Features with flat partial dependence plots up to a certain level: not important?

This paper focuses on partial dependence plots which are often used when modeling with machine learning techniques in order to better understand the effects of the features on the conditional expectation of the response variable. However, these plots must be interpreted with caution. Indeed, they can easily lead to wrong interpretations in case the analyst is not enough familiar with these plots. As noticed in a previous FAQctuary, a typical situation is the case where a feature is important because of its interactions with others while its partial dependence plot is flat. In this FAQctuary, we go one step further and we consider a very simple example with a three-way interaction effect and we show that only looking at partial dependence plots for each feature and for two features may indeed lead the analyst to wrong conclusions.