An observation asks what tends to accompany X. An intervention asks what happens when the procedure producing X is changed. Even a complete table of observations can leave those questions with different answers.
Three mechanisms, one observed distribution
The demonstration below uses two binary devices, X and Y. A fair source bit U and an independent noise bit N are supplied. N equals one with probability e. XOR means that a noise value of one reverses the input. Three invented mechanisms are available: forward transmission X = U, Y = X XOR N; reverse transmission Y = U, X = Y XOR N; and common preparation X = U, Y = U XOR N.
All three generate the same observational table: P(00) = P(11) = (1 − e)/2 and P(01) = P(10) = e/2. At e = 0.1 the devices agree in 90% of trials. Neither the full table nor unlimited repetitions of that unchanged experiment selects one of the three mechanisms.
Now replace X's usual rule by a clamp setting X = 1. Forward transmission predicts Y = 1 with probability 1 − e. The other two models predict one half because Y's generating rule remains untouched. Clamping Y instead separates reverse transmission from the other models. The intervention protocol supplies information absent from the observational table.
Try observing and setting
EXACT PROBABILITIES · THREE SPECIFIED MODELS
X = U; Y = X XOR N. U is fair; N flips with probability 0.10.
In the forward model, clamping X changes the input to Y. Observation and intervention agree for this comparison.
View the common observational table
| X, Y | Probability |
|---|---|
| 0, 0 | 0.450 |
| 0, 1 | 0.050 |
| 1, 0 | 0.050 |
| 1, 1 | 0.450 |
At e = 0.5, every displayed conditional or interventional probability becomes one half. A structural arrow can then have zero detectable average effect in these tests. A null result is a statement about the specified response and preparation, not an automatic proof that no underlying connection exists.
What is held fixed?
The calculation treats the clamp as replacing one equation while preserving the other mechanisms and the distribution of U and N. That is a modelling assumption about the apparatus. A real clamp might also heat the receiver, alter a shared power supply or change which trials are recorded. Those side effects require extra variables or a revised experimental design.
Random assignment can make the chosen setting independent of prior disturbances in a suitable experiment. It does not guarantee perfect implementation, eliminate measurement error or make results transferable to every population and environment. The question is always which intervention was performed and which outcome was measured.
From a diagram to a testable claim
Pearl's structural framework formalizes interventions by modifying a model's generating rules. The small examples here are original calculations illustrating that distinction. Their curves are exact probabilities of stipulated mechanisms, not data from a laboratory and not estimates fitted to the website's research.
For SMM/DTT, this suggests an explicit task: define how a preparation or coupling can be changed, which inner rules survive that change, and how completed readouts respond. A claim that two phenomena share a source becomes informative when it restricts these responses. The expanded book's tests with explicit alternatives describes the need to compare specified models using outcomes beyond those used for calibration.
After trying both directions of intervention, continue to counterfactuals. Knowing how an ensemble responds does not always identify what would have happened to one particular member under the other setting.
Sources and further reading
- Judea Pearl, Causal Inference, JMLR Workshop and Conference Proceedings 6 (2010), 39–58.
- Correlation and causation: the earlier example of aggregation and competing comparisons.