Does correlation imply causation?
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In episode 55 of The Effort Report podcast, Roger Peng makes an interesting observation: correlation implies causation all the time! That’s how we know most of things in science, including that smoking causes lung cancer.
(around 00:37:30)
Roger is not wrong, but the informal language makes his statement ambiguous.
In formal logic, we almost never say “A implies B” \((A \Rightarrow B)\), except as a shorthand for something else.
Instead, we say A implies B under a set of assumptions Γ, or
\[\Gamma \vdash A \Rightarrow B.\]
Therefore, the statement “A implies B” can be valid under some Γ and invalid under others.
When we say “correlation does not imply causation”, we implicitly mean “without any additional assumptions”, that is,
\[\varnothing \nvdash \text{X correlates with Y}\Rightarrow\text{X causes Y}.\]
But under a suitable set of causal assumptions Γ, correlation may very well imply causation, which is one way to interpret Roger’s claim:
\[\Gamma \vdash \text{X correlates with Y}\Rightarrow\text{X causes Y}.\]
Elizabeth Matsui actually points to this difference when she says “correlation does not equal causation”.
To learn about causal assumptions, read the excellent overview of causal inference by Judea Pearl or one of his many other works.