# Pythagorean Mean Rank Metrics

The mean rank (MR) and mean reciprocal rank (MRR) are among the most popular metrics reported for the evaluation of knowledge graph embedding models in the link prediction task. While they are reported on very different intervals ($\text{MR} \in [1,\infty)$ and $\text{MRR} \in (0,1]$, their deep theoretical connection can be elegantly described through the lens of Pythagorean means. This blog post describes ideas Max Berrendorf shared with me that I recently implemented in PyKEEN and later wrote up as a full manuscript.

If you want to cite some of the ideas in this post, you can use:

```
@article{hoyt2022metrics,
archivePrefix = {arXiv},
arxivId = {2203.07544},
author = {Hoyt, Charles Tapley and Berrendorf, Max and Gaklin, Mikhail and Tresp, Volker and Gyori, Benjamin M.},
eprint = {2203.07544},
month = {mar},
title = {A Unified Framework for Rank-based Evaluation Metrics for Link Prediction in Knowledge Graphs},
url = {https://arxiv.org/abs/2203.07544},
year = {2022}
}
```

## The Formulation of the Link Prediction Task

The link prediction task in knowledge graphs is effectively a binary classification task for each potential triple $(h,
r, t)$ on whether it is true or not. While the accuracy, precision, recall, $F_1$,
Matthews correlation coefficient (MCC), the area under
the precision-recall curve,
and area under the receiver operating characteristic curve
(AUC-ROC or AUROC) are typically used as metrics for binary classifications, the link prediction task has the added
twist that there are only positive examples in a knowledge graph. This means that during training, potential negative
triples need to be sampled from the finite set of all possible triples. This is called negative sampling within the
knowledge graph embedding model literature but more generally called *positive unlabeled learning* in the wider machine
learning community. This works because there’s a much higher likelihood that a triple not already in the knowledge graph
is negative than positive. Even if a triple is sampled that should be positive, the model and training algorithm should
remain robust.

This also means that alternative metrics need to be introduced that don’t rely on the existence or non-existence of negative examples - enter hits@k , mean rank, and mean reciprocal rank. Each of these relies on the concept of a ranking (not to be confused with the rank of a matrix) that is calculated based on combining the true and sampled triples into a list, then sorting based on their scores (calculated by the trained knowledge graph embedding model). The ranks are the positions of the positive triples in this list, where lower ranks correspond to better performance. For the molecular biologists reading, this is very similar to the rank-based visual inspection of GSEA results on differential gene expression experiments. The hits@k, mean rank, and mean reciprocal rank are all summary statistics over the set of ranks (again, the positions in the list of all true triples + sampled triples, sorted by score).

## Definitions of Rank-based Metrics

In the following definitions, I’m going to use $\mathcal{I}$ to denote the set of all ranks for true triples.

### Hits @ K

While this post isn’t about hits@k, it’s worth summarizing the alternative approach that it takes to summarizing the rank list because it’s a much more application-driven metric. Effectively, 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 between $(0, 1]$ where closer to 1 is better.

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:

\[\text{MR} =\frac{1}{|\mathcal{I}|} \sum \limits_{r \in \mathcal{I}} r\]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 $\text{MR} \in [1,\infty)$ where lower is better.

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 large candidate set size of 1,000,000, but incredibly poor performance for a small candidate set size of 20.

### Mean Reciprocal Rank

The mean reciprocal rank (MRR) is the arithmetic mean of reciprocal ranks, which can alternatively be formulated as the inverse of the harmonic mean of ranks. It is defined as:

\[\text{MRR} =\frac{1}{|\mathcal{I}|} \sum_{r \in \mathcal{I}} r^{-1} = \bigg(\frac{|\mathcal{I}|}{ \sum_{r \in \mathcal{I}} r^{-1}}\bigg)^{-1}\]It has been argued that the mean reciprocal rank has theoretical flaws by
Fuhr (2018). However, this opinion is not
undisputed, cf. Sakai (2021).
Despite its *potential* 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
$\text{MRR} \in (0, 1]$ where closer to 1 is better.

### Inverse Arithmetic Mean Rank and Harmonic Mean Rank

It’s odd that the mean rank and mean reciprocal rank are formulated in a way such that $\text{MR} \in [1,\infty)$ and $\text{MRR} \in (0,1]$. Since the mean reciprocal rank is just the inverse of the harmonic mean rank, then it would make sense to report the harmonic mean rank (HMR) as well, defined by:

\[\text{HMR} = \frac{|\mathcal{I}|}{ \sum_{r \in \mathcal{I}} r^{-1}} = \frac{1}{\text{MRR}}\]It has the benefit that it’s more easily comparable to the mean rank because $\text{HMR} \in [1,\infty)$. Alternatively, the inverse arithmetic mean rank (IAMR) could be defined as the inverse of the mean rank (which, remember, is really the arithmetic mean rank) by:

\[\text{IAMR} = \bigg(\frac{1}{|\mathcal{I}|} \sum \limits_{r \in \mathcal{I}} r \bigg)^{-1} = \frac{1}{\text{MR}}\]This has the benefit that it’s more comparable to the mean reciprocal rank because $\text{IAMR} \in (0,1]$.

## Demistying the Metrics

Since we’re thinking about arithmetic means and harmonic means over the ranks, it would make sense to investigate the third Pythagorean mean: the geometric mean.

This figure (from Wikimedia) shows the relationship between the arithmetic mean (A), geometric mean (G), and harmonic mean (H) on a triangle circumscribed by a circle of diameter $a+b$.

### Getting Geometrified

For sets of positive numbers (which ranks always are), there is a nice property that the arithmetic mean is the biggest, geometric mean is next, and the harmonic mean is the smallest. Since the mean rank (based on the arithmetic mean) tends to be biased towards low ranks, and the mean reciprocal rank (based on the harmonic mean) tends to be biased towards high ranks, it could be the case that geometric mean of ranks might balance these two biases. We can define the geometric mean rank (GMR) as:

\[\text{GMR} = \sqrt[\|\mathcal{I}\|]{\prod \limits_{r \in \mathcal{I}} r}\]Like the MR and HMR, $\text{GMR} \in [1,\infty)$. Like with the other metrics, its inverse, the inverse geometric mean rank (IGMR) can be defined as:

\[\text{IGMR} = \frac{1}{\text{GMR}} = \bigg(\sqrt[\|\mathcal{I}\|]{\prod \limits_{r \in \mathcal{I}} r}\bigg)^{-1}\]where $\text{IGMR} \in (0,1]$.

### Additional Statistics

While we proposed the geometric mean rank and its inverse after examining the relationships between the mean rank and mean reciprocal rank, the idea of reporting aggregations over the set of ranks could be extended much further. For example, it makes sense to report other aggregate statistics over the ranks, such as the median rank. This could prove to be more robust to outliers, but it is not clear whether outlier ranks are a major issue in the utility of the existing evaluation metrics or for the newly proposed one. Reporting the standard deviation of ranks, the variance of ranks, and the median absolute deviation of ranks could even enable statistical testing between the results from two different models trained and evaluated on the same dataset, which has been sorely lacking in previous benchmarking studies on using knowledge graph embedding models for link prediction.

## Usage in PyKEEN

PyKEEN has always been focused on easy usage. If you want to get these metrics, do the following:

```
from pykeen.pipeline import pipeline
results = pipeline(model='PairRE', dataset='FB15k-237', ...)
igmr = results.get_metric('inverse_geometric_mean_rank')
```

The rules for `get_metric()`

are outlined here.
Usually, you should write out the metric name in full, with underscores in the middle. The common ones like `mrr`

,
`mr`

, and `igmr`

can be accessed by synonym.

We’ve implemented all of these statistics in PyKEEN pull request #381, reported for the left sided, right sided, and two-sided evaluation as well as for the optimistic, pessimistic, and realistic rankings. This blog post was adapted and extended from the PyKEEN documentation.