10 Apr 2022 (Last Modified 27 Jun 2022)

## Bradford Hill Criteria

Determing causality is difficult, in part because the word *causality* has multiple meanings.

- Prediction (aka Sufficiency). X causes Y to happen in the sense that if I do X Y happens more than I would expect by chance.
- Explanation (aka Necessity). X causes Y to happen in the sense that Y cannot happen without X. This suggests that X is integral to Y’s realization.

In graduate school, I remember being taught that one can say that one thing causes another only if both things are true. Requiring (1) and (2) to both be true is equivalent to enforcing a biconditional.

$$
\begin{aligned}
X &\rightarrow Y\\
\neg X &\rightarrow \neg Y \quad \textrm{by contrapositive } Y \rightarrow X

& \therefore X \leftrightarrow Y
\end{aligned}
$$

The notation brings out a limitation of this reasoning. It is first-order logic and brooks no conditionality.

#### Bradford-Hill Criteria

The Bradford-Hill Criteria are guidelines that approximate a second-order logic generalization of the above formalism.

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