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June 10, 2018

# How to evaluate a classifier

Practitioners in quantum machine learning should not only build their skills in quantum algorithms, and having some basic notions of statistics and data science won’t hurt. In the following the see some ways to evaluate a classifier. What does it means in practice? Imagine you have a medical test that is able to tell if a patient is sick or not. You might want to consider the behavior of your classier with respect to the following parameters: the cost of identifying a sick patient as healthy is high, and the cost of identifying a healthy patient as sick. For example, if the patient is a zombie and it contaminates all the rest of the humanity you want to minimize the occurrences of the first case, while if the cure for “zombiness” is lethal for a human patient, you want to minimize the occurrences of the second case. With P and N we count the number of patients tested Positively or Negatively. This is formalized in the following definitions, which consists in statistics to be calculated on the test set of a data analysis.

• TP True positives (statistical power) : are those labeled as sick that are actually sick.

• FP False positives (type I error): are those labeled as sick but that actually are healthy

• FN False negatives (type II error) : are those labeled as healthy but that are actually sick.

• TN True negative: are those labeled as healthy that are healthy.

Given this simple intuition, we can take a binary classifier and imagine to do an experiment over a data set. Then we can measure:

• True Positive Rate (TPR) = Recall = Sensitivity: is the ratio of correctly identified elements among all the elements identified as sick. It answer the question: “how are we good at detecting sick people?”. $\frac{ TP }{ TP + FN} + \frac{TP }{P} \simeq P(test=1|sick=1)$ This is an estimator of the probability of a positive test given a sick individual.

• True Negative Rate (TNR) = Specificity is a measure that tells you how many are labeled as healthy but that are actually sick. $\frac{ TN }{ TN + FP} = p(test = 0 | sick =0)$ How many healthy patients will test negatively to the test? How are we good at avoiding false alarms?

• False Positive Rate = Fallout $FPR = \frac{ FP }{ FP + TN } = 1 - TNR$

• False Negative Rate = Miss Rate $FNR = \frac{ FN }{ FN + TP } = 1 - TPR$

• Precision, Positive Predictive Value (PPV): $\frac{ TP }{ TP + FP} \simeq p(sick=1 | positive=1)$ How many positive to the test are actually sick?

• $F_1$ score is a more compressed index of performance which is a possible measure of performance of a binary classifier. Is simply the harmonic mean of Precision and Sensitivity: $F_1 = 2\frac{Precision \times Sensitivity }{Precision + Sensitivity }$

• Receiver Operating Characteristic (ROC) Evaluate the TRP and FPR at all the scores returned by a classifier by changing a parameter. It is a plot of the true positive rate against the false positive rate for the different possible value (cutpoints) of a test or experiment.

• The confusion matrix generalize these 4 combination of (TP TN FP FN) to multiple classes: is a $l \times l$ where at row $i$ and column $j$ you have the number of elements from the class$i$ that have been classified as elements of class $j$.

Bref. This post because I always forgot about these terms and I wasn’t able to find them described in a concise way with the same formalism without googling more time than that I spent writing this post. Other links: here