Quantifying Intrinsic Causal Influence
By quantifying intrinsic causal influence, we answer the question:
How strong is the causal influence of a source node to a target node that is not inherited from the parents of the source node?
Naturally, descendants will have a zero intrinsic influence on the target node.
How to use it
To see how the method works, let us generate some data.
>>> import numpy as np, pandas as pd, networkx as nx
>>> from dowhy import gcm
>>> from dowhy.gcm.uncertainty import estimate_variance
>>> np.random.seed(10) # to reproduce these results
>>> X = np.random.normal(loc=0, scale=1, size=1000)
>>> Y = 2*X + np.random.normal(loc=0, scale=1, size=1000)
>>> Z = 3*Y + np.random.normal(loc=0, scale=1, size=1000)
>>> data = pd.DataFrame(data=dict(X=X, Y=Y, Z=Z))
Next, we will model cause-effect relationships as a structural causal model and fit it to the data.
>>> causal_model = gcm.StructuralCausalModel(nx.DiGraph([('X', 'Y'), ('Y', 'Z')])) # X -> Y -> Z
>>> causal_model.set_causal_mechanism('X', gcm.EmpiricalDistribution())
>>> causal_model.set_causal_mechanism('Y', gcm.AdditiveNoiseModel(gcm.ml.create_linear_regressor()))
>>> causal_model.set_causal_mechanism('Z', gcm.AdditiveNoiseModel(gcm.ml.create_linear_regressor()))
>>> gcm.fit(causal_model, data)
Finally, we can ask for the intrinsic causal influences of ancestors to a node of interest (e.g., \(Z\)).
>>> contributions = gcm.intrinsic_causal_influence(causal_model, 'Z',
>>> gcm.ml.create_linear_regressor(),
>>> lambda x, _: estimate_variance(x))
>>> contributions
{'X': 33.34300732332951, 'Y': 9.599478688607254, 'Z': 0.9750701113403872}
Interpreting the results: We estimated the intrinsic influence of ancestors of \(Z\), including itself, to its variance. These contributions sum up to the variance of \(Z\). We observe that ~76% of the variance of \(Z\) comes from \(X\).
Understanding the method
Consider the following example to get the intuition behind the notion of “intrinsic” causal influence we seek to measure here.
A charity event is organised to collect funds to help an orphanage. At the end of the event, a donation box is passed around to each participant. Since the donation is voluntary, some may not donate for various reasons. For instance, they may not have the cash. In this scenario, a participant that simply passes the donation box to the other participant does not contribute anything to the collective donation after all. Each person’s contribution then is simply the amount they donated.
To measure the intrinsic causal influence of a source node to a target node, we need a functional causal model. In particular, we assume that the causal model of each node follows an additive noise model (ANM), i.e. \(X_j := f_j (\textrm{PA}_j) + N_j\), where \(\textrm{PA}_j\) are the parents of node \(X_j\) in the causal graph, and \(N_j\) is the independent unobserved noise term. To compute the “intrinsic” contribution of ancestors of \(X_n\) to some property (e.g. entropy, variance) of the marginal distribution of \(X_n\), we first have to set up our causal graph, and learn the causal model of each node from the dataset.
Consider a causal graph \(X \rightarrow Y \rightarrow Z\) as in the code example above, induced by the following ANMs.
where \(N_w \sim \mathcal{N}(0, 1)\), for all \(w \in \{X, Y, Z\}\), are standard Normal noise variables.
Suppose that we are interested in the contribution of each variable to the variance of the target \(Z\), i.e. \(\mathrm{Var}[Z]\). If there were no noise variables, everything can be contributed to the root node \(X\) as all other variables would then be its deterministic function. The intrinsic contribution of each variable to the target quantity \(\mathrm{Var}[Z]\) is then really the contribution of corresponding noise term.
To compute “intrinsic” contribution, we also require conditional distributions of \(Z\) given subsets of noise variables \(N_T\), i.e., \(P_{Z \mid N_T}\), where \(T \subseteq \{X, Y, Z\}\). We estimate them using an ANM. To this end, we have to specify the prediction model from a subset of noise variables to the target. Below, we quantify the intrinsic causal influence of \(X, Y\) and \(Z\) to \(\mathrm{Var}[Z]\) using a linear prediction model from noise variables to \(Z\).
>>> from dowhy.gcm.uncertainty import estimate_variance
>>> prediction_model_from_noises_to_target = gcm.ml.create_linear_regressor()
>>> node_to_contribution = gcm.intrinsic_causal_influence(causal_model, 'Z',
>>> prediction_model_from_noises_to_target,
>>> lambda x, _: estimate_variance(x))
Note
While using variance as uncertainty estimator gives valuable information about the contribution of nodes to the squared deviations in the target, one might be rather interested in other quantities, such as absolute deviations. This can also be simply computed by replacing the uncertainty estimator with a custom function:
>>> mean_absolute_deviation_estimator = lambda x: np.mean(abs(x))
>>> node_to_contribution = gcm.intrinsic_causal_influence(causal_model, 'Z',
>>> prediction_model_from_noises_to_target,
>>> mean_absolute_deviation_estimator)
If the choice of a prediction model is unclear, the prediction model parameter can also be set to “auto”.