Precision And Recall
Precision and recall are metrics used to describe the performance of binary predictors.
Terminology Note
Precision and Recall are used extensively in Inforation retreival. In this context,
 relevant document = true instance
 retrieved document = true prediction
Precision¶
Given a set of True / False predictions and corresponding True / False instances, precision represents the accuracy rate of True predictions. It is sometimes referred to as positive prediction value.
Formula¶
Example¶
pred truth 
::
True True  < true positive
True True  < true positive
False True 
False True 
False False 
True False  < false positive
False False 
False False 
 \(\text{True Positives} = 2\)
 \(\text{False Positives} = 1\)
Recall¶
Given a set of True/False predictions and corresponding True/False instances, recall represents the accuracy rate of predictions on True instances. It is sometimes referred to as sensitivity.
Formula¶
Example¶
pred truth 
::
True True  < true positive
True True  < true positive
False True  < false negative
False True  < false negative
False False 
True False 
False False 
False False 
 \(\text{True Positives} = 2\)
 \(\text{False Negatives} = 2\)
FScore¶
FScore combines precision and recall into a single score in the range [0, 1], where
 0 indicates that all positive predictions are incorrect
 1 indicates that all true instances are correctly predicted with zero false positives
Formula¶
F_{1} Score (Balanced FScore)
Here, precision and recall are considered equally important.
F_{B}Score (Generalized FScore)
Here, recall is considered \(\beta\) times as important as precision.
Example¶
Calculate F_{0.5}, F_{1}, and F_{2} for the following data.
pred truth 
::
True True 
True True 
False True 
False True 
False False 
True False 
False False 
False False 
 \(\text{precision} = 0.67\)
 \(\text{recall} = 0.50\)
PrecisionRecall Curve¶
The PrecisionRecall Curve plots precision vs recall for a range of classification thresholds. For example, suppose we have the following predicted scores and truths.
 pred_score  truth 
::::
 0.349  True 
 1.084  False 
 0.270  False 
 0.360  True 
 0.898  True 
 1.923  True 
 0.552  True 
 2.273  False 
 1.986  False 
 0.122  True 
 1.738  False 
 3.082  False 
If we assume a classification threshold of 0 (also known as a cutoff value), we can convert each prediction score to a prediction class.
\(\text{pred_class} = \text{pred_score} \ge 0\)
 pred_score  pred_class  truth 
::::::
 0.349  True  True 
 1.084  False  False 
 0.270  False  False 
 0.360  True  True 
 0.898  True  True 
 1.923  False  True 
 0.552  True  True 
 2.273  False  False 
 1.986  False  False 
 0.122  False  True 
 1.738  False  False 
 3.082  False  False 
Here, recall = 0.67 and precision = 1.00. This pair of values, (0.67, 1.00), is one point on the PrecisionRecall curve. Repeating this process for a range of classification thresholds, we can produce a PrecisionRecall curve like the one below.
Plot data
 threshold  precision  recall 
::::::
 0.898  1.00  0.17 
 0.552  1.00  0.33 
 0.360  1.00  0.50 
 0.349  1.00  0.67 
 0.122  1.00  0.83 
 0.270  0.83  0.83 
 1.084  0.71  0.83 
 1.738  0.62  0.83 
 1.923  0.67  1.00 
 1.986  0.60  1.00 
 2.273  0.55  1.00 
 3.082  0.50  1.00 
It doesn't look much like a curve in this tiny example, but PrecisionRecall curves applied to larger data usually look more like this.
R code to reproduce this plot
library(data.table)
library(ggplot2)
trues < rnorm(3000)
falses < rnorm(5000, 2)
dt < data.table(
pred_score = round(c(trues, falses), 3),
truth = c(rep(T, length(trues)), rep(F, length(falses)))
)
# sort the data from highest likelihood of True to lowest likelihood of True
setorderv(dt, "pred_score", order = 1)
# calculate Precision and Recall for each threshold
# (we predict true where score >= threshold)
dt[, `:=`(
TP = cumsum(truth == T),
FP = cumsum(truth == F),
FN = c(tail(rev(cumsum(rev(truth))), 1), 0)
)]
dt[, `:=`(
precision = TP/(TP + FP),
recall = TP/(TP + FN)
)]
ggplot(dt, aes(x = recall, y = precision))+
geom_line(size = 1.5, color = "light blue")+
labs(x = "Recall", y = "Precision", title = "PrecisionRecall Curve")+
ylim(0,1)+
xlim(0,1)+
theme_bw()+
theme(text = element_text(size=16))
Note
Precision generally decreases as recall increases, but this relationship is not guaranteed.
Perfectly ranked prediction scores are reflected by a PrecisionRecall curve like this.
Average Precision¶
Consider predictions from two models A & B compared to known thruths.
 preds_A  preds_B  truths 
::::::
 True  True  True 
 True  True  True 
 True  True  False 
 True  True  False 
 False  False  True 
 False  False  True 
 False  False  False 
 False  False  False 
A and B have identical prediction classes, and therefore have identical precision, recall, and Fscores. However, suppose we reveal A's and B's prediction scores (in this case, probabilities).
 scores_A  scores_B  truths 
::::::
 0.95  0.55  True  < A is confident and correct
 0.85  0.59  True  < A is confident and correct
 0.73  0.88  False  < B is confident and incorrect
 0.62  0.97  False  < B is confident and incorrect
 0.48  0.20  True  < A is confident and correct
 0.39  0.09  True  < A is confident and correct
 0.12  0.43  False  < B is confident and incorrect
 0.04  0.32  False  < B is confident and incorrect
The classification threshold in this example is 0.5. That is, preds_A = scores_A >= 0.5
and preds_B = scores_B >= 0.5
.
Observe
 When model A was confident in its predictions, it was correct.
 When model B was confident in its predictions, it was incorrect.
Intuitively, this suggests model A better than model B. This is depicted visually observing that A's PrecisionRecall curve is consistently higher than B's PrecisionRecall curve.
This desire to score ranked predictions leads to Average Precision which measures the average precision value on the PrecisionRecall curve over the range [0, 1].
Formula¶
Average Precision equals area under the PrecisionRecall curve.
where \(p(r)\) is the precision at recall \(r\).
This integral can be discretized as
where \(k\) is the rank in the sequence of prediction scores (high to low), \(n\) is the total number of samples, \(P(k)\) is the precision at cutoff \(k\) in the list, and \(\Delta r(k)\) is the change in recall from items \(k1\) to \(k\).
Geometrically, this formula represents a Riemann Sum approximation to the PrecisionRecall curve using rectangles with height \(P(k)\) and width \(\Delta r(k)\).
R code to reproduce this plot
library(data.table)
library(ggplot2)
set.seed(4)
trues < rnorm(6)
falses < rnorm(10, 1)
dt < data.table(
pred_score = round(c(trues, falses), 3),
truth = c(rep(T, length(trues)), rep(F, length(falses)))
)
setorderv(dt, "pred_score", order = 1)
dt[, `:=`(
TP = cumsum(truth == T),
FP = cumsum(truth == F),
FN = c(tail(rev(cumsum(rev(truth))), 1), 0)
)]
dt[, `:=`(
precision = TP/(TP + FP),
recall = TP/(TP + FN)
)]
dt[, recall_prev := shift(recall, type = "lag", fill = 0)]
ggplot(dt, aes(x = recall, y = precision))+
geom_rect(aes(xmin = recall_prev, xmax = recall, ymin = 0, ymax = precision), alpha = 0.1, color = "black")+
geom_line(size = 1.5, color = "light blue")+
geom_point(size = 2, color = "red")+
labs(x = "Recall", y = "Precision", title = "PrecisionRecall Curve", subtitle = "Riemman sum approximation to area under the PR curve")+
ylim(0,1)+
xlim(0,1)+
theme_bw()+
theme(text = element_text(size=16))
Since recall ranges from 0 to 1, this can be interpreted as a weighted sum of Precisions whose weights are the widths of the rectangles (i.e. the changes in recall from threshold to threshold), hence the name Average Precision.
Furthermore, the width of each nonzerowidth rectangle is the same. Alternatively stated, each positive change in recall is equivalent. Thus, Average Precision can be described as an arithmetic mean of Precision values restricted to the set of True instances (relevant documents).
where \(\operatorname {rel} (k)\) is an indicator function equaling 1 if the item at rank \(k\) is a true instance, 0 otherwise.
Example¶
Using the toy data above,
 scores_A  truths  precision_k  delta_recall_k  rel_k 
::::::::::
 0.95  TRUE  1.00  0.25  1  < include
 0.85  TRUE  1.00  0.25  1  < include
 0.73  FALSE  0.67  0.00  0 
 0.62  FALSE  0.50  0.00  0 
 0.48  TRUE  0.60  0.25  1  < include
 0.39  TRUE  0.67  0.25  1  < include
 0.12  FALSE  0.57  0.00  0 
 0.04  FALSE  0.50  0.00  0 
Formula 1
Formula 2
 scores_A  truths  precision_k  delta_recall_k  rel_k 
::::::::::
 0.62  FALSE  0.00  0.00  0 
 0.73  FALSE  0.00  0.00  0 
 0.85  TRUE  0.33  0.25  1  < include
 0.95  TRUE  0.50  0.25  1  < include
 0.12  FALSE  0.40  0.00  0 
 0.04  FALSE  0.33  0.00  0 
 0.48  TRUE  0.43  0.25  1  < include
 0.39  TRUE  0.50  0.25  1  < include
Formula 1
Formula 2
Variations¶
 Some authors prefer to estimate Area Under the Curve using trapezoidal approximation.

Some authors use interpolated precision whereby the PrecisionRecall curve is transformed such that the precision at recall \(r\) is taken to be the \(max(precision)\) at all \(recall \ge r\).
Precision at k (P@k)¶
Precision at k measures what percent of the top k ranked predictions which are true instances?. This is useful in settings like information retrieval where one only cares about the top k ranked documents returned by a search query.
Example¶
Suppose we build a system that predicts which movies are relevant to a particular search query (e.g. "dog movies"). Our database contains 12 movies which receive the following prediction scores and prediction classes.
# pred_class = True implies we *predict* the movie is relevant.
# truth = True implies the movie *is* relevant.
 k  pred_score  pred_class  truth 
::::::::
 1  0.94  True  False 
 2  0.91  True  False 
 3  0.90  True  True 
 4  0.66  True  True 
 5  0.63  True  True 
 6  0.57  True  True 
 7  0.37  False  False 
 8  0.27  False  True 
 9  0.21  False  True 
 10  0.20  False  True 
 11  0.18  False  False 
 12  0.06  False  False 

Filter the the top 5 instances by
pred_score
. k  pred_score  pred_class  truth  ::::::::  1  0.94  True  False   2  0.91  True  False   3  0.90  True  True   4  0.66  True  True   5  0.63  True  True 

Count the true instances (i.e. relevant movies).
 k  pred_score  pred_class  truth  ::::::::  1  0.94  True  False   2  0.91  True  False   3  0.90  True  True  < True instance  4  0.66  True  True  < True instance  5  0.63  True  True  < True instance

Divide by k
\[ \text{Precision @ 5} =\frac{3}{5} = 0.6 \]
Edge Cases¶

What if there are fewer than \(k\) True instances (relevant documents)?
This scenario presents a drawback to \(\text{Precision @ k}\). If there are a total of \(n\) True instances (relevant documents) where \(n < k\), the model will have at most \(\text{Precision @ k} = \frac{n}{k} < 1\).

What if there are fewer than \(k\) True predictions (retrieved documents)?
If a model returns \(m < k\) True predictions (retrieved documents), one possible implementation is to treat the next \(k  m\) predictions as False Positive predictions (irrelevant documents).
Average Precision at k (AP@k)¶
Average Precision at k represents Average Precision amongst the top k classification thresholds (cutoffs). It can also be described as area below the Precision Recall curve, restricted to the top k thresholds, and normalized by the lesser of k and the total number of true instances (relevant documents).
Ambiguity
The term Average Precision at k usually implies (Average Precision) @ k, not Average (Precision @ K). These are distinctly different!
Formula¶
where \(\operatorname {rel} (i)\) is an indicator function equaling 1 if the item at rank \(i\) is a true instance, 0 otherwise.
Example¶
Given the following predictions and truths, calculate \(\operatorname {AP@3}\).
 k  pred_score  truth 
::::::
 1  0.840  TRUE 
 2  0.085  TRUE 
 3  0.264  FALSE 
 4  0.463  TRUE 
 5  0.551  TRUE 
 6  0.579  FALSE 
 7  0.708  FALSE 
 8  1.108  FALSE 
 9  1.170  FALSE 
 10  1.593  FALSE 

Filter the top k=3 entries by highest prediction score.
 k  pred_score  truth  ::::::  1  0.840  TRUE   2  0.085  TRUE   3  0.264  FALSE 

Calculate precision at each of these cutoffs.
 k  pred_score  truth  precision  ::::::::  1  0.840  TRUE  1.00   2  0.085  TRUE  1.00   3  0.264  FALSE  0.67 

Sum the precisions of relevant documents (true instances)
 k  pred_score  truth  precision  ::::::::  1  0.840  TRUE  1.00  < include  2  0.085  TRUE  1.00  < include  3  0.264  FALSE  0.67 
\[ \sum _{i=1}^{3}P(i)\times \operatorname {rel} (i) = 1.00 + 1.00 = 2.00 \] 
Normalize by the lesser of \(k\) and the total number of relevant documents (true instances)
In this example, the total number of true instances is 4.
\[ \operatorname {AP@k} ={\frac {\sum _{i=1}^{3}P(i)\times \operatorname {rel} (i)}{\min(k, \text{total true instances})}} = \frac{2.00}{\min(3, 4)} = \frac{2}{3} = 0.67 \]
Intuition
Observe the PrecisionRecall curve for this example.
Average precision measures the area under the curve, which we can approximate with a Reimann sum.
Average Precision at 3 corresponds to area beneath the precision recall curve restricted to the first three cutoffs in the ordered predictions. In other words, the area beneath the curve to the left of the third point.
However the rectangle widths are rescaled to have width \(\frac{1}{3}\) so that the maximum achievable area beneath the curve in this range is 1. This is the purpose of the denominator, \(\min(k, \text{total true instances})\) in the formula for \(\operatorname {AP@k}\).
Mean Average Precision (MAP)¶
Mean Average Precision calculates Average Precision for Q queries and averages their results.
Formula¶
Mean Average Precision at k (MAP@k)¶
Mean Average Precision at k calculates Average Precision at k for Q queries and averages their results.