The Cause of Time

Actually, there is a theorem which states that two variables X and Y are independent if and only if f(X) and g(Y) are uncorrelated for any two functions f,g. That means that you cannot bring things into correlation by any independent transformations whatsoever exactly when the variables are independent. Specifically regarding affine transformations, you would have to add some multiple of X to Y in order to get a correlation, which is obviously silly.

Thanks for the link to RCCP, I hadn't seen the history of d-sep before. You should definitely check out the causality section of Highly Advanced Epistemology 101 for Beginners and possibly Judea Pearl's book Causality, which contain a more up-to-date discussion and address many of your concerns much better than I could.

I like your outside view of insight. That was part of the reason I pointed out that this is not a new insight, and has been found many times by other people before.

Edit after reading pragmatist's comment: Knowing your background, I'll add some technical meat.

First, regarding correlation versus dependence, consider any functions f(T) and g(e). The exact same argument made in the post still applies: without time, there is no ordering of the points, so we cannot establish any correlation. Since there is no correlation for any functions f,g the variables are independent. The argument could be made that the correlation is undefined rather than zero, but if we take a Bayesian approach then we should probably be summing over all permutations (since there is no reason to prefer any particular permutation). Intuitively, that seems like it ought to go to zero given enough data, but I'm not sure if it's identically zero for smaller data sets.

Regarding quantum, RCCP works under the MWI which everyone seems to love around here (the world-branch becomes a hidden common cause). But setting that argument aside, we can happily restrict ourselves to non-chaotic macroscopic situations.

What the thermometer and tuning fork really have in common in this example is a person (or something) looking at a clock and then recording T_n and e_n. So the example already is a slightly more complex thermostrainometer. It's interesting how much we take accurate measurements of time for granted; in the good old days astronomers had to invent precise definitions and mechanisms for measuring time in order to correlate the motions of the heavenly bodies with pseudo-periodic observations.

We don't actually write down (t_0, e_0) and (t_2,T_2), we write down (clock-step_x, e_x), etc. Even if we're using an atomic clock we're really just counting the number of times a sine wave generator has cycled since we started it and not some nebulous substance called "time".

I was hoping someone would bring this up! This is why I was careful to specify that the temperature was taken outside my window, and the strain was measured in a tuning fork in some unspecified location. In that situation, time really is the only correspondence between the points.

But your example brings up a much more general (and much more interesting) problem of identifying points. I'll illustrate with another example. Suppose we measure a bunch of physiological variables in mice. We get a bunch of tuples mapping mice to the relevant variables, and we find lots of correlations. But then we lose our mouse id's! Suddenly we have no idea which mouse each measurement came from. As before, everything gets scrambled and correlations disappear. We conclude that the measurements cause the mouse, or more accurately, the measurements cause the id of the mouse.

In the mouse example, notice that giving the mice actual names or id numbers wasn't really necessary. We could just identify each mouse by its tuple of measurements. The identity of the mouse is mathematically just a mapping to match up the data points from different sensors.

Going back to your aptly-named thermostrainometer, we see a similar situation. Time is no longer the variable used to identify data points with each other. Instead, T and e points are associated through both space and time, and the whole mapping is conveniently handled inside the sensor itself and given to us in a convenient tuple structure. But the sensor itself still needs to associate the T and e values somehow, which is where space and time come in.

Yup, you're right. That's the right way to handle it, and it yields time as the common cause of temperature and strain, as we'd expect.

Now that I'm knee-deep in it, I do think this crazy concept of separating sets of values of variables from the mappings between the sets has something to it. It isn't necessary for the example in the main post, but I think the example I gave with mice in one of the other comments still applies. The mapping between points is legitimately a variable unto itself, so it seems like it should be possible to handle it like other variables. It might even be useful to do so, since the mapping is nonparametric.

Anyway, thanks for giving a proper analysis of the problem.

Name one.

Assuming you're comfortable with d-separation and causality, I suspect the confusion is mostly my poor writing, since I did not spend any time editing this or adding diagrams (which it desperately needs). Anyway, if you draw out the causal diagram, we have the three named variables (t, T, and e). t is the only source of ordering and correspondence between T and e, from which everything else follows. It may take some thinking to believe that the information conveyed by t is exactly the ordering of the data. I'm still trying to come up with a less hand-wavy way to explain that part.

If you want to brush up on d-separation and causality, check out the causality section of Highly Advanced Epistemology 101 for Beginners .

Causal networks do not make an iid assumption. Consider one of the simplest examples, in which we examine experimental data. Some of the variables are chosen by the experimenter. They can be chosen any way the experimenter pleases, so long as they vary. The process is the same, but that does not imply iid observations. It just means that time dependence must enter through the variables. As you say, it is not built in to the framework.

The problem is to reduce the phrase "the different measurements of each variable are associated because they come from the same sample of the causal process." What is a sample? How do we know two numbers (or other strings) came from the same sample? Since the association contains information separate from the values themselves, how can we incorporate that information into the framework explicitly? How can we handle uncertainty in the association apart from uncertainty in the values of the variables?

Yeah, when I wrote the post I hadn't thought through it enough to put it in clear terms. I hadn't even realized that there was an open question in here. Bouncing back and forth with commenters helped a lot. (Thankyou!)

At this point it's pretty clear that we can't treat the associations as variables in the same way that we usually treat variables in a causal net (as I did in the original post). As you say, we can't treat it as d-separation in the usual way. But we still need some way to integrate this information when trying to learn causal nets. So I guess the interesting question is how. I may write another discussion post with a better example and a clean formulation of the question.