# Accuracy in multi-label classification [duplicate]

In a multi-label classification, the accuracy is commonly defined as  $$\text{Accuracy}(\boldsymbol{Y},\, \boldsymbol{Z}) := \frac{1}{n} \sum_{i=1}^n \frac{\lvert Y_i \cap Z_i \rvert}{\lvert Y_i \cup Z_i \rvert},$$

where $Y_i$ is the set of predicted labels for sample $n$ and $Z_i$ is the corresponding set of ground-truth labels. If I get things right, this means that we divide the number of correctly classified labels per sample by the number of labels that are either predicted or given as ground-truth per sample and average over. For example, if $\boldsymbol{Y} := ([0,0,1],\,[1,0,1])$ and $\boldsymbol{Z} := ([0,0,0],\, [1,0,1])$, this would give $$\text{Accuracy}(\boldsymbol{Y},\, \boldsymbol{Z}) = \frac{1}{2}\left( \frac{0}{1} + \frac{2}{2} \right) = \frac{1}{2}.$$

To me, this is unintuitive, since the classwise accuracies are $1$, $1$ and $\frac{1}{2}$ (in that order). The problem is that the definition of multi-label accuracy does not reflect correctly classified absence of labels, whereas the classwise accuracy does. The same is true for multi-label recall and precision as defined in .

Is it really like that or do I get the definitions wrong? Can you recommend any multi-label measures that suit better for problems, where correctly classified absence of labels is equally important as presence (apart from the Hamming Loss)?

 Sorower, Mohammad S. "A literature survey on algorithms for multi-label learning." Oregon State University, Corvallis 18 (2010).

• Although this is an interesting thread, it does not really help in my situation, because that thread is about single-label classification. Like they state: "we assign equal cost to false positives and false negatives". This is exactly not, what we have here, because the definition only assigns cost to false negatives, not to false positives.. Jul 18 '18 at 14:22
• My answer at the duplicate thread applies equally well to multi-label classification. Jul 18 '18 at 15:23