Is it a creditable approach to use Random Forrest Variable importance for causal inference? I recently ran into a discussion with a college who used Random Forrest Variable importance to discover causal links between some actions of web users and their characteristics. As I come from econometrics, I was wondering whether this is a good approach, since I intuitively would have fitted a Logistic Regression (the action is binary) and analyze its coefficients for this task. To make it more concrete consider the following problem setting:  
Lets say we have a data set $D = (X,y)$ which consists of an output variable $y$ and $d$ explanatory Variables $X = (X_1, \dots, X_d)$. Furthermore, the goal is not to predict new data points $y$, but only to find causal links between $X$ and $y$. 
Would it be good idea to fit a Random Forrest to the data and extract Variable importance in this setting?
If yes, what are pros and cons of this comparing it to Linear/Logistic Regression coefficient analysis?
 A: The analogous concept to parameter estimates in the context of tree based models are not the variable importances, they are the partial dependency plots.
Recall the a parameter estimate in a regression can be interpreted as the effect of on the predictions of varying a feature, when all other features are held constant.  
In a regression (without basis expansions or interactions) the effect of this change is linear with respect to the change in the feature (or linear on the link scale for a glm), and so it can be summarized by a single number.  In a more complex model, this effect is no longer linear, it is instead a curve.  These curves can be plotted, and the result is called a partial dependence plot.
So, for example, if you make partial dependence plots from a fit linear regression model (with no basis expansions or interactions) the resulting curves are lines whose slopes are the parameter estimates.
Here's a quick overview of the idea, with some code to get started.
A: Yes, it is possible to extract causal interpretations from an RF model but certain requirements will need to be met by the model for the extraction to be valid. Similar requirements are also necessary before extracting causal inferences from a simple linear model. The issues around doing so are discussed in Zhao & Hastie (2017), "Causal Interpretations of Black Box Models".
Beginning of paper's conclusion, "In contrast to the conventional view that machine learning algorithms are just blackbox predictive models, we have argued that it is possible to extract causal information from these models using the partial dependence plots (PDP) and the individual conditional expectation (ICE) plots. In summary, a successful attempt of causal interpretation requires: (1) A good predictive model, so the black-box function g is (hopefully) close to the law of nature f. (2) Some domain knowledge about the causal structure to assure the back-door condition is satisfied.(3) Visualization tools such as the PDP and its extension ICE."
