# How is strength of an arrow in graphical causal models related to variance of target variable?

From Causal Inference In Statistics, A Primer by Judea Pearl, I learnt that for linear SCMs (Structural Causal Model), the arrow strengths can be related to the appropriate linear regression coefficients. For example, in SCM -

$$Z:=U_1$$
$$X:=aZ+U_2$$
$$Y:=bX+cZ+U_3$$, where $$U_1,U_2,U_3$$ are assumed to be independent of each other,

with causal graph - the OLS (ordinary least squares) linear regression coefficient of $$X$$ when $$Y$$ is regressed on $$X$$ and $$Z$$, gives the coefficient $$b$$ in the SCM, i.e., the strength of the arrow $$X \rightarrow Y$$. The interpretation of this arrow's strength is that if $$X$$ is increased (via intervention) by 1 unit, $$Y$$ is supposed to increase by $$b$$ units.

In contrast to this interpretation, in the user guide for dowhy library, the interpretation is given in terms of KL-divergence and increment of variance; absence of the arrow $$X \rightarrow Y$$ increases the variance of $$Y$$ by certain units.

What is the significance/intuition behind this second form of interpretation and how does it relate to the first type (which is quite intuitive)?

Note: I am okay with references being provided to standard sources, in case I am lacking some major fundamental background. An explanation here itself, even if filled with mathematical notations, is even more preferable.

• KL-divergence could be interpreted as information gain in Bayesianism, thus we could treat the post-cutting distribution of $Y$ as prior distribution, and after computing the KL-divergence you arrive at its posterior conditional probability measure assuming above causal path diagram holds. Thus the direct (linear) causal strength of $Z$ upon $Y$ is this information gain. Feb 19 at 23:44

## DoWhy library documentation

They seem to be using the methodology proposed in the paper Quantifying Causal Influences according to the source code.

## Regression coefficients can capture causal influence but they are not universal

For regression coefficients to capture causal edge weights, the model needs to be correctly specified using the right variables and functional form (e.g. linear, as in your example). In general, it may be very difficult to come up with a correctly specified model or to even define causal influence. Imagine a non-linear causal model where the influence of an edge would depend on the range of the inputs - there is no clear answer how the causal influence should be expressed by a single number in this case. Additionally one could argue, for example, that the position in the causal order of the nodes connected by an edge should affect its causal influence. So really there is not one way to measure "causal influence". Edge weights are simply a convenient and intuitive thing to look at in linear models.

## Using differences between distributions to estimate causal influence

To estimate causal influence regardless of the functional form and model specification, DoWhy uses the method presented in Quantifying Causal Influences. This approach builds upon comparing the distribution of the edge's target node before and after "cutting" the edge using difference of variance, or KL-divergence.

This is how I understand their variance example:

Let the causal model involving $$X$$, $$Y$$ connected by an edge with weight $$w_{X\to Y}$$, and some other variables in a graph $$G$$. Consider a cut of $$X \to Y$$.

1. Compute pre-cut unexplained variance $$v_{\text{pre_cut}} = Var(Y_{\text{Pa}_G(Y)}\mid \text{Pa}_G(Y))$$
2. Let $$X'$$ be a random shuffle of $$X$$
3. Compute post-cut unexplained variance $$v_{\text{post_cut}} = Var(Y_{\text{Pa}_G(Y)\setminus{X}} + w_{X\to Y} X' \mid \text{Pa}_G(Y) \setminus X)$$
4. Consider as the causal influence of $$A \to B$$: $$\quad v_{\text{post_cut}} - v_{\text{pre_cut}}$$

I'm not sure that's exactly what they implement, but hopefully it captures the idea. The main intuition is that the edge is removed in some way, the distributions are re-computed, and compared to the original (e.g. using difference of variance, KL-divergence, ...).