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Causality and graphical methods

This is post 1 in a sequence exploring Judea Pearls book Causality entitled “Pearl’s formalisation of causality”

Sometimes the topics that we use intuitively day to day aren’t the easiest to formalise, like causality. There’s a famous saying that, “Correlation does not imply causality” which expresses the basic difficulty of formalising causal processes. To simplify, this saying means that just because two events always come one after the other, it doesn’t mean that the first event caused the second. This book club is going to explore Judea Pearl’s Causality: Models, Reasoning, and Inference, an attempt to formalise causality and will basically take the form of the notes I make while reading the book.

While others learn differently, I find that, when exploring a formal topic, it helps to have an informal understanding first. That informal view may turn out to be simplified or even incorrect but having that overview helps me to have a conceptual view on which to hang new ideas. So I’m going to start my reading of Causality with the epilogue which is the text of a less formal speech summarising Pearl’s views.

The problems of causality

Pearl starts off by exploring the problems of causation. The important point is this: It’s hard to separate out causality from correlation (as Pearl says, the roster crowing does not cause the sun to rise though the two things may be correlated).

In trying to find a formalised, mathematical theory that distinguishes causation from correlation Pearl suggests two principle problems that need solving.

  1. How can we view the environment and legitimately determine cause and effect relationships (and similarly, how can we program an AI to do so)?
  2. How can we process causal information that is given to us (or, similarly, how can an AI process causal information that we give to it)?

As an example of the potential difficulty of solving the second, Pearl points out that, if we tell a robot both:

  1. If the grass is wet, then it rained; and
  2. If we break this bottle, the grass will get wet.

It may conclude that if it breaks the bottle then it rained.

The use of graphical methods

Pearl tries to solve this second problem first. How can we process causal information given to us? Pearl argues that this must be done graphically rather than through algebraic notation by exploring how the two are different with the following example:

A system has two components, a multiplier which multiplies any number given to it and an adder which adds something to any number given to it.

Imagine you start with a number, X, it then goes through an plus-1 adder which outputs Y followed by a times-2 multiplier which outputs Z. In equations, you might represent this as:

Y = X + 1

Z = 2Y

And in graphs as:

Are these two ways of representing the system the same? Pearl says not and demonstrates it as follows. If we rewrite the equations above to change the subjects we get.

X = Y – 1

Y = Z/2

Graphically, this would be represented as:

The two sets of equations are equivalent which means, if the algebra and the graphs are representing the same thing, then the two graphical diagrams should also be equivalent. However, they are not. If we manipulated the value of Y in the first diagram it would affect the value of Z (and leave the value of X unchanged). In the second diagram, on the other hand, it would affect the value of X (and leave the value of Z unchanged). The diagrams are doing something different to the equations.

According to Pearl, the diagrams are doing something more useful. This is because, if we make a change to Y (called surgery, an example would be changing it’s value to 0 regardless of what calculations were made before reaching Y), the diagram lets us see which equations need to be modified. So, if we surgically altered Y to the value of 0, the diagrams make it clear which equations needs to be deleted (and replaced with a new equation Y =  0). In the first case, the equation Y = X + 1 needs to be changed to Y = 0. In the second, the equation Y = Z/2 needs to be changed to Y=0.

The two different graphs are both represented by the same set of equations. Given this, the algebraic notation can’t contain the information about which equation needs to be changed because different equations will need to be changed in different circumstances, even when the algebraic system is the same or equivalent.

So graphs are the route to causality ala Pearl.

The next post will explore how these graphs allow us to answer the second of Pearl’s causal questions: How can we process causal information?

The next post is “A causal calculus: Processing causal information”

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