Counterfactuals, thick and thin

Sum­mary: There’s a “thin” con­cept of coun­ter­fac­tual that’s easy to for­mal­ize and a “thick” con­cept that’s harder to for­mal­ize.

Sup­pose you’re try­ing to guess the out­come of a coin­flip. You guess heads, and the coin lands tails. Now you can ask how the coin would have landed if you had guessed tails. The ob­vi­ous an­swer is that it would still have landed tails. One way to think about this is that we have two vari­ables, your guess and the coin , that are in­de­pen­dent in some sense; so we can coun­ter­fac­tu­ally vary while keep­ing con­stant.

But con­sider the vari­able XOR . If we change to tails and keep the same, we con­clude that if we had guessed tails, the coin would have landed heads!

Now this is clearly silly. In real life, we have a causal model of the world that tells us that the first coun­ter­fac­tual is cor­rect. But we don’t have any­thing like that for log­i­cal un­cer­tainty; the best we have is log­i­cal in­duc­tion, which just give us a joint dis­tri­bu­tion. Given a joint dis­tri­bu­tion over , there’s no rea­son to pre­fer hold­ing con­stant rather than hold­ing XOR con­stant. I want a thin con­cept of coun­ter­fac­tu­als that in­cludes both choices. Here are a few defi­ni­tions, in in­creas­ing gen­er­al­ity:

1. Given in­de­pen­dent dis­crete ran­dom vari­ables and , such that is uniform, a thin coun­ter­fac­tual is a choice of per­mu­ta­tion of for ev­ery .

2. Given a joint dis­tri­bu­tion over and , a thin coun­ter­fac­tual is a ran­dom vari­able in­de­pen­dent of and an iso­mor­phism of prob­a­bil­ity spaces that com­mutes with the pro­jec­tion to .

3. Given a prob­a­bil­ity space and a prob­a­bil­ity ker­nel , a thin coun­ter­fac­tual is a prob­a­bil­ity space and a ker­nel such that .

There are of­ten mul­ti­ple choices of thin coun­ter­fac­tual. When we say that one of the thin coun­ter­fac­tu­als is more nat­u­ral or bet­ter than the oth­ers, we are us­ing a thick con­cept of coun­ter­fac­tu­als. Pearl’s con­cept of coun­ter­fac­tu­als is a thick one. No one has yet for­mal­ized a thick con­cept of coun­ter­fac­tu­als in the set­ting of log­i­cal un­cer­tainty.