# Understanding the Evaluation

This part of the tutorial is aimed to help you understand the evaluation of knowledge graph embeddings. In particular it explains rank-based evaluation metrics reported in pykeen.evaluation.RankBasedMetricResults.

Knowledge graph embedding are usually evaluated on the task of link prediction. To this end, an evaluation set of triples $$\mathcal{T}_{eval} \subset \mathcal{E} \times \mathcal{R} \times \mathcal{E}$$ is provided, and for each triple $$(h, r, t) \in \mathcal{T}_{eval}$$ in this set, two tasks are solved:

• Right-Side In the right-side prediction task, a pair of head entity and relation are given and aim to predict the tail, i.e. $$(h, r, ?)$$. To this end, the knowledge graph embedding model is used to score each of the possible choices $$(h, r, e)$$ for $$e \in \mathcal{E}$$. Higher scores indicate higher plausibility.

• Left-Side Analogously, in the left-side prediction task, a pair of relation and tail entity are provided and aim to predict the head, i.e. $$(?, r, t)$$. Again, each possible choice $$(e, r, t)$$ for $$e \in \mathcal{E}$$ is scored according to the knowledge graph embedding model.

Note

Practically, many embedding models allow fast computation of all scores $$(e, r, t)$$ for all $$e \in \mathcal{E}$$, than just passing the triples through the model’s score function. As an example, consider DistMult with the score function $$score(h,r,t)=\sum_{i=1}^d \mathbf{h}_i \cdot \mathbf{r}_i \cdot \mathbf{t}_i$$. Here, all entities can be scored as candidate heads for a given tail and relation by first computing the element-wise product of tail and relation, and then performing a matrix multiplication with the matrix of all entity embeddings. # TODO: Link to section explaining this concept.

In the rank-based evaluation protocol, the scores are used to sort the list of possible choices by decreasing score, and determine the rank of the true choice, i.e. the index in the sorted list. Smaller ranks indicate better performance. Based on these individual ranks, which are obtained for each evaluation triple and each side of the prediction (left/right), there exist several aggregation measures to quantify the performance of a model in a single number.

Note

There are theoretical implications based on whether the indexing is 0-based or 1-based (natural). PyKEEN uses 1-based indexing to conform with related work.

## Rank-Based Metrics

Given the set of individual rank scores $$\mathcal{I}$$, the following scores are commonly used as aggregation.

### Hits @ K

The hits @ k describes the fraction of true entities that appear in the first $$k$$ entities of the sorted rank list. It is given as:

$\text{score}_k = \frac{1}{|\mathcal{I}|} \sum \limits_{r \in \mathcal{I}} \mathbb{I}[r \leq k]$

For example, if Google shows 20 results on the first page, then the percentage of results that are relevant is the hits @ 20. The hits @ k, regardless of $$k$$, lies on the $$[0, 1]$$ where closer to 1 is better.

Warning

This metric does not differentiate between cases when the rank is larger than $$k$$. This means that a miss with rank $$k+1$$ and $$k+d$$ where $$d \gg 1$$ have the same effect on the final score. Therefore, it is less suitable for the comparison of different models.

### Mean Rank

The mean rank (MR) computes the arithmetic mean over all individual ranks. It is given as:

$\text{score} =\frac{1}{|\mathcal{I}|} \sum \limits_{r \in \mathcal{I}} r$

It has the advantage over hits @ k that it is sensitive to any model performance changes, not only what occurs under a certain cutoff and therefore reflects average performance. With PyKEEN’s standard 1-based indexing, the mean rank lies on the interval $$[1, \infty)$$ where lower is better.

Warning

While it remains interpretable, the mean rank is dependent on the number of candidates. A mean rank of 10 might indicate strong performance for a candidate set size of 1,000,000, but incredibly poor performance for a candidate set size of 20.

### Mean Reciprocal Rank

The mean reciprocal rank (MRR) is the arithmetic mean of reciprocal ranks, and thus the inverse of the harmonic mean of the ranks. It is defined as:

$\text{score} =\frac{1}{|\mathcal{I}|} \sum_{r \in \mathcal{I}} r^{-1}$

Warning

It has been argued that the mean reciprocal rank has theoretical flaws by [fuhr2018]. However, this opinion is not undisputed, cf. [sakai2021].

Despite its flaws, MRR is still often used during early stopping due to its behavior related to low rank values. While the hits @ k ignores changes among high rank values completely and the mean rank changes uniformly across the full value range, the mean reciprocal rank is more affected by changes of low rank values than high ones (without disregarding them completely like hits @ k does for low rank values) Therefore, it can be considered as soft a version of hits @ k that is less sensitive to outliers. It is bound on $$(0, 1]$$ where closer to 1 is better.

### Inverse Geometric Mean Rank

The mean rank corresponds to the arithmetic mean, and tends to be more affected by high rank values. The mean reciprocal rank corresponds to the harmonic mean, and tends to be more affected by low rank values. The remaining Pythagorean mean, the geometric mean, lies in the center and therefore could better balance these biases. Therefore, the inverse geometric mean rank (IGMR) is defined as:

$\text{score} = \sqrt[\|\mathcal{I}\|]{\prod \limits_{r \in \mathcal{I}} r}$

Note

This metric is novel as of its implementation in PyKEEN and was proposed by Max Berrendorf

Typical rank-based metrics are affected by the number of entities in knowledge graphs, therefore making results not comparable. The following adjustments were proposed or inspired by [berrendorf2020] in order to make the metrics invariant to number of entities.

The expectation and variance of a discrete uniform variable $$X \sim \mathcal{U}(a, b)$$ are respectively $$\mathbb{E}\left[X\right] = \frac{b+a}{2}$$ and $$\text{Var}\left[X\right] = \frac{\left( b-a+1\right)^2}{12}$$. We assume discrete uniform distribution over the ranks such that $$r_i \sim \mathcal{U}(1, N_i) \in [1,\ldots,N_i]$$. While the upper bound $$N_i$$ may vary by ranking task $$i$$, e.g., due to filtered evaluation, we assume it remains constant throughout the following derivations such that $$\forall i: N_i = n$$. We use $$\doteq$$ to denote equivalences asserted under this assumption.

The expectation of an inverse-uniform distributed variable $$\frac{1}{X} \sim \mathcal{U}(\frac{1}{a},\frac{1}{b})$$ is $$\mathbb{E}\left[\frac{1}{X}\right] = \frac{\ln b - \ln a}{b - a}$$. Given our uniformly distributed variable $$r_i$$ with parameters $$a=1$$ and $$b=N_i$$ and its corresponding inverse-uniform distributed variable $$r_i^{-1}$$, we get:

$\mathbb{E}\left[r_i^{-1}\right] = \frac{\ln 1 - \ln N_i}{N_i - 1} = \frac{\ln N_i}{N_i - 1} \doteq \frac{\ln n}{n - 1}$

The expected value of the mean rank is then derived like:

$\mathbb{E}\left[\text{MRR}\right] = \mathbb{E}\left[\frac{1}{n} \sum \limits_{i=1}^n r_i^{-1}\right] = \frac{1}{n} \sum \limits_{i=1}^n \mathbb{E}\left[r_i^{-1}\right] = \mathbb{E}\left[r_i^{-1}\right] \doteq \frac{\ln n}{n - 1}$

The adjusted mean rank (AMR) was introduced by [berrendorf2020]. It is defined as the ratio of the mean rank to the expected mean rank

$\text{MRR}^{*}(r_1,\ldots,r_n) = \frac{\text{MRR}(r_1,\ldots,r_n)}{\mathbb{E}\left[\text{MRR}\right] }$

It lies on the open interval $$(0, 2)$$ where lower is better.

The adjusted mean rank index (AMRI) was introduced by [berrendorf2020] to make the AMR more intuitive.

$\text{score} = 1 - \frac{MR - 1}{\mathbb{E}\left[MR - 1\right]} = \frac{2 \sum_{i=1}^{n} (r_{i} - 1)}{\sum_{i=1}^{n} (|\mathcal{S}_i|)}$

The AMR has a bounded value range of $$[-1, 1]$$ where closer to 1 is better.

## Ranking Types

While the aforementioned definition of the rank as “the index in the sorted list” is intuitive, it does not specify what happens when there are multiple choices with exactly the same score. Therefore, in previous work, different variants have been implemented, which yield different results in the presence of equal scores.

• The optimistic rank assumes that the true choice is on the first position of all those with equal score.

• The pessimistic rank assumes that the true choice is on the last position of all those with equal score.

• The realistic rank is the mean of the optimistic and the pessimistic rank, and moreover the expected value over all permutations respecting the sort order.

• The non-deterministic rank delegates the decision to the sort algorithm. Thus, the result depends on the internal tie breaking mechanism of the sort algorithm’s implementation.

PyKEEN supports the first three: optimistic, pessimistic and realistic. When only using a single score, the realistic score should be reported. The pessimistic and optimistic rank, or more specific the deviation between both, can be used to detect whether a model predicts exactly equal scores for many choices. There are a few causes such as:

• finite-precision arithmetic in conjunction with explicitly using sigmoid activation

• clamping of scores, e.g. by using a ReLU activation or similar.

## Ranking Sidedness

Besides the different rank definitions, PyKEEN also report scores for the individual side predictions.

Side

Explanation

The rank-based metric evaluated only for the head / left-side prediction.

tail

The rank-based metric evaluated only for the tail / right-side prediction.

both

The rank-based metric evaluated on both predictions.

By default, “both” is often used in publications. The side-specific scores can however often give access to interesting insights, such as the difference in difficulty of predicting a head/tail given the rest, or the model’s incapability to solve of one the tasks.

## Filtering

The rank-based evaluation allows using the “filtered setting”, proposed by [bordes2013], which is enabled by default. When evaluating the tail prediction for a triple $$(h, r, t)$$, i.e. scoring all triples $$(h, r, e)$$, there may be additional known triples $$(h, r, t')$$ for $$t \neq t'$$. If the model predicts a higher score for $$(h, r, t')$$, the rank will increase, and hence the measured model performance will decrease. However, giving $$(h, r, t')$$ a high score (and thus a low rank) is desirable since it is a true triple as well. Thus, the filtered evaluation setting ignores for a given triple $$(h, r, t)$$ the scores of all other known true triples $$(h, r, t')$$.

Below, we present the philosophy from [bordes2013] and how it is implemented in PyKEEN:

### HPO Scenario

During training/optimization with pykeen.hpo.hpo_pipeline(), the set of known positive triples comprises the training and validation sets. After optimization is finished and the final evaluation is done, the set of known positive triples comprises the training, validation, and testing set. PyKEEN explicitly does not use test triples for filtering during HPO to avoid any test leakage.

### Early Stopper Scenario

When early stopping is used during training, it periodically uses the validation set for calculating the loss and evaluation metrics. During this evaluation, the set of known positive triples comprises the training and validation sets. When final evaluation is done with the testing set, the set of known positive triples comprises the training, validation, and testing set. PyKEEN explicitly does not use test triples for filtering when early stopping is being used to avoid any test leakage.

### Pipeline Scenario

During vanilla training with the pykeen.pipeline.pipeline() that has no optimization, no early stopping, nor any post-hoc choices using the validation set, the set of known positive triples comprises the training and testing sets. This scenario is very atypical, and regardless, should be augmented with the validation triples to make more comparable to other published results that do not consider this scenario.

### Custom Training Loops

In case the validation triples should not be filtered when evaluating the test dataset, the argument filter_validation_when_testing=False can be passed to either the pykeen.hpo.hpo_pipeline() or pykeen.pipeline.pipeline().

If you’re rolling your own pipeline, you should keep the following in mind: the pykeen.evaluation.Evaluator when in the filtered setting with filtered=True will always use the evaluation set (regardless of whether it is the testing set or validation set) for filtering. Any other triples that should be filtered should be passed to additional_filter_triples in pykeen.evaluation.Evaluator.evaluate(). Typically, this minimally includes the training triples. With the [bordes2013] technique where the testing set is used for evaluation, the additional_filter_triples should include both the training triples and validation triples as in the following example:

from pykeen.datasets import FB15k237
from pykeen.evaluation import RankBasedEvaluator
from pykeen.models import TransE

# Get FB15k-237 dataset
dataset = FB15k237()

# Define model
model = TransE(
triples_factory=dataset.training,
)

# Train your model (code is omitted for brevity)
...

# Define evaluator
evaluator = RankBasedEvaluator(
filtered=True,  # Note: this is True by default; we're just being explicit
)

# Evaluate your model with not only testing triples,
# but also filter on validation triples
results = evaluator.evaluate(
model=model,
mapped_triples=dataset.testing.mapped_triples,
dataset.training.mapped_triples,
dataset.validation.mapped_triples,
],
)


## Entity and Relation Restriction

Sometimes, we are only interested in a certain set of entities and/or relations, $$\mathcal{E}_I \subset \mathcal{E}$$ and $$\mathcal{R}_I \subset \mathcal{R}$$ respectively, but have additional information available in form of triples with other entities/relations. As example, we would like to predict whether an actor stars in a movie. Thus, we are only interested in the relation stars_in between entities which are actors/movies. However, we may have additional information available, e.g. who directed the movie, or the movie’s language, which may help in the prediction task. Thus, we would like to train the model on the full dataset including all available relations and entities, but restrict the evaluation to the task we are aiming at.

In order to restrict the evaluation, we proceed as follows:

1. We filter the evaluation triples $$\mathcal{T}_{eval}$$ to contain only those triples which are of interest, i.e. $$\mathcal{T}_{eval}' = \{(h, r, t) \in \mathcal{T}_{eval} \mid h, t \in \mathcal{E}_I, r \in \mathcal{R}_I\}$$

2. During tail prediction/evaluation for a triple $$(h, r, t)$$, we restrict the candidate tail entity $$t'$$ to $$t' \in \mathcal{E}_{eval}$$. Similarly for head prediction/evaluation, we restrict the candidate head entity $$h'$$ to $$h' \in \mathcal{E}_{eval}$$

### Example

The pykeen.datasets.Hetionet is a biomedical knowledge graph containing drugs, genes, diseases, other biological entities, and their interrelations. It was described by Himmelstein et al. in Systematic integration of biomedical knowledge prioritizes drugs for repurposing to support drug repositioning, which translates to the link prediction task between drug and disease nodes.

The edges in the graph are listed here, but we will focus on only the compound treat disease (CtD) and compound palliates disease (CpD) relations during evaluation. This can be done with the following:

from pykeen.pipeline import pipeline

evaluation_relation_whitelist = {'CtD', 'CpD'}
pipeline_result = pipeline(
dataset='Hetionet',
model='RotatE',
evaluation_relation_whitelist=evaluation_relation_whitelist,
)


By restricting evaluation to the edges of interest, models more appropriate for drug repositioning can be identified during hyper-parameter optimization instead of models that are good at predicting all types of relations. The HPO pipeline accepts the same arguments:

from pykeen.hpo import hpo_pipeline

evaluation_relation_whitelist = {'CtD', 'CpD'}
hpo_pipeline_result = hpo_pipeline(
n_trials=30,
dataset='Hetionet',
model='RotatE',
evaluation_relation_whitelist=evaluation_relation_whitelist,
)