Causal and Bayesian Networks

In his Nobel Acceptance Lecture, Sir C.J.W. Granger mentions that he did not realize people had so many conceptions of causality, nor that his proposed test would be so controversial – resulting in its being confined to a special category “Granger Causality.’

That’s an astute observation – people harbor many conceptions and shades of meaning for the idea of causality. It’s in this regard that renewed efforts recently – motivated by machine learning – to operationalize the idea of causality, linking it with both directed graphs and equation systems, is nothing less than heroic.

However, despite the confusion engendered by quantum theory and perhaps other “new science,” the identification of “cause” can be materially important in the real world. For example, if you are diagnosed with metastatic cancer, it is important for doctors to discover where in the body the cancer originated – in the lungs, in the breast, and so forth. This can be challenging, because cancer mutates, but making this identification can be crucial for selecting chemotherapy agents. In general, medicine is full of problems of identifying causal nexus, cause and effect.

In economics, Herbert Simon, also a Nobel Prize recipient, actively promoted causal analysis and its representation in graphs and equations. In Causal Ordering and Identifiability, Simon writes,


For example, we cannot reverse the causal chain poor growing weather → small wheat crops → increase in price of wheat by an attribution increase in price of wheat → poor growing weather.

Simon then proposes that the weather to price causal system might be represented by a series of linear, simultaneous equations, as follows:


This example can be solved recursively, first by solving for x1, then by using this value of x1 to solve for x2, and then using the so-obtained values of x1 and x2 to solve for x3. So the system is self-contained, and Simon discusses other conditions. Probably the most important is assymmetry and the direct relationship between variables.

Readers interested in the milestones in this discourse, leading to the present, need to be aware of Pearl’s seminal 1998 article, which begins,

It is an embarrassing but inescapable fact that probability theory, the official mathematical language of many empirical sciences, does not permit us to express sentences such as “”Mud does not cause rain”; all we can say is that the two events are mutually correlated, or dependent – meaning that if we find one, we can expect to encounter the other.”

Positive Impacts of Machine Learning

So far as I can tell, the efforts of Simon and even perhaps Pearl would have been lost in endless and confusing controversy, were it not for the emergence of machine learning as a distinct specialization

A nice, more recent discussion of causality, graphs, and equations is Denver Dash’s A Note on the Correctness of the Causal Ordering Algorithm. Dash links equations with directed graphs, as in the following example.

DAGandEQS Dash shows that Simon’s causal ordering algorithm (COA) to match equations to a cluster graph is consistent with more recent methods of constructing directed causal graphs from the same equation set.

My reading suggests a direct line of development, involving attention to the vertices and nodes of directed acyclic graphs (DAG’s) – or graphs without any backward connections or loops – and evolution to Bayesian networks – which are directed graphs with associated probabilities.

Here is are two examples of Bayesian networks.

First, another contribution from Dash and others


So clearly Bayesian networks are closely akin to expert systems, combining elements of causal reasoning, directed graphs, and conditional probabilities.

The scale of Bayesian networks can be much larger, or societal-wide, as this example from Using Influence Nets in Financial Informatics: A Case Study of Pakistan.


The development of machine systems capable of responding to their environment – robots, for example – are a driver of this work currently. This leads to the distinction between identifying causal relations by observation or from existing data, and from intervention, action, or manipulation. Uncovering mechanisms by actively energizing nodes in a directed graph, one-by-one, is, in some sense, an ideal approach. However, there are clearly circumstances – again medical research provides excellent examples – where full-scale experimentation is simply not possible or allowable.

At some point, combinatorial analysis is almost always involved in developing accurate causal networks, and certainly in developing Bayesian networks. But this means that full implementation of these methods must stay confined to smaller systems, cut corners in various ways, or wait for development (one hopes) of quantum computers.

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