All metrics are wrong, but some are useful
On the insufficient and approximate nature of metrics.
George Box stated that:
"All models are wrong, but some are useful".
It primarily applies to statistical models, but it also applies to all scientific models.
The model, in the statistical sense and the general sense
A model, in the general sense, is created to approximately describe a system or a natural process. The approximate nature of the model must always be borne in mind, Box said.
A statistical model is a set of equations that describe a process generating the observed data. One statistical model can represent the data more accurately than others.
In the same sense, a map is wrong in terms of describing the world (lack of height and depth), but it’s useful for you to navigate around.
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How about metrics?
A metric is a numerical measurement that is computed from the properties of the system. A metric is used to rate how well a process is working, and to compare systems to one another. Metrics are usually designed to be monotonic to be useful and easy to understand.
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A metric is inherently insufficient
A metric can be thought of as an attempt to distill the knowledge of how well the system works down to one single number. It’s a lossy process. It gives a biased view of how the system works.
A machine learning model that is optimized for accuracy will neglect the recall rate.
For example, we have a disease that occurs in 1% of cases. A machine learning model can achieve 99% accuracy just by predicting NO all the time. As you can see, it’s useless (and reckless).
The F1 score can be used to judge the system if the cost of false-negative and false-positive is roughly the same. F1 is blind to the asymmetry of the cost of being wrong. One should build the cost of misclassifying into the metric.
Another example, using a single statistic like mean, median or mode to represent a collective neglects its variance.
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But it’s useful
A metric is not perfect in describing the quality of the process or system, but in many cases, it’s all that we have, due to the limitation of our analysis, our visualization, or even our mental capacity. One can gauge the distribution plots, but distilling the whole thing remains hard.
A metric designed for machines should aim for precision and good numerical properties. A metric designed for humans should be clear, intuitive and easy to understand. The one who uses the metric should know its strengths and limitations, in order not to be fooled.