.. _library_bradley_terry_ranker:

`bradley_terry_ranker`

Bradley-Terry pairwise preference ranker. Uses a deterministic minorization-maximization update to estimate one relative strength parameter per item from weighted pairwise wins and losses.

The library implements the ranker_protocol defined in the ranking_protocols library. It provides predicates for learning a ranker from pairwise preferences, using it to order candidate items, and exporting it as a list of predicate clauses or to a file.

Datasets are represented as objects implementing the pairwise_ranking_dataset_protocol protocol from the ranking_protocols library. See the test_datasets directory for examples. The training dataset must declare each ranked item once, enumerate positive-weight pairwise preferences between distinct declared items, induce a connected undirected comparison graph, and induce a strongly connected directed win graph so that a finite Bradley-Terry maximum-likelihood estimate exists.

API documentation

Open the `../../apis/library_index.html#bradley_terry_ranker <../../apis/library_index.html#bradley_terry_ranker>`__ link in a web browser.

Loading

To load this library, load the loader.lgt file:

| ?- logtalk_load(bradley_terry_ranker(loader)).

Testing

To test this library predicates, load the tester.lgt file:

| ?- logtalk_load(bradley_terry_ranker(tester)).

To run the performance benchmark suite, load the tester_performance.lgt file:

| ?- logtalk_load(bradley_terry_ranker(tester_performance)).

Features

Pairwise Preference Learning: Learns relative item strengths from weighted head-to-head outcomes.
Original MM Fidelity: Rejects datasets without a finite Bradley-Terry maximum-likelihood estimate instead of masking them with implicit strength flooring or other hidden regularization.
Deterministic Ranking: Orders candidate items by learned strength with deterministic tie-breaking.
Strict Dataset Validation: Rejects duplicate items, undeclared items, self-preferences, non-positive weights, disconnected comparison graphs, and pairwise datasets that do not admit a finite Bradley-Terry maximum- likelihood estimate.
Training Diagnostics: Learned rankers include convergence and dataset summary metadata that can be accessed using the diagnostics/2 predicate.
Ranker Export: Learned rankers can be exported as self-contained terms.
Sparse Comparison Processing: Training aggregates weighted comparisons into sparse adjacency lists so iteration cost scales with observed pairwise comparisons instead of the full dense item cross-product.

Dataset requirements

This implementation requires more than undirected connectedness. In order to admit a finite Bradley-Terry maximum-likelihood estimate, the directed win graph induced by the preferences must be strongly connected. Datasets that leave one or more items isolated, split the undirected comparison graph into multiple components, or create dominance partitions with no directed path back to stronger items are rejected instead of producing degenerate strengths.

For a related MAP formulation that keeps the same pairwise-preference setting but uses an explicit Gamma prior to admit connected datasets whose directed win graph is not strongly connected, see the regularized_bradley_terry_ranker library.

Usage

Learning a ranker ~~~~~~~~~~~~~~~~~

% Learn from a pairwise ranking dataset object | ?- bradley_terry_ranker::learn(my_dataset, Ranker). ...

% Learn with custom iteration and convergence options | ?- bradley_terry_ranker::learn(my_dataset, Ranker, [maximum_iterations(500), tolerance(1.0e-7)]). ...

Inspecting diagnostics ~~~~~~~~~~~~~~~~~~~~~~

% Inspect convergence and dataset summary metadata | ?- bradley_terry_ranker::learn(my_dataset, Ranker), bradley_terry_ranker::diagnostics(Ranker, Diagnostics). Diagnostics = [...] ...

Diagnostics syntax

The diagnostics/2 predicate returns a list of metadata terms with the form:

[ model(bradley_terry_ranker), options(Options), convergence(Status), iterations(Iterations), final_delta(FinalDelta), dataset_summary(DatasetSummary) ]

Where:

model(bradley_terry_ranker) identifies the learning algorithm that produced the ranker.
options(Options) stores the effective learning options after merging the user options with the library defaults.
convergence(Status) records the training stop condition. The current values are converged and maximum_iterations_exhausted.
iterations(Iterations) stores the number of update iterations that were executed.
final_delta(FinalDelta) stores the maximum absolute strength update in the last iteration.
dataset_summary(DatasetSummary) stores a summary list describing the validated training dataset.
The current dataset_summary/1 payload has the form:

::

[ items(NumberOfItems), preferences(NumberOfPreferences), connected_components(NumberOfComponents), isolated_items(IsolatedItems) ]

Where IsolatedItems is the list of declared items that have no comparisons. For valid Bradley-Terry training datasets this list is expected to be empty, because disconnected datasets are rejected.

For example, learning from the head_to_head test dataset currently returns diagnostics with the structure:

::

[ model(bradley_terry_ranker), options([maximum_iterations(5000), tolerance(1.0e-6)]), convergence(converged), iterations(...), final_delta(...), dataset_summary([ items(4), preferences(6), connected_components(1), isolated_items([]) ]) ]

Use the ranking_protocols diagnostic/2 and ranker_options/2 helper predicates when you only need a single metadata term or the effective options.

Ranking candidate items ~~~~~~~~~~~~~~~~~~~~~~~

::

% Rank a candidate set from most preferred to least preferred | ?- bradley_terry_ranker::learn(my_dataset, Ranker), bradley_terry_ranker::rank(Ranker, [item_a, item_b, item_c], Ranking). Ranking = [...] ...

Candidate lists must be proper lists of unique, ground items declared by the training dataset. Invalid ranker terms, duplicate candidates, and candidates containing variables are rejected with errors instead of being silently accepted.

Exporting the ranker ~~~~~~~~~~~~~~~~~~~~

Learned rankers can be exported as a list of clauses or to a file for later use.

::

% Export as predicate clauses | ?- bradley_terry_ranker::learn(my_dataset, Ranker), bradley_terry_ranker::export_to_clauses(my_dataset, Ranker, my_ranker, Clauses). Clauses = [my_ranker(bt_ranker(...))] ...

% Export to a file | ?- bradley_terry_ranker::learn(my_dataset, Ranker), bradley_terry_ranker::export_to_file(my_dataset, Ranker, my_ranker, 'ranker.pl'). ...

Options

The following options can be passed to the learn/3 predicate:

maximum_iterations(MaximumIterations): Positive integer iteration bound.
tolerance(Tolerance): Positive convergence tolerance.

Ranker representation

The learned ranker is represented by a compound term of the form:

bt_ranker(Items, Strengths, Diagnostics)

Where:

Items: List of ranked items.
Scores: List of Item-Strength pairs.
Diagnostics: List of metadata terms, including the effective options, convergence status, iteration count, final update delta, and dataset summary.
When exported using export_to_clauses/4 or export_to_file/4, this ranker term is serialized directly as the single argument of the generated predicate clause so that the exported model can be loaded and reused as-is.

References

Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39(3/4), 324-345.
Hunter, D. R. (2004). MM algorithms for generalized Bradley-Terry models. The Annals of Statistics, 32(1), 384-406.