Judging a classifier honestly

A spam filter is tested on 1000 messages. It is right 960 times, so it scores 96 percent. The team is pleased.

Then somebody finds the filter says "not spam" to every single message. Forty of the 1000 really were spam. One score hid the whole fault.

Accuracy counts every right answer the same. That is fine when the labels are about equally common.

It fails when the table is imbalanced. If 40 messages in 1000 are spam, a model that never says "spam" is right 960 times. It is right 96 percent of the time and finds no spam at all.

docker exec jupyter python3 -c "t=[1]*40+[0]*960; g=[0]*1000; print(sum(a==b for a,b in zip(t,g))/1000)"

Spam is the thing we look for, so call it positive. Each message lands in one of four boxes.

Caught: spam, flagged. False alarm: not spam, flagged. Missed: spam, not flagged. Right rejection: not spam, not flagged.

The four counts together are the confusion matrix.

docker exec jupyter python3 -c "from sklearn.metrics import confusion_matrix, precision_score, recall_score; t=[1]*40+[0]*960; g=[1]*30+[0]*10+[1]*20+[0]*940; print(confusion_matrix(t,g)); print(round(precision_score(t,g),3), round(recall_score(t,g),3), (30+940)/1000)"

Precision asks: when the filter says spam, how often is it right? Here that is 30 of the 50 flagged, which is 0.6.

Recall asks: of the real spam, how much did the filter find? Here that is 30 of 40, which is 0.75. The accuracy was 0.97. The filter is better than the one that never says spam, but accuracy hardly shows how much.

Most models give a score and a threshold turns it into a label. Lower the threshold and more messages are flagged. More spam is caught, but more good mail is flagged too.

So precision and recall pull against each other, and a person has to choose. Run the code and watch the counts move.

# Ten messages: the score a model gave (0 to 1) and whether it really was spam.
messages = [
    (0.95, 1), (0.9, 1), (0.8, 0), (0.7, 1), (0.6, 0),
    (0.5, 1), (0.4, 0), (0.3, 0), (0.2, 0), (0.1, 0),
]
for limit in [0.9, 0.7, 0.5, 0.3]:
    caught = 0
    false_alarms = 0
    missed = 0
    for score, spam in messages:
        flagged = score >= limit
        if flagged and spam == 1:
            caught = caught + 1
        if flagged and spam == 0:
            false_alarms = false_alarms + 1
        if not flagged and spam == 1:
            missed = missed + 1
    print("limit", limit, "caught", caught, "false alarms", false_alarms, "missed", missed)
# Which limit would you choose for a mailbox? Which for a bank?

Say a missed spam costs 1 minute of reading and a false alarm costs 5 minutes looking for lost mail. Use the table of four limits.

Work out the minutes lost at each limit. Which limit costs least? Would it change if a false alarm cost only 1 minute?