# Interpretation of 'recall' as a conditional probability P( X=>+ | X=+ )

In the context of performance measures for classification, I have a question about recall and precision.

Looking at the definition of recall:-

$recall = \frac{T_p}{T_p+F_n}$

When I look at this, it sounds like a $conditional$ probability to me -- probability that a test instance will be classified as positive, given that it is indeed positive:-

$recall = Pr(X\hspace{1mm}is\hspace{1mm}predicted\hspace{1mm}as\hspace{1mm}positive | X=positive ) = \frac{T_p}{T_p+F_n}$

Here I am taking liberty to think (as it goes from the definition of $F_n$ ) that $F_n$ in the denominator is actually count of the test instances which are positive but got mis-classified as negative.

In the same light, if I think of precision now, this is how I am thinking of it as a probabilitty that a new test instance actually being positive given that it is predicted as a positive:-

$precision = \frac{T_p}{T_p+F_p} = Pr(X=positive | X\hspace{1mm}is\hspace{1mm}predicted\hspace{1mm}as\hspace{1mm}positive)$

Is this interpretation of precision and recall correct?

• Yes, it is correct. Commented Apr 4, 2017 at 8:15
• Please post it as an answer, so I can accept it as an answer. Just formality. Thx Commented Apr 4, 2017 at 8:17
• Tried posting confirmation myself that yes it is correct interpretation but looks like this site doesn't allow such a short answer. So I guess I will just have to leave this as it is then. Commented Apr 5, 2017 at 23:02

$$F_n$$ represents the number of false negatives whereas $$F_p$$ represents the number of false positives.

# Recall

answers the question: when presented a positive example, how often does the classifier get it right?

aliases: sensitivity, true positive rate (TPR)

# Precision

answers the question: out of all the examples the classifier thought were positive, how often were the examples actually positive?

aliases: positive predictive value (PPV)

Another interesting isight is, that Bayes lemma connects both precision and recall when they are viewed as estimates of contional probabilities: $$P(\hat{C}=P|C=P):=\frac{P(C=P,\hat{C}=P)}{P(C=P)}=\frac{P(C=P|\hat{C}=P)P(\hat{C}=P)}{P(C=P)},$$

with

• $$\hat{C}$$ predicted class
• $$C$$ true class (e.g. $$C=P$$ means that the true class is positive)
• $$P(\hat{C}=P|C=P)$$ Recall
• $$P(C=P|\hat{C}=P$$ Precision
• $$P(C=P)$$ prevalence
• $$P(\hat{C}=P)$$ probability predicted positive (implicitly depending on thresholds), estimated by fraction of predicted positive cases

The implicit dependence of $$P(\hat{C}=P)$$ on the thresholds is the reason why the precision recall curve is not just a straight line.