# `plackett_luce_last_ranker`

Tie-aware Plackett-Luce-last grouped-ranking ranker. It processes each group as a sequence of last-choice eliminations from lowest relevance to highest relevance, using grouped tie blocks and a deterministic fixed-point update on positive item strengths.

The library implements the `ranker_protocol` defined in the `ranking_protocols` library. It provides predicates for learning a ranker from grouped rankings, 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 `ranking_dataset_protocol` protocol from the `ranking_protocols` library. See the `test_datasets` directory for examples. The training dataset must declare each group once, use only declared groups and items in relevance judgments, assign non-negative integer relevance values, and induce a strongly connected directed strict-order graph across groups so that a finite Plackett-Luce-last maximum-likelihood estimate exists.

## API documentation

Open the ../../apis/library_index.html#plackett_luce_last_ranker link in a web browser.

## Loading

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

    | ?- logtalk_load(plackett_luce_last_ranker(loader)).

## Testing

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

    | ?- logtalk_load(plackett_luce_last_ranker(tester)).

## Features

- Grouped Last-Choice Learning: Learns positive item strengths from grouped rankings by processing each group from lowest relevance to highest relevance.

- Tie-Aware Likelihood: Uses grouped tie blocks so equal relevance judgments are handled as unordered last-choice blocks instead of being broken arbitrarily. Each tie block contributes a size-constrained last-choice likelihood term against the remaining higher-relevance items.

- Deterministic Ranking: Orders candidate items by learned strength with deterministic tie-breaking.

- Strict Dataset Validation: Rejects malformed grouped datasets, unsupported options, duplicate candidates, and invalid ranker terms.

- Regular MLE Fidelity: Rejects grouped datasets whose strict-order graph does not admit a finite Plackett-Luce-last maximum-likelihood estimate instead of masking non-identifiability with implicit regularization.

- Missing relevance semantics: Missing relevance facts are treated as zero by default using the `missing_relevance(zero)` option and can be rejected using `missing_relevance(error)`.

- Training Diagnostics: Learned rankers include convergence, iteration, final update delta, and dataset summary metadata accessible using the `diagnostics/2` predicate.

- Ranker Export: Learned rankers can be exported as self-contained terms.

- Shared Grouped Infrastructure: Reuses the shared grouped tie-block representation and iterative positive-strength scaffolding from the `ranking_protocols` library.

## Dataset requirements

This implementation requires more than grouped-dataset well-formedness. In order to admit a finite Plackett-Luce-last maximum-likelihood estimate, the directed strict-order graph induced by the grouped rankings must be strongly connected. Intuitively, no partition of the items may dominate all others in only one direction across the observed groups.

Unlike `borda_ranker`, this model therefore rejects grouped datasets that consist of disconnected query universes or one-way dominance chains, because those data do not identify a finite global strength scale.

## Usage

### Learning a ranker

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

    % Learn with custom iteration and missing-relevance options
    | ?- plackett_luce_last_ranker::learn(my_dataset, Ranker, [maximum_iterations(500), tolerance(1.0e-7), missing_relevance(error)]).
    ...

### Inspecting diagnostics

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

## Diagnostics syntax

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

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

Where:

- `model(plackett_luce_last_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.

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
    | ?- plackett_luce_last_ranker::learn(my_dataset, Ranker),
         plackett_luce_last_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
    | ?- plackett_luce_last_ranker::learn(my_dataset, Ranker),
         plackett_luce_last_ranker::export_to_clauses(my_dataset, Ranker, my_ranker, Clauses).
    Clauses = [my_ranker(plackett_luce_last_ranker(...))]
    ...

    % Export to a file
    | ?- plackett_luce_last_ranker::learn(my_dataset, Ranker),
         plackett_luce_last_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.

- `missing_relevance(zero|error)`: Policy used when a declared item in a group has no explicit relevance judgment.

## Ranker representation

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

    plackett_luce_last_ranker(Items, Strengths, Diagnostics)

Where:

- `Items`: List of ranked items.

- `Scores`: List of normalized `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.

## See also

For the complementary grouped top-choice variant over the same dataset protocol, see the `plackett_luce_ranker` library. For a deterministic non-probabilistic grouped-ranking baseline over the same dataset protocol, see the `borda_ranker` library.

## References

1. Plackett, R. L. (1975). The analysis of permutations. Applied Statistics, 24(2), 193-202.

2. Luce, R. D. (1959). Individual Choice Behavior: A Theoretical Analysis. Wiley.

3. Hunter, D. R. (2004). MM algorithms for generalized Bradley-Terry models. The Annals of Statistics, 32(1), 384-406.