I agree with pragmatist’s explanation. But let me add a bit more detail to illustrate that a temporal ordering will not save you here. Imagine instead of two variables we have three variables : rain (R), my grass being wet (G1), and my neighbor’s grass being wet (G2). Clearly R preceeds both G1, and G2, and G1 and G2 are contemporaneous. In fact, we can even consider G2 to be my neighbor’s grass 1 hour in the future (so clearly G1 preceeds G2!).
Also clearly, p(R = yes | G1 = wet) is high, and p(R = yes | G2 = wet) is high, also p(G1 = wet | R = yes) is high, and p(G2 = wet | R = yes) is high.
So by hosing my grass I am making it more likely than my neighbor’s grass one hour from now will be wet?
Yeah, well, I’ve heard somewhere that correlation does not equal causation :-)
I agree that causal models are useful—if only because they make explicit certain relationships which are implicit in plain-vanilla regular models and so trip up people on a regular basis.What I’m not convinced of is that you can’t re-express that joint density on the outcomes in a conventional way even if it turns out to look a bit awkward.
Lumifer : “can we not express cause effect relationships via conditioning probabilities?”
me : “No: [example].”
Lumifer : “Ah, but this is silly because of time ordering information.”
me : “Time ordering doesn’t matter: [slight modification of example].”
Lumifer : “Yeah… causal models are useful, but it’s not clear they cannot be expressed via conditioning probabilities.”
I guess you can lead a horse to water, but you can’t make him drink. I have given you everything, all you have to do is update and move on. Or not, it’s up to you.
I agree with pragmatist’s explanation. But let me add a bit more detail to illustrate that a temporal ordering will not save you here. Imagine instead of two variables we have three variables : rain (R), my grass being wet (G1), and my neighbor’s grass being wet (G2). Clearly R preceeds both G1, and G2, and G1 and G2 are contemporaneous. In fact, we can even consider G2 to be my neighbor’s grass 1 hour in the future (so clearly G1 preceeds G2!).
Also clearly, p(R = yes | G1 = wet) is high, and p(R = yes | G2 = wet) is high, also p(G1 = wet | R = yes) is high, and p(G2 = wet | R = yes) is high.
So by hosing my grass I am making it more likely than my neighbor’s grass one hour from now will be wet?
Or, to be more succinct : http://www.smbc-comics.com/index.php?db=comics&id=1994#comic
Yeah, well, I’ve heard somewhere that correlation does not equal causation :-)
I agree that causal models are useful—if only because they make explicit certain relationships which are implicit in plain-vanilla regular models and so trip up people on a regular basis.What I’m not convinced of is that you can’t re-express that joint density on the outcomes in a conventional way even if it turns out to look a bit awkward.
Here’s how this conversation played out.
Lumifer : “can we not express cause effect relationships via conditioning probabilities?”
me : “No: [example].”
Lumifer : “Ah, but this is silly because of time ordering information.”
me : “Time ordering doesn’t matter: [slight modification of example].”
Lumifer : “Yeah… causal models are useful, but it’s not clear they cannot be expressed via conditioning probabilities.”
I guess you can lead a horse to water, but you can’t make him drink. I have given you everything, all you have to do is update and move on. Or not, it’s up to you.
Yes, I’m a picky sort of a horse :-) Thanks for the effort, though.