Precision and recall

In

.

Precision (also called

positive predictive value

) is the fraction of relevant instances among the retrieved instances. Written as a formula:

$${\displaystyle {\text{Precision}}={\frac {\text{Relevant retrieved instances}}{\text{All retrieved instances}}}}$$

Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved. Written as a formula:

$${\displaystyle {\text{Recall}}={\frac {\text{Relevant retrieved instances}}{\text{All relevant instances}}}}$$

Both precision and recall are therefore based on relevance.

Consider a computer program for recognizing dogs (the relevant element) in a digital photograph. Upon processing a picture which contains ten cats and twelve dogs, the program identifies eight dogs. Of the eight elements identified as dogs, only five actually are dogs (true positives), while the other three are cats (false positives). Seven dogs were missed (false negatives), and seven cats were correctly excluded (true negatives). The program's precision is then 5/8 (true positives / selected elements) while its recall is 5/12 (true positives / relevant elements).

Adopting a

More generally, recall is simply the complement of the type II error rate (i.e., one minus the type II error rate). Precision is related to the type I error rate, but in a slightly more complicated way, as it also depends upon the

The above cat and dog example contained 8 − 5 = 3 type I errors (false positives) out of 10 total cats (true negatives), for a type I error rate of 3/10, and 12 − 5 = 7 type II errors (false negatives), for a type II error rate of 7/12. Precision can be seen as a measure of quality, and recall as a measure of quantity. Higher precision means that an algorithm returns more relevant results than irrelevant ones, and high recall means that an algorithm returns most of the relevant results (whether or not irrelevant ones are also returned).

Introduction

In a

For classification tasks, the terms true positives, true negatives, false positives, and false negatives (see Type I and type II errors for definitions) compare the results of the classifier under test with trusted external judgments. The terms positive and negative refer to the classifier's prediction (sometimes known as the expectation), and the terms true and false refer to whether that prediction corresponds to the external judgment (sometimes known as the observation).

Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

Precision and recall are then defined as:^[13]

$${\displaystyle {\begin{aligned}{\text{Precision}}&={\frac {tp}{tp+fp}}\\{\text{Recall}}&={\frac {tp}{tp+fn}}\,\end{aligned}}}$$

Recall in this context is also referred to as the true positive rate or

$${\displaystyle {\text{True negative rate}}={\frac {tn}{tn+fp}}\,}$$

Precision vs. Recall

Both precision and recall may be useful in cases where there is imbalanced data. However, it may be valuable to prioritize one over the other in cases where the outcome of a false positive or false negative is costly. For example, in medical diagnosis, a false positive test can lead to unnecessary treatment and expenses. In this situation, it is useful to value precision over recall. In other cases, the cost of a false negative is high. For instance, the cost of a false negative in fraud detection is high, as failing to detect a fraudulent transaction can result in significant financial loss. ^[14]

Probabilistic Definition

Precision and recall can be interpreted as (estimated) conditional probabilities:^[15] Precision is given by ${\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)}$ while recall is given by ${\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)}$,^[16] where ${\displaystyle {\hat {C}}}$ is the predicted class and ${\displaystyle C}$ is the actual class (i.e. ${\displaystyle C=P}$ means the actual class is positive). Both quantities are, therefore, connected by Bayes' theorem.

No-Skill Classifiers

The probabilistic interpretation allows to easily derive how a no-skill classifier would perform. A no-skill classifiers is defined by the property that the joint probability ${\displaystyle \mathbb {P} (C=P,{\hat {C}}=P)=\mathbb {P} (C=P)\mathbb {P} ({\hat {C}}=P)}$ is just the product of the unconditional probabilites since the classification and the presence of the class are

For example the precision of a no-skill classifier is simply a constant ${\displaystyle \mathbb {P} (C=P|{\hat {C}}=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} ({\hat {C}}=P)}}=\mathbb {P} (C=P),}$ i.e. determined by the probability/frequency with which the class P occurs.

A similar argument can be made for the recall: ${\displaystyle \mathbb {P} ({\hat {C}}=P|C=P)={\frac {\mathbb {P} (C=P,{\hat {C}}=P)}{\mathbb {P} (C=P)}}=\mathbb {P} ({\hat {C}}=P)}$ which is just (the typically threshold dependent) probability for a positive classification.

Some very specific no-skill classifiers are implemented in sklearn and are named dummy classifiers there.^[17]

Imbalanced data

$${\displaystyle {\text{Accuracy}}={\frac {TP+TN}{TP+TN+FP+FN}}\,}$$

Accuracy can be a misleading metric for imbalanced data sets. Consider a sample with 95 negative and 5 positive values. Classifying all values as negative in this case gives 0.95 accuracy score. There are many metrics that don't suffer from this problem. For example, balanced accuracy^[18] (bACC) normalizes true positive and true negative predictions by the number of positive and negative samples, respectively, and divides their sum by two:

$${\displaystyle {\text{Balanced accuracy}}={\frac {TPR+TNR}{2}}\,}$$

For the previous example (95 negative and 5 positive samples), classifying all as negative gives 0.5 balanced accuracy score (the maximum bACC score is one), which is equivalent to the expected value of a random guess in a balanced data set. Balanced accuracy can serve as an overall performance metric for a model, whether or not the true labels are imbalanced in the data, assuming the cost of FN is the same as FP.

The TPR and FPR are a property of a given classifier operating at a specific threshold. However, the overall number of TPs, FPs etc depend on the class imbalance in the data via the class ratio ${\textstyle r=P/N}$. As the recall (or TPR) depends only on positive cases, it is not affected by ${\textstyle r}$, but the precision is. We have that

${\displaystyle {\text{Precision}}={\frac {TP}{TP+FP}}={\frac {P\cdot TPR}{P\cdot TPR+N\cdot FPR}}={\frac {TPR}{TPR+{\frac {1}{r}}FPR}}.}$

Thus the precision has an explicit dependence on ${\textstyle r}$.^[19] Starting with balanced classes at ${\textstyle r=1}$ and gradually decreasing ${\textstyle r}$, the corresponding precision will decrease, because the denominator increases.

Another metric is the predicted positive condition rate (PPCR), which identifies the percentage of the total population that is flagged. For example, for a search engine that returns 30 results (retrieved documents) out of 1,000,000 documents, the PPCR is 0.003%.

$${\displaystyle {\text{Predicted positive condition rate}}={\frac {TP+FP}{TP+FP+TN+FN}}\,}$$

According to Saito and Rehmsmeier, precision-recall plots are more informative than ROC plots when evaluating binary classifiers on imbalanced data. In such scenarios, ROC plots may be visually deceptive with respect to conclusions about the reliability of classification performance.^[20]

Different from the above approaches, if an imbalance scaling is applied directly by weighting the confusion matrix elements, the standard metrics definitions still apply even in the case of imbalanced datasets.^[21] The weighting procedure relates the confusion matrix elements to the support set of each considered class.

F-measure

A measure that combines precision and recall is the harmonic mean of precision and recall, the traditional F-measure or balanced F-score:

$${\displaystyle F=2\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}}$$

This measure is approximately the average of the two when they are close, and is more generally the harmonic mean, which, for the case of two numbers, coincides with the square of the geometric mean divided by the arithmetic mean. There are several reasons that the F-score can be criticized, in particular circumstances, due to its bias as an evaluation metric.^[1] This is also known as the ${\displaystyle F_{1}}$ measure, because recall and precision are evenly weighted.

It is a special case of the general ${\displaystyle F_{\beta }}$ measure (for non-negative real values of ${\displaystyle \beta }$):

$${\displaystyle F_{\beta }=(1+\beta ^{2})\cdot {\frac {\mathrm {precision} \cdot \mathrm {recall} }{\beta ^{2}\cdot \mathrm {precision} +\mathrm {recall} }}}$$

Two other commonly used ${\displaystyle F}$ measures are the ${\displaystyle F_{2}}$ measure, which weights recall higher than precision, and the ${\displaystyle F_{0.5}}$ measure, which puts more emphasis on precision than recall.

The F-measure was derived by van Rijsbergen (1979) so that ${\displaystyle F_{\beta }}$ "measures the effectiveness of retrieval with respect to a user who attaches ${\displaystyle \beta }$ times as much importance to recall as precision". It is based on van Rijsbergen's effectiveness measure ${\displaystyle E_{\alpha }=1-{\frac {1}{{\frac {\alpha }{P}}+{\frac {1-\alpha }{R}}}}}$, the second term being the weighted harmonic mean of precision and recall with weights ${\displaystyle (\alpha ,1-\alpha )}$. Their relationship is ${\displaystyle F_{\beta }=1-E_{\alpha }}$ where ${\displaystyle \alpha ={\frac {1}{1+\beta ^{2}}}}$.

Limitations as goals

There are other parameters and strategies for performance metric of information retrieval system, such as the area under the

• Uncertainty coefficient, also called proficiency
• Sensitivity and specificity
• Confusion matrix
• Scoring rule
• Base rate fallacy

References