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Gleaning Insights from Ordinal Preferences

You've gone through the hard work of collecting a survey with ranked choices. You appreciate the beauty of this format, by which participants relay their preferences without getting caught up in allocating abstract units of utility. What next? It would be a shame to throw away any of the population dynamics conveyed through these rankings.

This package was developed with the single aim of capturing the bevy of high-order relations between cohort preferences. It provides a flexible Bayesian model, implements two samplers, and exposes an arsenal of procedures to interpret resulting posteriors. Key features include mixtures of full-rank multivariate Gaussian cohorts, and the ability for one model to accommodate rankings between distinct sets of items.

Idealistically, our elections should run on this kind of algorithm. See this blog post that elaborates further on the subject.


pkg> add RankedChoices

(press ] in the REPL to invoke the package manager on your current project)


Here I detail how to parse rankings, handle the Gibbs sampler, and interpret the results. For instance,

# (n_respondents x ranking_size)
rankings = [
  "B" "C" "D";
  "A" "D" "C";
  "B" "C" "E";
  "A" "B" "" ; # denote missing entries with empty strings
  "D" "A" "B";
  "A" ""  "" ] # e.g...

# there are other ways to extract IssueVote structures
votes, candidates = parse_matrix(rankings)

n_candidates = length(candidates) # 5, per above
n_cohorts = 3
n_trials = 40_000

prior = make_impartial_prior(n_candidates, n_cohorts,
  precision_scale=1e1, precision_dof=1e2,
  mean_scale=1e0, dirichlet_weight=1e1)

simulation = HamiltonianSim(
  n_trajectories=1, collision_limit=128)

result = simulate(prior, simulation, votes, n_trials,
  n_burnin=10_000, seed=1337)


The main data type to carry your observations is the IssueVote, which contains an array choices of RankedChoice immutable objects. They correspond to a participant's ranking of 1:n_candidates items. All RankedChoices of each IssueVote are static integer vectors of common length R, the maximum ranking size. A model actually consumes MultiIssueVote objects, which wrap an IssueVote tuple.


You produce IssueVote(choices, n_candidates) objects by manual construction or through...

  • parse_matrix(votes::Matrix{String})
  • the more versatile parse_matrix(votes, candidates) where you list off the candidate strings in your desired order, and unrecognized strings are tossed into the second return value.
  • read_csv(filename, candidates)
  • read_xlsx(filename, candidates)


Two simulation tactics are exposed: RejectionSim and HamiltonianSim. The original and simpler implementation, RejectionSim(n_sample_attempts::Int), converges in fewer iterations because its samples are independent. It is, however, drastically slower in handling large rankings. Please ensure that n_sample_attempts is large enough that the resultant n_failures / (n_voters * n_iterations) is close to zero.

The snazzy HamiltonianSim(n_trajectories::Int, collision_limit::Int) is much more complex, but converges quickly even on large rankings. You can set parameter n_trajectories := 1 if you don't mind consecutive samples being correlated. Additionally, collision_limit can be set to a relatively high number like 128. It determines how many times a ball can bounce between the walls of a linearly constrained Gaussian before we give up early.

Conjugate Priors

My sampler is Bayesian, and takes advantage of a number of conjugate priors. We typically seek to treat all cohorts and respondents equally a priori. Hence, I expose a method

prior = make_impartial_prior(n_total_candidates, n_cohorts;
  precision_scale, precision_dof, mean_scale, dirichlet_weight)


  • precision_scale, usually around or above unit, scales the diagonal matrix that parametrizes a Wishart prior for the precision matrix.
  • precision_dof sets the strength of the Wishart prior for influencing precision matrices.
  • mean_scale, which can stay unit, constrains the spread of the multivariate cohort means. In particular, it determines how much less the mean is spread out compared to the spread of the actual utilities.
  • dirichlet_weight, typically greater than unit, determines how strongly to favor cohorts of equal proportion


result = simulate(prior, simulation, votes, n_trials;
  n_burnin, seed, indifference=false, verbose=false)

The boolean flag indifference tells the sampler whether to treat incomplete rankings as implying that all unlisted candidates are lower than those listed (indifference := false) or not (indifference := true).

More details on the core methods coming soon.