## Everything Unlikely => Nothing Surprising?

William M. Briggs  misses the main point when he claims that the recent sequence of high temperature months is no cause for alarm (on the grounds that a sequence of thirteen high months is no less improbable than any other particular sequence).

SteveBrooklineMA’s sarcastic comment (about refusing to adjust one’s aim in the face of consistently hitting too high) is actually right on target.  Given a sequence of consistently high results it would make sense to consider the possibility that there *might* be a systematic effect happening (and especially so when there is a good theoretical reason to expect that there really is such an effect!).

The difference from Briggs’  blade of grass story is in the fact that, although all blades are equally unlikely, if his golf ball hit the one blade that *I* had picked out *in advance* then I would be very surprised and suspect a trick in a way that I would not  if everyone in the world had chosen their own favourite and some other person’s favourite got hit.

Similarly, while I wouldn’t be surprised to learn that someone somewhere in the world threw 13 Heads in a row with a fair coin yesterday, I would have good reason to be very suspicious of a coin which gave *me* that result in the first time I tried it.

In the same way, the particular unlikely sequence of thirteen high temperature months in a row is different from other equally unlikely but less interesting sequences because none of the others correspond to a simple hypothesis that could have been (and was!) made in advance, and it is as foolish to ignore this as to deny the possibility that a sequence of thirteen high shots at a target is consistent with the simple plausible hypothesis that I am pulling the barrel up as I shoot.

What deserves our attention is not just the occurrence of a low probability event but the occurrence of a particular low probability event that we have specified in advance. For example, I would bet \$100 that someone’s selection of numbers will win the lotto jackpot at least once in the coming year but I would also bet \$100 that none of the selections *I* make will win.

Similarly, given that we have a theoretical reason to expect that there might be global warming, events such as having more than even just five *specified* current months (eg the next five starting now if month to month correlation can be ignored) in the “top third” bracket of the long-term historical record would make me suspicious of anyone who insists that the predicted warming is not happening.

Of course if those five aren’t all in the top third we wouldn’t be free to just keep trying until we found five that were, but there are definitely other ways of looking at the data which avoid the pitfalls of “cherry picking”.

The significance of that string of 13 high temperature months depends on whether it was specified in advance, and if it wasn’t then how many other strings of 13 months could have been chosen *without* a high proportion of high temp cases.

Even with no warming at all, if we wait long enough we’ll almost certainly eventually see a string of 13 high-temp months, and if we choose just those then the probability of them all being high is 1.  So to some extent, Briggs does have a valid point about the quoted figure. But we *didn’t* have to wait forever, and so although the 1 in 1.6 million figure is bogus, I think the extent to which it needs to be discounted is not really so great after all.

Assuming independence, the probability of a RANDOMLY CHOSEN sequence of 13 data values all being in the top third IS (1/3)^13 but that is NOT the same as the probability of there being such a sequence somewhere in a larger sample. If we wait long enough we will certainly eventually see such a sequence, and the month after we see 13 high months in a row is NOT a randomly chosen place to end a sample. Over the thirty years or so during which climate scientists have been actively looking for a trend, there have been 3587 opportunities to see the  the last 13months  of a 100years of data  all be in the top third so one might argue very roughly that there should be something like a 3600 in 1.6million chance of seeing the 13 top thirds in a row by now (though I won’t be standing behind anything like that figure without more thought about the matter).

Of course, this is still very low and should be sufficient to cause a reasonable person to be suspicious of a claim that these high values occurred as a result of chance variation rather than an actual increasing trend – which is why I say that Briggs missed the point in his response tot he report. But the chance of making that point (and explaining its implications properly) may well have been blown by now because of the manifest foolish exaggeration of the original 1.6 million claim. The problem is not with those who will buy anything that fits with their political prejudices, but with the kind of people who are persuadable by reason but reluctant to be stampeded and who now have good reason to suspect that climate scientists are cavalier about the use of probability and statistics.

It seems to me a bit ironic that the problem of  “after-the-fact” selection which Briggs correctly noted as tainting the claims of the climate science crew also underlies the error in his dismissal of their actual evidence.