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 causal influence on the target node. This method is based on the paper:
Dominik Janzing, Patrick Blöbaum, Lenon Minorics, Philipp Faller, Atalanti Mastakouri. Quantifying intrinsic causal contributions via structure preserving interventions arXiv:2007.00714, 2021
Let’s consider an example from the paper to understand the type of influence being measured here. Imagine a schedule of
three trains, Train A, Train B and Train C, where the departure time of Train C depends on the arrival time of Train B,
and the departure time of Train B depends on the arrival time of Train A. Suppose Train A typically experiences much
longer delays than Train B and Train C. The question we want to answer is: How strong is the influence of each train
on the delay of Train C?
While there are various definitions of influence in the literature, we are interested in the intrinsic causal influence,
which measures the influence of a node that has not been inherited from its parents, that is, the influence of the noise
of a node. The reason for this is that, while Train C has to wait for Train B, Train B mostly inherits the delay from
Train A. Thus, Train A should be identified as the node that contributes the most to the delay of Train C.
See the Understanding the method section for another example and more details.
How to use it
To see how the method works, let us generate some data following the example above:
>>> import numpy as np, pandas as pd, networkx as nx
>>> from dowhy import gcm
>>> X = abs(np.random.normal(loc=0, scale=5, size=1000))
>>> Y = X + abs(np.random.normal(loc=0, scale=1, size=1000))
>>> Z = Y + abs(np.random.normal(loc=0, scale=1, size=1000))
>>> data = pd.DataFrame(data=dict(X=X, Y=Y, Z=Z))
Note the larger standard deviation of the ‘noise’ in \(X\).
Next, we will model cause-effect relationships as a structural causal model and fit it to the data. Here, we are using the auto module to automatically assign causal mechanisms:
>>> causal_model = gcm.StructuralCausalModel(nx.DiGraph([('X', 'Y'), ('Y', 'Z')])) # X -> Y -> Z
>>> gcm.auto.assign_causal_mechanisms(causal_model, data)
>>> 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')
>>> contributions
{'X': 8.736841722582117, 'Y': 0.4491606897202768, 'Z': 0.35377942123477574}
Note that, although we use a linear relationship here, the method can also handle arbitrary non-linear relationships.
Interpreting the results: We estimated the intrinsic causal influence of ancestors of \(Z\), including itself, to its variance (the default measure). These contributions sum up to the variance of \(Z\). As we see here, we observe that ~92% of the variance of \(Z\) comes from \(X\).
Understanding the method
Let’s look at a different example to explain the intuition behind the notion of “intrinsic” causal influence further:
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. For instance, we can 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. variance or entropy) 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,
>>> attribution_func=lambda x, _: estimate_variance(x))
Here, we explicitly defined the variance in the parameter attribution_func as the property we are interested in.
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 attribution_func 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,
>>> attribution_func=mean_absolute_deviation_estimator)
If the choice of a prediction model is unclear, the prediction model parameter can also be set to “auto”.
Remark on using the mean for the attribution: Although the attribution_func can be customized for a given use
case, not all definitions make sense. For instance,
using the mean does not provide any meaningful results. This is because the way influences are estimated is based
on the concept of Shapley values. To understand this better, we can look at a general property of Shapley values, which
states that the sum of Shapley values, in our case the sum of the attributions, adds up to \(\nu(T) - \nu(\{\})\).
Here, \(\nu\) is a set function (in our case, the expectation of the attribution_func), and \(T\) is the full
set of all players (in our case, all noise variables).
Now, if we use the mean, \(\nu(T)\) becomes \(\mathbb{E}_\mathbf{N}[\mathbb{E}[Y | \mathbf{N}]] = \mathbb{E}[Y]\), because the target variable \(Y\) depends deterministically on all noise variables \(\mathbf{N}\) in the graphical causal model. Similarly, \(\nu(\{\})\) becomes \(\mathbb{E}[Y | \{\}] = \mathbb{E}[Y]\). This would result in \(\mathbb{E}_\mathbb{N}[\mathbb{E}[Y | \mathbb{N}]] - \mathbb{E}[Y | \{\}] = 0\), i.e. the resulting attributions are close to 0. For more details, see Section 3.3 of the paper.