A counterfactual is statement saying what would be the case if its conditional were true. Like, “Germany would have won WWII if Hitler did not invade Russia.” Or, “The temperature at our spot would be X if no city existed.” Counterfactuals do not make statements about what really is, but only what might have been given something that wasn’t true was true.
They are sometimes practical. Credit card firms face counterfactuals each time they deny a loan and say, “This person will default if we issue him a card.” Since the decision to issue a card is based on some model or other decision process, the company can never directly verify whether its model is skillful, because they will never issue the card to find out whether or not its holder defaults. In short, counterfactuals can be interesting, but they cannot change what physically happened.
However, probability can handle counterfactuals, so it is not a mistake to seek their quantification. That is, we can assign easily a probability to the Hitler, credit card, or temperature question (given additional information about models, etc.).
Asking what the temperature would be at our spot had there not been a city is certainly a counterfactual. Another is to ask what the temperature of the field would have been given there was a city. This also is a strange question to ask.
Why would we want to know what the temperature of a non-existent city would have been? Usually, to ask how much more humans who don’t live in the city at this moment might have influenced the temperature in the city now. Confusing? The idea is if we had a long series in one spot, surrounded by a city that was constant in size and make up, we could tell if there were a trend in that series, a trend that was caused by factors not directly associated with our city (but was related to, say, the rest of the Earth’s population).
But since the city around our spot has changed, if we want to estimate this external influence, we have to guess what the temperature would have been if either the city was always there or always wasn’t. Either way, we are guessing a counterfactual.
The thing to take away is that the guess is complicated and surrounded by many uncertainties. It is certainly not as clear cut as we normally hear. Importantly, just as with the credit card example, we can never verify whether our temperature guess is accurate or not.
Intermission: uncertainty bounds and global average temperature
This guess would—should!—have a plus and minus attached to it, some guidance of how certain we are of the guess. Technically, we want the predictive uncertainty of the guess, and not the parametric uncertainty. The predictive uncertainty tells us the plus and minus bounds in the units of actual temperature. Parametric uncertainty states those bounds in terms of the parameters of the statistical model. Near as I can tell (which means I might be wrong), GHCN and, inter alia, Mann use parametric uncertainty to state their results: the gist being that they are, in the end, too confident of themselves.
(See this post for a distinction between the two; the predictive uncertainty is always larger than the parametric, usually by two to ten times as much. Also see this marvelous collection of class notes.)
OK. We have our guess of what the temperature might have been had the city not been there (or if the city was always there), and we have said that that guess should come attached with plus/minus bounds of its uncertainty. These bounds should be super-glued to the guess, and coated with kryptonite so that even Superman couldn’t detach them.
Alas, they are usually tied loosely with cheap string from a dollar store. The bounds fall off at the lightest touch. This is bad news.
It is bad because our guess of the temperature is then given to others who use it to compute, among other things, the global average temperature (GAT). The GAT is itself a conglomeration of measurements from sites all over (a very small—and changing—portion) of the globe. Sometimes the GAT is a straight average, sometimes not, but the resulting GAT is itself uncertain.
Even if we ignored the plus/minus bounds from our guessed temperatures, and also ignored it from all the other spots that go into the GAT, the act of calculating the GAT ensures that it must carry its own plus/minus bounds—which should always be stated (and such that they are with respect to the predictive, and not parametric uncertainty).
But if the bounds from our guessed temperature aren’t attached, then the eventual bounds of the GAT will be far, far, too narrow. The gist: we will be way too certain of ourselves.
We haven’t even started on why the GAT is such a poor estimate for the global average temperature. We’ll come to these objections another day, but for now remember two admonitions. No thing experiences GAT, physical objects can only experience the temperature of where they are. Since the GAT contains a large (but not large enough) number of stations, any individual station—as Dick Lindzen is always reminding us—is, at best, only weakly correlated with the GAT.
But enough of this, save we should remember that these admonitions hold whatever homogenization scenario we are in.
My travails of a week ago, battling the air at thirty-eight thousand feet was such a life-affirming experience that I have decided to repeat it. From this afternoon, I will be out of contact for a day or so.