xAI aims at explaining how predictions are being made. In general, xAI != causality. xAI methods measure which variables are used for predictions by the algorithm, or how far variables improve predictions. The important point to note here: If a variable causes something, we could also expect that it helps predicting the very thing. The opposite, however, is not generally true - very often it is possible that a variable that doesn’t cause anything can predict something.
In statistics courses (in particular our course: Advanced Biostatistics), we discuss the issue of causality at full length. Here, we don’t want to go into the details, but again, you should in general resist to interpret indicators of importance in xAI as causal effects. They tell you something about what’s going on in the algorithm, not about what’s going on in reality.
Methods for causal inference depend on whether we have dynamic or static data. The latter is the more common case. With static data, the problem is confounding. If you have several correlated predictors, you can get spurious correlations between a given predictor and the response, although there is no causal effect in general.
Multiple regression and few other methods are able to correct for other predictors and thus isolate the causal effect. The same is not necessarily true for machine learning algorithms and xAI methods. This is not a bug, but a feature - for making good predictions, it is often no problem, but rather an advantage to also use non-causal predictors.
Here an example for the indicators of variable importance in the random forest algorithm. The purpose of this script is to show that random forest variable importance will split importance values for collinear variables evenly, even if collinearity is low enough so that variables are separable and would be correctly separated by an lm / ANOVA.
We first simulate a data set with 2 predictors that are strongly correlated, but only one of them has an effect on the response.
library(randomForest)randomForest 4.7-1.2Type rfNews() to see new features/changes/bug fixes.set.seed(123)
# Simulation parameters.
n = 1000
col = 0.7
# Create collinear predictors.
x1 = runif(n)
x2 = col * x1 + (1-col) * runif(n)
# Response is only influenced by x1.
y = x1 + rnorm(n)lm / anova correctly identify \(x1\) as causal variable.
summary(lm(y ~ x1 + x2))
Call:
lm(formula = y ~ x1 + x2)
Residuals:
Min 1Q Median 3Q Max
-3.0709 -0.6939 0.0102 0.6976 3.3373
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.02837 0.08705 0.326 0.744536
x1 1.07383 0.27819 3.860 0.000121 ***
x2 -0.04547 0.37370 -0.122 0.903186
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.011 on 997 degrees of freedom
Multiple R-squared: 0.08104, Adjusted R-squared: 0.0792
F-statistic: 43.96 on 2 and 997 DF, p-value: < 2.2e-16Fit random forest and show variable importance:
set.seed(123)
fit = randomForest(y ~ x1 + x2, importance = TRUE)
varImpPlot(fit)
Variable importance is now split nearly evenly.
Task: understand why this is - remember:
We found that (D)NN can better separate collinearity than the other classical ML algorithms. Probably, this is one of the advantages of (D)NN over RF or BRT for tabular data for two reasons:
library(cito)
nn.fit = dnn(y~x1+x2, loss = "mse", data = data.frame(y = y, x1=x1, x2=x2), verbose = F, plot = F)Registered S3 methods overwritten by 'reformulas':
method from
head.call cito
head.formula cito
head.name citosummary(nn.fit)Summary of Deep Neural Network Model
Feature Importance:
variable importance_1
1 x1 0.13864303
2 x2 0.00346932
Average Conditional Effects:
Response_1
x1 0.9445299
x2 0.1204029
Standard Deviation of Conditional Effects:
Response_1
x1 0.04281256
x2 0.06451134But how to get the causal graph? In statistics, it is common to “guess” it and afterwards do residual checks, in the same way as we guess the structure of a regression. For more complicated problems, however, this is unsatisfying. Some groups therefore work on so-called causal discovery algorithms, i.e. algorithms that automatically generate causal graphs from data. One of the most classic algorithms of this sort is the PC algorithm. Here an example using the pcalg package:
library(pcalg)Loading the data:
data("gmG", package = "pcalg") # Loads data sets gmG and gmG8.
suffStat = list(C = cor(gmG8$x), n = nrow(gmG8$x))
varNames = gmG8$g@nodesFirst, the skeleton algorithm creates a basic graph without connections (a skeleton of the graph).
skel.gmG8 = skeleton(suffStat, indepTest = gaussCItest,
labels = varNames, alpha = 0.01)
Rgraphviz::plot(skel.gmG8@graph)What is missing here is the direction of the errors. The PC algorithm now makes tests for conditional independence, which allows fixing a part (but typically not all) of the directions of the causal arrows.
pc.gmG8 = pc(suffStat, indepTest = gaussCItest,
labels = varNames, alpha = 0.01)
Rgraphviz::plot(pc.gmG8@graph )When working with dynamic data, we can use an additional piece of information - the cause usually precedes the effect, which means that we can test for a time-lag between cause and effect to determine the direction of causality. This way of testing for causality is known as Granger causality, or Granger methods. Here an example:
library(lmtest)
## What came first: the chicken or the egg?
data(ChickEgg)
grangertest(egg ~ chicken, order = 3, data = ChickEgg)Granger causality test
Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3)
Model 2: egg ~ Lags(egg, 1:3)
Res.Df Df F Pr(>F)
1 44
2 47 -3 0.5916 0.6238grangertest(chicken ~ egg, order = 3, data = ChickEgg)Granger causality test
Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3)
Model 2: chicken ~ Lags(chicken, 1:3)
Res.Df Df F Pr(>F)
1 44
2 47 -3 5.405 0.002966 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As we have seen, there are already a few methods / algorithms for discovering causality from large data sets, but the systematic transfer of these concepts to machine learning, in particular deep learning, is still at its infancy. At the moment, this field is actively researched and changes extremely fast, so we recommend using Google to see what is currently going on. Particular in business and industry, there is a large interest in learning about causal effect from large data sets. In our opinion, a great topic for young scientists to specialize on.