Counterfactuals … a key Digital Transformation tool
Dr. PG Madhavan
Digital Twin maker: Causality & Data Science --> TwinARC - the "INSIGHT Digital Twin"!
In this note, I frame Counterfactuals and Causality in a fresh manner to make these hard concepts as simple as possible (but not any simpler!). Causality as a technical subject of study has seen rapid development in the past few decades. Counterfactual experiment or simulation is less well understood.
But before we start, why is this understanding important? Answer: Digital Transformation!
In essence, Digital Transformation (DX) offers the opportunity to displace “fire-and-forget” practices with continuous assessment and frequent improvements. For improving performance, counterfactual simulation is an essential tool to perform (1) outcome analysis and (2) root-cause analysis. Outcome analysis is done to improve performance by calculating outcomes when changes are made in the system – think of it as “tweaking the knobs” and observing the outcome (underlying causal model is like a “mechanism”). Root-cause analysis allows us to go beyond treating the symptoms and solving the problems once and for all. DX within this framework is fully discussed in “Digital Transformation – Digital Twin’s Role” (Madhavan, Aug 2022).
Let us start from Newton’s 2nd Law and see how we can regard it as a “causal graph”. Then we discuss a “counterfactual experiment” using the 2nd Law and generate some insights.
Causality – start with the basics
When confronting a demanding subject, a Laplacian program to develop a deep understanding is a well-tested approach: idealize the situation significantly, develop scientific understanding and then layer on complexities one step at a time.
Take the case of an object on a lab bench subjected force. At its most idealized form, Newton’s 2nd Law gives you the acceleration from which final position and velocity of the object can be ascertained. But if you wanted to send a craft to the Moon and land it on Shackleton peak, 2nd Law is simply not enough! One needs to add the effects of friction, air currents in the Earth’s atmosphere, relativistic effects due to gravitation of the Earth, Moon, Sun, etc. This is the opposite of peeling an onion layer by layer – start simple and add on complexity!
We will follow such an approach to Causality via the Laplacian program of building complexity step-by-step on idealized model. Let us start at the beginning . . . Newton’s 2nd Law.
“The acceleration of an object depends on the mass of the object and the amount of force applied.” Or -
F = m a . . . (1)
Where ‘F’ is the applied force, ‘a’ is the acceleration of the object and ‘m’ is the mass of the object. This is clearly an IDEALIZED equation. But let us look at equation (1) from a different perspective.
If we consider an object of known mass, ‘m’, and apply a random force to it, its acceleration is given by –
a = (1/m) * F . . . (2)
Another way to state equation (2) is that ‘F’ is the CAUSE and ‘a’ is the EFFECT with 1/m as the Causal Factor! This is a Causality statement.
If we drew a “Causal Graph” corresponding to equation (2), it will look like figure 1.
In Causality lingo, equation (2) is a Structural Causal Model (SCM) or equation; ‘structural’ because there is no time dependency in the equation (relationships are instantaneous – this distinction becomes salient when considering timeseries).
We can also plot equation (2). Fix ‘m’, take a bunch of random ‘F’, calculate ‘a’ per equation (2) and plot ‘a’ versus ‘F’; we get figure 2.
In many other situations, where basic equations are not available, there is an initial step of identifying the causal graph - “Causal Discovery”; which are the variables (nodes) and causal connections (links)? This is typically obtained from subject-matter experts and/or obtained through offline analysis. Causal factors that are the links can then be estimated from the data already collected. This is the subject matter of causal estimation (a few comments in the Appendix).
Counterfactual simulation experiments
Counterfactual simulations are uniquely enabled by Causality; without knowing cause-effect relationships, such simulations cannot be performed. Counterfactual simulations allow you to perform (1) outcome analysis and (2) root-cause analysis, as we noted earlier.
So what exactly is “counterfactual”? Counter to the factual situation – something that did NOT happen but you would like to explore in a “what-if” mindset. Let us stretch Newton’s 2nd Law example . . . let us say, we have already performed the lab bench experiment and plotted the data so obtained as in figure 2.
Counter to all the facts we know from classical mechanics (no “spooky” quantum stuff here!), let us ask, “What if mass of the object increases linearly as the force applied to it increases?” Clearly, a (nonsensical) counterfactual. In other words,
m = constant * F
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When performing this counterfactual simulation, the Causal Factor shown in figure 1 is changed to the one in the dotted box in figure 3.
Simulation result can be seen from equation (2) for acceleration in the counterfactual simulation by substituting the new ‘m’ (=constant * F).
a = (1/m)*F = 1/(constant)
Figure (2) for this “unusual” case will be just a horizontal line! The counterfactual simulation tells us that if the mass of the object is proportional to the force applied on the object, the object will have a steady acceleration – it cannot be accelerated to higher or lower velocities! This result of our what-if analysis makes intuitive sense . . . even if objects whose masses increase with the amount force applied don't exist!
This is indeed the power of Counterfactual Simulation – if we have a causal law (equation (1)) or a Directed Graph (figure (1)), we can perform valuable (or “wild”!) what-if analyses . . .
We painstakingly went through this toy example to make crystal clear what counterfactual is . . . when a directed graph has many more nodes and links, the same fundamentals apply – think of such link or node changes as “fiddling with the knobs in a mechanism”!
Importance of Causality in Digital Transformation:
Work in Causality is accelerating under topics such as Causal AI, Causal ML, etc. The importance of Counterfactual simulations in achieving measurable digital transformation results in industries and businesses leading to significant productivity improvements cannot be overstated.
As mentioned, for improving performance, counterfactual simulation is the necessary tool to perform (1) outcome analysis and (2) root-cause analysis: by calculating outcomes when changes are made in the system by “tweaking the knobs” of the causal “mechanism” and observing the outcome. This outcome is NOT a statistical prediction but a *prescription” that tells you what to do if you want to alter the outcomes!
Digital Transformation of process, business model, domain and cultural/ organizational systems are currently underway. “Low-hanging fruits” have well-developed Deterministic and Statistical ML solutions mostly. To reach up to the high-end fruits, Causal estimation and Counterfactual simulations are going to be essential ML tools!
Appendix - Causal estimation
Estimating the parameters of an SCM (structural causal model) from measured data requires fairly sophisticated but well-understood stochastic process algorithms.
SCM estimation requires the following formal idealizing assumptions.
1. Acyclic: X causing Y and Y causing X in reverse is not allowed.
2. Markov: Assumed that a node has no bearing on nodes which do not descend from it.
3. Faithfulness: Causal influence is not hidden by coincidental cancelations.
These are simplified explanations of the assumptions; the actual definitions are more strict (for example, Markov assumption is – “Every node is conditionally independent of its non-descendants, given its parents”).
Given that these 3 assumptions are satisfied (which they are at least approximately in a wide variety of situations), we can start to develop a Structural Causal Model (SCM) and estimate its model parameters from measured data. The technical details are available but not at the right level for this note. There are many significant publications that explain causal estimation and I have made them accessible to the “digital transformation” technical audience in this article: Structural & Granger CAUSALITY for IoT Digital Twin (Madhavan, March 2022).
Dr. PG Madhavan
IoT Digital Twin, Data Science, Causality