passing on arguments to compute_mallows.

Hypothesizing that we may not need more than 10 clusters to find a useful partitioning of the assessors, we start by doing test runs with 1, 4, 7, and 10 mixture components in order to assess convergence. We set the number of Monte Carlo samples to 5,000, and since this is a test run, we do not save within-cluster distances from each MCMC iteration and hence set include_wcd = FALSE.

library("parallel")
cl <- makeCluster(detectCores())
bmm <- compute_mallows_mixtures(
  n_clusters = c(1, 4, 7, 10),
  data = sushi_data,
  compute_options = set_compute_options(nmc = 5000, include_wcd = FALSE),
  cl = cl)
stopCluster(cl)

The function assess_convergence automatically creates a grid plot when given an object of class BayesMallowsMixtures, so we can check the convergence of \(\alpha\) with the command

assess_convergence(bmm)

Trace plots for scale parameters.

The resulting plot shows that all the chains seem to be close to convergence quite quickly. We can also make sure that the posterior distributions of the cluster probabilities \(\tau_{c}\), \((c = 1, \dots, C)\) have converged properly, by setting parameter = "cluster_probs".

assess_convergence(bmm, parameter = "cluster_probs")

Trace plots for cluster assignment probabilities.

Note that with only one cluster, the cluster probability is fixed at the value 1, while for other number of mixture components, the chains seem to be mixing well.

Deciding on the number of mixture components

Given the convergence assessment of the previous section, we are fairly confident that a burn-in of 1,000 is sufficient. We run 40,000 additional iterations, and try from 1 to 10 mixture components. Our goal is now to determine the number of mixture components to use, and in order to create an elbow plot, we set include_wcd = TRUE to compute the within-cluster distances in each step of the MCMC algorithm. Since the posterior distributions of \(\rho_{c}\) (\(c = 1,\dots,C\)) are highly peaked, we save some memory by only saving every 10th value of \(\boldsymbol{\rho}\) by setting rho_thinning = 10.

cl <- makeCluster(detectCores())
bmm <- compute_mallows_mixtures(
  n_clusters = 1:10, 
  data = sushi_data,
  compute_options = 
    set_compute_options(nmc = 11000, burnin = 1000,
                        rho_thinning = 10, include_wcd = TRUE),
  cl = cl)
stopCluster(cl)

We then create an elbow plot:

plot_elbow(bmm)

Elbow plot for deciding on the number of mixture components.

Although not clear-cut, we see that the within-cluster sum of distances levels off at around 5 clusters, and hence we choose to use 5 clusters in our model.

Posterior distributions

Having chosen 5 mixture components, we go on to fit a final model, still running 10,000 iterations after burnin. This time we call compute_mallows and set n_clusters = 5. We also set clus_thinning = 10 to save the cluster assignments of each assessor in every 10th iteration, and rho_thinning = 10 to save the estimated latent rank every 10th iteration. Note that thinning is done only for because saving the values at every iteration would result in very large objects being stored in memory, thus slowing down computation. For statistical efficiency, it is best to avoid thinning.

bmm <- compute_mallows(
  data = sushi_data,
  model_options = set_model_options(n_cluster = 5),
  compute_options = set_compute_options(
    nmc = 11000, burnin = 1000, clus_thinning = 10, rho_thinning = 10)
)

We can plot the posterior distributions of \(\alpha\) and \(\boldsymbol{\rho}\) in each cluster using plot.BayesMallows as shown previously for the potato data.

plot(bmm)

Posterior of scale parameter in each cluster.

Since there are five clusters, the easiest way of visualizing posterior rankings is by choosing a single item.

plot(bmm, parameter = "rho", items = 1)

Posterior of item 1 in each cluster.

We can also show the posterior distributions of the cluster probabilities.

plot(bmm, parameter = "cluster_probs")

Posterior for cluster probabilities.

Using the argument parameter = "cluster_assignment", we can visualize the posterior probability for each assessor of belonging to each cluster:

plot(bmm, parameter = "cluster_assignment")

Posterior for cluster assignment.

The number underlying the plot can be found using assign_cluster.

We can find clusterwise consensus rankings using compute_consensus.

cp_consensus <- compute_consensus(bmm)
reshape(
  cp_consensus, 
  direction = "wide", 
  idvar = "ranking",
  timevar = "cluster",
  varying = list(unique(cp_consensus$cluster)),
  drop = "cumprob"
  )
#>    ranking     Cluster 1     Cluster 2     Cluster 3     Cluster 4     Cluster 5
#> 1        1        shrimp    fatty tuna    fatty tuna    sea urchin    fatty tuna
#> 2        2       sea eel          tuna    salmon roe    fatty tuna    sea urchin
#> 3        3         squid       sea eel    sea urchin    salmon roe          tuna
#> 4        4           egg        shrimp          tuna       sea eel    salmon roe
#> 5        5    fatty tuna     tuna roll        shrimp        shrimp       sea eel
#> 6        6          tuna         squid     tuna roll          tuna     tuna roll
#> 7        7     tuna roll           egg         squid         squid        shrimp
#> 8        8 cucumber roll cucumber roll       sea eel     tuna roll         squid
#> 9        9    salmon roe    salmon roe           egg           egg           egg
#> 10      10    sea urchin    sea urchin cucumber roll cucumber roll cucumber roll

Note that for estimating cluster specific parameters, label switching is a potential problem that needs to be handled. BayesMallows ignores label switching issues inside the MCMC, because it has been shown that this approach is better for ensuring full convergence of the chain (Jasra, Holmes, and Stephens 2005; Celeux, Hurn, and Robert 2000). MCMC iterations can be re-ordered after convergence is achieved, for example by using the implementation of Stephens’ algorithm (Stephens 2000) provided by the R package label.switching (Papastamoulis 2016). A full example of how to assess label switching is provided in the examples for the compute_mallows function.