Here is some handy linkage on coincidences. Thanks to a coincidence. I was reading the The #Skeptic’s Daily News and in it, by coincidence, were two separate papers on the subject of coincidence. Though only one was labeled such; the other, just happened by coincidence to be about the same thing.

I have written on coincidences before. How they mess with the heads of some epistemologists when they try to make sense of Gettier Problems (where coincidence can coincidentally cause you to believe a true statement for what is only technically a justifiable reason). And they have an epistemological and methodological role in Bayesian reasoning—for example, because effects “by coincidence” are less probable than “effects that are predictably caused,” and a lot of attempts to deny causation rely on pretending coincidences are more likely. So you have to be able to know when that’s not true.

Although, sometimes, coincidences are just as likely as causation, or near enough as to make no visible difference in our math, or even more likely the case. And thus we can’t rule them out. But sometimes we actually can. So you have to know when is which. Like when we look for evidence of meaningful literary emulation in ancient texts (Proving History, pp. 192-204). Or when some hucksters tried to claim we found the tomb of Jesus. Or when we look for evidence that the Jewish scholar Philo understood a character named Jesus in Zechariah 6 to be the same archangel Paul thinks his Jesus is, by noting that the alternative explanation requires so many coincidences to have occurred as to be extraordinarily improbable (On the Historicity of Jesus, pp. 200-05), including the fact that Paul and Philo assign all the same unusual attributes to the same figure, and the fact that Philo said he made the connection because the archangel in question was already known to him as the Son of God and the High Priest, and the only person in the Zechariah passage he quotes who is identified as the Son of God and the High Priest, is Jesus. Or how coincidence actually better explains the conversion of the Apostle Paul than the Christian thesis that he “really saw Jesus.”

Coincidences are also an important hypothesis to test and understand when criticizing pseudoscience, conspiracy theories, paranormalism, “miracle claims,” and all sorts of things of interest to atheists and skeptics.

So the two papers that have come up lately will interest you, if you are interested in any of those things!

An Actual Study of Coincidences

The first I’ll mention, actually about coincidences, was noted by Dr. Len Fisher, who caught notice of a noting by Steve Strogatz, of a research article by Persi Diaconis and Frederick Mosteller. What a lot of coincidences! Or not. As Fisher says:

[Coincidence is] a very important part of scientific thinking, because distinguishing coincidence from cause and effect is vital. But to distinguish the two is no mean task.

He then illustrates this point with an anecdote atheists will find most amusing:

Pascal and Fermat…laid the foundations for the modern theory of probability that scientists now use to distinguish random chance from real cause and effect. Here was certainty, far removed from statistics, but Pascal converted it into statistics through his famous ‘wager’, which is the argument that ‘it is in one’s own best interest to behave as if God exists, since the possibility of eternal punishment in hell outweighs any advantage in believing otherwise’. The argument sounds convincing, and has attracted much philosophical comment. None has been more trenchant, though, than that of Homer Simpson: ‘Suppose we’ve chosen the wrong god. Every time we go to church we’re just making him madder and madder’. Which just goes to show the dangers of statistics.

The Diaconis & Mosteller paper summarize research on the varieties of documented “coincidence” reporting and break down the results into four categories, or “kinds,” of coincidence, which I shall re-summarize from the original paper:

[1.] Hidden Cause. Much of scientific discovery depends on finding the cause of a perplexing coincidence. Changes in the world can create coincidences; likewise, changes in our own behavior such as a new pattern of reading or eating can create a pattern. Frequency of forecasting the same dire event improves the chances of simultaneity of forecast and outcome. Forgetting many failed predictions makes success seem more surprising. At the same time, vast numbers of coincidences arise from hidden causes that are never discovered. At the moment, we have no measure of the size of this body of occurrences. Similarly, we have no general way to allow for misrepresentation, mistaken or deliberate, that may lead to many reports of coincidences that never occurred.

So, as they remark, one hidden cause of coincidences is simply just fabricating a claim of a coincidence. Obviously. I’ve even mentioned the problem of scientific fraud before, and why “statistical significance” testing does not even account for its frequency. And accounting for the frequency of such things is what Diaconis & Mosteller are concerned about: how often do observed coincidences turn out to have other never-discovered causes instead, like fraud…though that isn’t even the only one to account for. For example, fabricated coincidences even sometimes happen by accident (e.g. the telephone game), like when Christian apologist Francis Beckwith mistakenly claimed there was a “case in which fifteen people all had different and unique reasons for being late to a church choir rehearsal, and because of this none were killed in an explosion that went off shortly after they were supposed to have arrived.” That actually turns out to be largely untrue—and as it happens, other causes were operating as well (that day was unusually cold, a common cause of low turnout at that church). So when you add them all up (all the different kinds of known hidden causes of observed coincidences, besides random chance), what percentage of observed correlations are the products of causes we simply won’t ever discover, or haven’t yet? It can’t be zero.

So it’s significant that there are many other hidden causes these authors are suggesting often turn out, besides fabrication and bad reporting. Constantly predicting an event (and failing to be right) will inevitably result in a correct prediction by chance alone (since, if it’s not too unusual, the predicted event must occur eventually). I’ve noted this problem before with vague Biblical prophecies of lost battles or falling nations, or even a “bird landing on our head.” But most commonly of all, coincidences often do have nonrandom causes: just not the causes we think. If people later behind the scenes arrange for an event to occur as prophesied, it wasn’t God fulfilling the prophecy, nor was the prophet really prescient. Likewise, a coincidence found in a body of data might have some other scientific cause than we first guess it to be (Is a universal human behavior caused by convergent cultural evolution or neurobiological evolution? Is a pattern in the star field caused by an anomaly in the Big Bang event or an instrumentation error?). Thus exploring alternative causes is just as important as ruling out randomness.

[2.] Psychology. What we perceive as coincidences and what we neglect as not notable depends on what we are sensitive to. Some research suggests that previous experience gives us hooks for identifying coincidences. Multiple events emphasize themselves, and without them we have no coincidence to recognize. The classical studies of remembering remind us that frequency, recency, intensity, familiarity, and relevance of experience strengthen recall and recognition. Thus classical psychology has much to teach us about coincidences because they depend so much on recall and recognition.

Indeed, a lot of cognitive biases are manifestations of this, the various ways we err in estimating frequencies by using the wrong psychological cues. “A lot of violent crime in the news” becomes “violent crime is on the rise,” when in fact it’s on an overall decline and has been for decades. Or our attention and notice is triggered by a coincidence (because, perhaps, we evolved to overdetect them, since overdetecting coincidences costs less than underdetecting them, as in the case of the Agency Overdetection I discussed earlier this month), but not triggered by all the times that same coincidence didn’t occur, and in result we over-estimate the frequency of the coincidence because we under-count the times it failed to transpire. For example, all the times a thought about mom coincided with her phoning you—you notice that, but your brain doesn’t record (as being wholly uninteresting) all the times you thought about mom and she didn’t call you, or all the times your mom called you when you weren’t thinking about her.

Interestingly, it has been proposed that we might have some evolved controls for this, such as our occasional preference for non-coincidental arrangements in our visual field, prompting us to move our head when encountering a coincidental alignment, so as to check if it’s real, or an accident of where we are standing (Ramachandran’s “Counter-Symmetry Effect,” Sense and Goodness without God, VI.2.6, p. 358). If only such control behavior had been universally wired into us when encountering every kind of coincidence there is. Then we’d make far fewer errors and more reliably build a better database of correct knowledge. Oh right, our brain wasn’t intelligently designed. Bummer that. Which is why we need to invent and install so many software patches to fix it.

[3.] Multiple Endpoints and the Cost of ‘Close’. In a world where close to identity counts, as it is often allowed to do in anecdotes, and ‘close’ is allowed to get fairly far away, as when Caesar spoke of a military victory as avenging the death in battle, 50 years earlier, of the grandfather of his father-in-law, as if it were a personal revenge…then the frequency of coincidences rises apace. Some formulas presented here emphasize the substantial effect that multiple endpoints can have.

This is what psychics and prophecy peddlers play on when they use tactics like retrofitting to convince people something impressive has happened when in fact it hasn’t. The more you widen the set of “what counts as a coincidence,” the more coincidences you will find—by random chance alone.

[4.] The Law of Truly Large Numbers. Events rare per person occur with high frequency in the presence of large numbers of people; therefore, even larger numbers of interactions occur between groups of people and between people and objects. We believe that this principle has not yet been adequately exploited, so we look forward to its further contribution.

This is the Law of Large Numbers that Christian apologist David Marshall once tried to claim didn’t exist. In order to ignore the fact that: the universe is so big and old, the extreme improbability of random biogenesis on a per-reaction basis actually becomes virtually 100% on cosmic sum. It is more formally referred to as the Infinite Monkey Theorem, as the Law of Large Numbers is also used to refer to what causes the Infinite Monkey Theorem to be true. So these authors coin a new way of referring to it, as The Law of Truly Large Numbers (I used The Law of Big Numbers for the same effect). The point is the same: the more occasions for a coincidence to occur, the more such coincidences will occur. And without a mathematical check, we cannot know from our isolated POV whether we are one of those coincidences or not. As the authors explain:

Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million…are bound to be plentiful in a population of 250 million people. If a coincidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year. Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incredibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.

This is the biggest math error most people make: they think amazing coincidences can’t be accidental. Well, guess what. Tons are. Because the world is so big, and so many things are happening in it. More than we actually have any real capacity to imagine (only to calculate—with one of those “software patches,” which we call mathematics). An even bigger problem this leads us to, is that science has grown in publication rate faster than it has adjusted its standards of evidence, and is now being overwhelmed by the multiple comparisons fallacy, often to the tune of a third or more of all peer reviewed science papers now being false. When you publish a thousand papers a year, you can no longer use a 1 in 20 failure rate as your standard, as that guarantees 50 false results a year, even if all your math is in order, but it’s been shown that in fact, the way the math is being done, the actual rate is so high at that standard that we are getting 300 false results a year. The problem of the Law of Large Numbers is a serious problem indeed. Coincidences are far too common now, for science to continue with such low standards of evidence anymore.

-:-

Their conclusion overall is even more interesting: they note that “coincidence” doesn’t exist outside the human mind. The very idea of a “coincidence” (as a proxy for “randomness”) is a hypothesis our mind builds, and then attempts to accept or reject, sometimes on unsound principles. Outside our mind, things just happen. They always have causes. Everything is caused. They just differ in what those causes are. Winning a lottery “the first and only ever time we played it” is not a “coincidence,” it’s just one more sequence of deterministic causal events. We assign the label of coincidence to it. We often then over-model its significance in our heads, because we attach too much meaning to certain events, and aren’t aware of the actual frequency of their expected conjunction.

The authors advise:

To get a better grip on coincidences that matter to people, it might be useful to employ a critical incidence study. The results might help us distinguish between those coincidences that genuinely move people and those that they regard as good fun though not affecting their lives.

For example, the UN picture at the top of this post: everyone is amused by the coincidence of where the soldier’s helmet just accidentally happened to be when the picture was taken; but no one thinks that’s anything other than a random thing, it isn’t God or the Universe sending us a message. It’s just funny. Because it’s ironic. And yet other coincidences people do regard as God or the Universe sending them a message, or even intervening in the course of events to favor or punish, or as evincing an amazing discovery or a grand conspiracy (the atheist versions of miraculism).

The authors continue (emphasis mine):

Such distinctions, if they are valid, would help focus further coincidence studies on matters people think are important. In a culture like ours based heavily on determinism and causation, we tend to look for causes, and we ask ‘What is the synchronous force creating all of these coincidences?’ We could equally well be looking for the stimuli that are driving so many people to look for the synchronous force. The coincidences are what drive us. And the world’s activity and our labeling of events generates the coincidences.

So learning what actually makes people choose between randomly placed UN helmets and the influence of gods or conspiracies would be of considerable use to us.

Placebos as Coincidences

Which brings us to the second paper (coincidentally paired with the first), which relates to exactly that point about coincidences: it shows that most of what people call “the placebo effect” is actually no such thing, but just an inevitable expectation of coincidence. At DC’s Improbable Science (tagline, “Truth, falsehood and evidence: investigations of dubious and dishonest science”) is a recent entry, “Placebo effects are weak: regression to the mean is the main reason ineffective treatments appear to work.” They begin with a quote from a researcher in 1983:

Statistical regression to the mean predicts that patients selected for abnormalcy will, on the average, tend to improve. We argue that most improvements attributed to the placebo effect are actually instances of statistical regression. Thus, we urge caution in interpreting patient improvements as causal effects of our actions and should avoid the conceit of assuming that our personal presence has strong healing powers.

Amen. The problem is this:

Consider the common experiment in which a new treatment is compared with a placebo, in a double-blind randomised controlled trial (RCT). It’s common to call the responses measured in the placebo group the placebo response. But that is very misleading, and here’s why. The responses seen in the group of patients that are treated with placebo arise from two quite different processes. One is the genuine psychosomatic placebo effect. This effect gives genuine (though small) benefit to the patient. The other contribution comes from the get-better-anyway effect. This is a statistical artefact and it provides no benefit whatsoever to patients. There is now increasing evidence that the latter effect is much bigger than the former.

And yes, studies have been done to tease these apart: “The only way to measure the size of genuine placebo effects is to compare…the effect of a dummy treatment with the effect of no treatment at all. Most trials don’t have a no-treatment arm, but enough do that estimates can be made.” The results of one such survey? “We did not find that placebo interventions have important clinical effects in general.” Only “in certain settings placebo interventions can influence patient-reported outcomes, especially pain and nausea,” though even then biased reporting is hard to rule out. But importantly, “In some cases, the placebo effect is barely there at all.”

DCIS concludes:

So the placebo effect, though a real phenomenon, seems to be quite small. In most cases it is so small that it would be barely perceptible to most patients. Most of the reason why so many people think that medicines work when they don’t isn’t a result of the placebo response, but it’s the result of a statistical artefact. … The get-better-anyway effect has a technical name, regression to the mean. … [And] when you think about it, it’s simply common sense. You tend to go for treatment when your condition is bad, and when you are at your worst, then a bit later you’re likely to be better.

In other words, hidden cause: your own natural immune system is already always on the job. Unlike your desk computer, which when it malfunctions can’t fix itself, your body has a number of systems always busily trying to do just that. You repair damaged tissues. Your body’s native responses (swelling; phlegm overproduction) dissipate inevitably as they are programmed to do. Your immune response overwhelms an invader. The invader just dies of its own accord (because it’s mortal, and can’t always successfully reproduce). There are so many things already happening naturally in your body, that with no treatment at all, you usually get better from things.

And the contrast in your perception is magnified when you “seek external treatment” at the highest point of discomfit (when the pain rises to its worst peak, etc.), precisely when a reversion to lower points is most expected to soon happen (obviously your pain will start to lighten after it reaches its worst point—that is almost a tautology; the only thing preventing it being so is when your natural responses can’t stop the stimulus, e.g. permanent damage, causing that set-point of pain). And yet that is precisely when most people seek external treatment.

Thus, what we have here is a hidden cause of a coincidence (natural biology, and pre-understood patterns of human behavior), magnified by a psychological cause (“noticing” the coincidence of an external treatment and an accidentally correlating improvement, but not remembering all the times such improvement happened without an external treatment), that becomes a multiple endpoints fallacy (any improvement is considered a cure, even though the condition often persists without real external treatment), compounded by the Law of Large Numbers (some people will just randomly get better faster or more often or from more stubborn ailments, simply because there are so many people on earth; yet if each assumes they are special, and ignores all the people like them who didn’t recover, it’s easy to conclude your recovery was caused by the external treatment, when it wasn’t).

Welcome to the power of coincidence.

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