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ICYMI: Indeterminacy

The ABA Blog has been in existence for almost 6 years, and there’s a lot of good content back in the archives that deserves an audience now that it might not have received way back when. So, semi-regularly we will bring some of that stuff back. Here’s one by Ted Floyd that has the honor of hosting one of the very first barn-burners in the comments section. 


Practically all birders come to realize, sooner or later, that bird identification is an uncertain affair. And most birders, I think it’s fair to say, would tell you that there are two major causes of this uncertainty. Here they are:

1. Any bird, if seen poorly or briefly, may be difficult or impossible to identify. You’re at the local landfill, and it’s getting on toward sundown. Ring-billed Gulls are swirling all over the place. One of the gulls catches your eye for being a bit darker and daintier than the others. You see the bird put down, and it looks good for Mew Gull—a rarity in your area. You train your scope on the bird and…it’s gone. Other gulls have settled down in front of it, and the bird is invisible. Honestly, you tell yourself, you don’t know what the bird was. The lighting was poor, after all. And you can’t have observed the bird for more than three or four seconds. You have to let it go. 

2. In other instances, you get a great view of the bird—but you still can’t affix a name to it. In this case, you’ve had ample opportunity to study the bird. Your photos are tack-sharp. You’ve discussed the bird with other folks, both in the field and via the internet. But nobody’s willing to pull the trigger and attach a label to the bird. Nobody can say for certain that it was a pure Glaucous-winged Gull—very rare in your state. Maybe it was a hybrid with a Herring Gull. And if so, was it a straightforward “F1” hybrid (one Herring Gull parent, one Glaucous-winged Gull parent), or was it some weird backcross (now we’re talkin’ grandma and grandpa, and then some)? Again, you let it go.

To sum up, we would seem to have two sources of uncertainty: (1) the observer; and (2) the bird itself. And that’s that. We’ve covered all our bases. End of story.

Not quite.

In both of the preceding scenarios, we’re making a particular assumption. At first blush, it’s an eminently reasonable assumption. Our assumption is that, regardless of what the bird really is, it most assuredly is something. In the first scenario, that bird seen briefly in the setting sun either was a rare Mew Gull or it was something else—most likely a very common Ring-billed Gull. In the second scenario, that tricky large gull either was a pure Glaucous-winged Gull or it wasn’t. If it wasn’t, then it was 50% Glaucous-winged (an “F1” hybrid) or 75% Glaucous-winged (an “F2” backcross) or conceivably 72.3187% Glaucous-winged (perfect knowledge of the bird’s entire genome courtesy of futuristic technology for DNA analysis). Regardless, it was one thing or another. It was either a pure (albeit aberrant) Glaucous-winged Gull, say, or it was an “F2” backcross. But it certainly couldn’t have been both of those things.

GWGu x HerG January 2004
This interesting gull visited Boulder County, Colorado, during January of 2004. Was it a pure but aberrant Glaucous-winged Gull? Or was it a hybrid? Photo by
© Bill Schmoker.

But what if our assumption has been wrong? Let go of your cherished notions of reality, for just a moment now, and consider the possibility that bird identification is better conceived in probabilistic terms than in familiar deterministic terms. Consider, then, a third scenario:

3. You’re out birding in coastal California, and you see the bird depicted below. It’s a Thayer’s Gull, right? Just then, your birding buddy from Pennsylvania e-mails you a photo of a bird he’s got under observation. It looks just like your bird. But your buddy isn’t sure his bird is a Thayer’s Gull; he thinks it might be an Iceland. Just for fun, let’s say we somehow know that your bird in California and your buddy’s in Pennsylvania hatched from the same clutch. In fact, we even have knowledge, somehow, that they’re monozygotic; they’re “identical twins.” The two birds are identical. Despite this new knowledge, you stick with your initial ID, and your buddy sticks with his. What’s going on here? Or, to be blunt about it: Which one of you is wrong?

Possible thayeri February 2007
This gull was found in Allegheny County, Pennsylvania, during February of 2007. Thayer’s or Iceland? Now what if the photo were from California? Would you change your mind? Photo by © Geoff Malosh.

I’m going to argue that you’re both right. Both IDs, seemingly mutually exclusive, are correct.

Let’s proceed. You’re still in California, but now you’re looking at the bird depicted below. Looks pretty good for Iceland Gull. In fact, I’d say it looks great for Iceland Gull. Now, as many birders know, Iceland Gulls are variable. Let’s say “only” 99% of Iceland Gulls are as pale overall as this one, but 1% are darker. Fine. This bird would appear to be among the 99% of Iceland Gulls that are, for want of a better word, “normal.”

Iceland Nebraska Dec 2004
If a bird is an Iceland Gull, there is perhaps a 99% probability it will look as “good” (as “typical”) as the bird depicted here. Is that enough to make this particular bird an Iceland Gull? Photo by
© Bill Schmoker.

Enter Thayer’s Gull. Thayer’s Gulls are variable, too—notoriously so. But let’s be reasonable. Let’s say fully 98% of Thayer’s Gulls are darker than the bird depicted here. (And let me acknowledge here that there is more to distinguishing Thayer’s from Iceland than overall “paleness.” I know that. But it doesn’t affect the argument. We can say that 99% of Iceland Gulls show the suite of characters shown by this bird, whereas 98% of Thayer’s Gulls do not. But so as not to get bogged down in terminology, I’ll stick with the simple pale vs. dark dichotomy.)

Anyhow, the case is looking awfully strong for Iceland Gull, you would think. We can’t be completely certain, of course, but I think most of us would lean pretty heavily toward Iceland. We won’t say it’s “definitely” or “positively” an Iceland Gull, but let’s say it’s “likely” or “probably” that species. I mean, 99% of Iceland Gulls look this way, and 98% of Thayer’s Gulls do not. Let’s split the difference, and say this bird has a 98.5% chance of being an Iceland Gull.

Not so fast.

Iceland Gulls are much rarer in California than Thayer’s Gulls. I’m going to say that for every Iceland Gull in California, there are about a thousand Thayer’s Gulls. I think a lot of folks would say I’m being pretty generous in that assessment. Maybe it’s more like five thousand to one. But let’s stick with one in a thousand. In a sample of one thousand Thayer’s and Iceland Gulls in California, only one is an Iceland.

I’ve now given you the three pieces of information that are necessary for determining the probability that this bird is an Iceland Gull. To recap: 99% of birds that look like this are Iceland Gulls; 98% of birds that don’t look like this are Thayer’s Gulls; and Iceland Gulls make up one-tenth of one percent of the California population of the Iceland and Thayer’s Gulls combined. Now what’s the probability that this bird is an Iceland Gull?

The answer, surprisingly, is 4.7%. (For the mathematically inclined birder, I work out the numbers in a footnote at the end of this post.) That’s it. That’s all. There’s only a 4.7% probability that the bird is an Iceland Gull. Statisticians generally say that probabilities below 5% tell you you’re looking at something else. Statisticians call that something else an “alternative hypothesis.” We birders call that something else a Thayer’s Gull. This bird is a Thayer’s Gull.

Unless you’re in Pennsylvania.

In Pennsylvania, Iceland Gulls outnumber Thayer’s Gulls by at least ten to one. And that’s being quite generous. If you do the math for Pennsylvania, there’s a 99.8% chance the same bird is an Iceland Gull. The bird in Pennsylvania is 10,118 times (yes, ten thousand, and then some) more likely than the same bird in California to be an Iceland Gull.

The only difference is the location of the photo. (If you’re a literalist, and you require two separate birds—one in California, one in Pennsylvania—then go back to my scenario of monozygotic twins. We have two birds now, identical to one another. The only difference is that one has strayed to California, whereas the other has wandered to Pennsylvania.)

This result is stunning. I’ve known about it for quite some time, but it still amazes me.

For starters, the result has consequences for how we identify birds. Let’s say we know from museum specimens that 4% of Species A’s outermost primaries are longer than 80 millimeters, whereas 20% of Species B’s outermost primaries are longer than 85 millimeters. Stop. Don’t go any further. We can’t do anything with those numbers until we know something about the relative probabilities of detecting Species A vs. Species B. Let’s say you’ve just netted a possible Alder Flycatcher in Utah. Now what? Well, you need two things. First, you need The Pyle Guide, with its heaps of data on wing formula and such. Second, you need to know the likelihood of detecting an Alder Flycatcher in Utah, as opposed to, say, New York.

Alder FlycatcherAlder or Willow? The “classical” method for answering that question would be to conduct a morphometric analysis of the bird’s physical characters. Many modern statisticians and other scientists would say a different approach is required. Photo courtesy of © The Geophysical Institute, Unversity of Alaska-Fairbanks.

Which brings us to something even more stunning. Reality is situational. Scientists, philosophers, and ethicists have been converging on that worldview for more than a century now. The general public, meanwhile, has been a bit more resistant.

But here’s a cheery thought. The whole time, we birders have known about it. Think back to the last time you said something along the lines of the following: “If I were in California, I’d surely call this bird a Thayer’s Gull.” There’s a lot more philosophy—and no small amount of statistics—in that statement than I think a great many of us give ourselves credit for.


Footnote: Possible Iceland Gull in California.

p(Iceland) = 0.001
p(Thayer’s) = 0.999
p(pale|Iceland) = 0.99
p(pale|Thayer’s) = 0.02

By Bayes’ Theorem:

p(Iceland|pale) = p(Iceland∩pale)/p(pale), where

p(Iceland∩pale) = 0.001×0.99 = 0.00099, and
p(pale) = p(Iceland∩pale)+p(Thayer’s∩pale) = 0.00099+0.999×0.02 = 0.02097


p(Iceland|pale) = 0.00099/0.02097 = 0.047

For Pennsylvania, update p(Iceland) to 0.9 and update p(Thayer’s) to 0.1. Following the same steps as above, we get p(Iceland|pale) = 0.998.

The relative odds of occurrence (Pennsylvania to California) are given by the cross-product ratio, (0.953×0.998)/(0.047×0.002) = 10,118.

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Ted Floyd

Ted Floyd

Editor, Birding magazine at American Birding Association
Ted Floyd is the Editor of Birding magazine, and he is broadly involved in other programs and initiatives of the ABA. He is the author of more than 100 magazine and journal articles, and has written four recent books, including an ABA title, the ABA Guide to Birds of Colorado. Floyd is a frequent speaker at birding festivals and state ornithological society meetings, and he has served on the boards of several nonprofit organizations. Mainly, he listens to birds at night.
Ted Floyd

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