From 786714e2a9acdbd650092bcaf351128dbebd160d Mon Sep 17 00:00:00 2001 From: Thibaut Horel Date: Sat, 26 Sep 2015 16:14:32 -0400 Subject: Add philosophical comments on causality --- supplements/main.tex | 57 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 57 insertions(+) (limited to 'supplements') diff --git a/supplements/main.tex b/supplements/main.tex index 1f3e69f..17a91bb 100644 --- a/supplements/main.tex +++ b/supplements/main.tex @@ -357,4 +357,61 @@ We also simulated contagions on the co-offending network. Since we are most inte \subsection{Comments on Causality} +The interpretation of the cascades obtained by applying the methodology of +Section~\ref{sec:meth} requires a nuanced understanding of the notion of +causality. In particular, the meaning of assigning a single cause to an +infection needs to be qualified. + +A common definition of causation is the so-called \emph{counter-factual} one +whose first formalization can be traced back to the philosopher David Hume +(CITE): +\begin{quote} + \emph{\ldots we may define a cause to be an object followed by another + [\ldots] where, if the first object had not been, the second never + had existed.} +\end{quote} +The limits of this definition have been the subject of many philosophical +works, starting from David Lewis' paper \emph{Causation} (CITE). In particular, +the problem of \emph{late preemption} is made apparent by the following thought +experiment due to Ned Hall (CITE): +\begin{quote} + \emph{Suzy and Billy, two friends, both throw rocks at a bottle. Suzy is + quicker, and consequently it is her rock, and not Billy’s, that breaks + the bottle. But Billy, though not as fast, is just as accurate: Had + Suzy not thrown, or had her rock somehow been interrupted mid-flight, + Billy’s rock would have broken the bottle moments later.} +\end{quote} +While Suzy's throw is obviously the cause of the bottle shattering, it fails +the counter-factual definition of causation, since had her throw not occurred +the bottle would have been shattered all the same by Billy's throw. + +Without entering the philosophical debate about defining causation, we note +that, at least for the purpose of assigning responsibility, Billy and Suzy +should be considered jointly responsible for the bottle breaking: they were +both throwing rocks at it, and the fact that Suzy's rock reached the bottle +first is coincidental. Of course, if Suzy was throwing rocks twice as +frequently as Billy, without observing whose rock hit the bottle first, it +would be tempting to say that it is more likely that a rock thrown by Suzy hit +the bottle first, or that she is ``more responsible''. But it is still a fact +that at the time the bottle was broken, it was the target of the combined +throws of Billy and Suzy. + +The Hawkes contagion model introduced in \eqref{eq:hawkes} can be +re-interpreted in light of this example: as they become infected, victim nodes +start throwing rocks at their associates with a frequency which decreases over +time and a node becomes infected when it is hit by a rock. Since we do not +observe whose rock hits first, the only thing we can say for certain is that at +the time it was infected, the individual was subject to the combined throws of +its neighbors who were infected in the past. This combined effect is expressed +mathematically by the sum in \eqref{eq:hawkes}. The Hawkes model also allows us +to order the nodes by the frequency at which they are throwing rocks. + +It is now clear which interpretation to give to the cascades extracted in +Section~\ref{sec:meth}: it is a simplification where we assign the +responsibility of an infection to the candidate cause which was throwing rocks +with the highest frequency at the time of the infection. This simplification is +acceptable in the sense that it is more likely that the infection was the +effect of a rock coming from this cause. But even though it makes tracing the +paths of violence easier, a more conservative statement would instead consider +all the throws to be jointly responsible for the infection. \end{document} -- cgit v1.2.3-70-g09d2