The Bayesian Case for UFOs

After recent reports in the New York Times, I believe there is a greater than 50% chance that we are being monitored by aliens.

If you think that’s crazy, you’re not alone. Neil Degrasse Tyson’s skepticism – “The evidence is so paltry for aliens to visit Earth, I have no further interest” – seems pretty common. “Call me when you have a dinner invite from an alien,” Tyson says. This is a frequentist attitude: the null hypothesis (aliens aren’t monitoring us) can’t be said, with certainty, to have been rejected. I, instead, will take a more Bayesian approach.


But first, let’s talk about those NYT articles.

You need to read them – both of them – if you haven’t yet. I have never cared much about UFO sightings, but this goes far beyond some farmer’s fuzzy photograph that turns out to be a weather balloon. There is video evidence, and multiple credible accounts, of some very bizarre events. And this, apparently, is the tip of the iceberg. A lot more information remains classified; in fact, the head of the Pentagon’s UFO program resigned to protest its excessive secrecy. That same man, Luis Elizondo – who does not come across as a nutcase – warns about “the many accounts from the Navy and other services of unusual aerial systems interfering with military weapon platforms and displaying beyond-next-generation capabilities.” The program he led was supported by three high-ranking U.S. Senators from both parties. It wrote, in an official Pentagon briefing, that “what was considered science fiction is now science fact.” If anyone in the Pentagon disputes this, they have yet to come forward.

By far the most fascinating revelation is detailed in the second article. Again, read it for the full picture. The sighted objects seem to defy all known limits of aviation.

For two weeks, the operator said, the Princeton had been tracking mysterious aircraft. The objects appeared suddenly at 80,000 feet, and then hurtled toward the sea, eventually stopping at 20,000 feet and hovering. Then they either dropped out of radar range or shot straight back up.

Popular Mechanics has a good summary, along with more details.

The Super Hornets flew to investigate the last known location of the object and to their surprise, found two objects. The first was large and just below the surface of the water, causing the water to churn. The second object hovered just 50 feet above the water, moving erratically.

The first object, according to a leaked 2007 account, was “much larger than a submarine.”

The second object suddenly rose up and flew towards the Super Hornets, with one pilot. Commander David Fravor, saying it appeared it was rising up to meet him. The Hornet turned towards the object to meet it and the object peeled away, accelerating, “like nothing I’ve ever seen,” Fravor later said.

The Super Hornets conferred with the USS Princeton and were vectored to a CAP point 60 miles away. Within seconds, the pilots were told by the Princeton that radar had picked up the object already at the CAP point. By the time the Super Hornets arrived however the object had already disappeared.

According to Popular Mechanics, this means the object “would have had to have flown in excess of 2,400 miles an hour.” More bizarre still, “the object did not emit hot jet exhaust typical of ordinary aircraft.”

According to Commander Fravor, the object “had no plumes, wings, or rotors and outran our F-18s.”

On top of the 2004 incident, there’s also video of a strange, rotating aircraft; the Navy pilots have no idea what they’re seeing, and one remarks that “there’s a whole fleet of them.” There’s also supposedly alloys recovered from UFOs sitting in a ranch, which some people claim has physically affected them. That would be the story here, if we had any details or even a photograph. All alloys are easily identifiable, so it’s not clear what “unknown alloy” means, or whether it’s something significant.

But let’s go back to the 2004 incident. It seems too incredible to believe, but there are multiple eyewitnesses and video evidence. Even skeptics don’t claim that it’s a hoax – just that there’s an explanation that doesn’t involve aliens.

One popular theory is that the objects were part of a secret government project. This is extremely unconvincing. It means that a small group of scientists, working secretly within the government, developed technology decades ahead of what’s available now – in 2004! And in that whole time, this technology – advanced enough to change society forever – has been kept hidden from the public (and, amazingly, with no leaks). Popular Science calls this “a less exciting rationale”; I call it a scandal to put Watergate to shame. Some UFO sightings turn out to be secret projects, but none even remotely as extraordinary. If any government has this technology, they ought to be using it regularly by now.

Another theory is that it was a totally normal object, and both the Navy’s equipment and the pilots themselves failed to realize this. Popular Mechanics notes how unlikely this is:

This is a discomforting explanation, because it assumes that the pilots and the Princeton’s crew were incompetent and unable to discern ordinary objects from extraordinary ones. It also assumes the guided missile cruiser’s radar malfunctioned. If this explanation is correct, none of these pilots should have been flying for the Navy, and the Princeton’s air defense radar has a previously undiagnosed flaw. Given the level of skill necessary to fly from a U.S. Navy carrier it seems extremely unlikely these pilots were prone to fantasy or misidentifying the sun as a white, tic-tac-shaped UFO hovering close to the water.

The other explanation is… honestly, those are the only two I could find. Everything else boils down to “we don’t know.” As Tyson puts it: “The universe brims with mysteries. Just because you don’t know what it is you’re looking at doesn’t mean it’s intelligent aliens visiting from another planet.” Astrophysicist Sara Seager, quoted in the first NYT article, echoes this sentiment: “what people sometimes don’t get about science is that we often have phenomena that remain unexplained.”

On the other hand, Scientific American quotes chemist Richard Sachleben, who’s skeptical for the exact opposite reason:

There’s not as many mysteries in science as people like to think. It’s not like we know everything – we don’t know everything. But most things we know enough about to know what we don’t know.

So, it’s probably not aliens because there’s a lot of mystery in science. But also, it’s probably not aliens because there isn’t a lot of mystery in science.

Neither of these contradictory explanations are convincing. Scientific mysteries typically involve new particles or oddly-behaving stars, not hyper-fast, physics-defying aircraft.

It is true, though, that hard evidence does not exist. We don’t know what those pilots saw; it could be almost anything. Luckily, Bayes’ Theorem gives us a use for such messy data.


First, you need a prior. Before the recent revelations, what odds would you have given to the presence of aliens or alien artifacts on Earth? It’s hard to judge this objectively, but Fermi’s paradox can help. Enrico Fermi calculated that, statistically, the galaxy should be teeming with intelligent alien life, and some of these ought to have visited Earth by now; so, he asked, “where is everybody?”

Let’s start with what we know. Only one planet – Earth (duh) – is known to harbor life. Is it an exception, or a rule? Well, it’s an exception within the Solar System – none of the other 7 planets seem inhabited. So, that gives life a 12.5% (1/8) chance of developing. Counting only rocky planets, that rises to 25%; counting large moons and dwarf planets, it falls to less than 5%. (Although I think we should look closer at Europa before counting it out.)

But Earth is very distinct from those places. In terms of Earth-like planets – planets within the habitable zone, with liquid water, an atmosphere, etc. – our sample size is only 1. Life arose pretty quickly on Earth – within one billion years of the planet’s formation, and possibly within just a few hundred million years. (Intelligent life took a lot longer, but let’s go one step at a time.)

So: for an Earth-like, Earth-age planet, our sample size is exactly 1. In such situations, our only guide is the principle of indifference. For all we know, Earth is unusual in how long it took for life to arise. Without more data, we should assume Earth-like planets have a 50% chance of harboring life.

But how many such planets exist in the Milky Way? According to the evidence: a lot, probably.

The Milky Way contains 100-400 billion stars. Most of these (75-85%) are red dwarves, which have some barriers to habitability – they’re very dim, fluctuate wildly, and so small that any planet would have to be tidally locked. We have no idea what extraterrestrial life might look like, so these might not be barriers at all. Still, to avoid getting too speculative, let’s count them out, along with giant stars, orange dwarves, and anything else that doesn’t resemble the Sun. That still leaves us with several billion candidates. (And there’s no good reason to discount orange dwarves, which make up about 10% of the galaxy; if anything, they ought to be more friendly to life than the Sun.) Many of these are younger than the Sun (Sun-like stars expand after 10 billion years; the Sun is 4.7 billion years old). And more than half orbit other stars, which creates further complications. Still, even the most stringent estimate gives you over 500 million stars to work with.

According to one study, 17% of Sun-like stars have an Earth-sized planet in an orbit closer than Mercury’s. That’s a smaller region than the Sun’s habitable zone, by most estimates. In another study, per the LA Times, “researchers surmised that about 22% of stars like the sun have planets in their habitable zone that are about the size of Earth.”

That gives us at least 100 million potentially habitable planets. Now, maybe they have the wrong chemistry. After all, Mars and Venus might have been habitable with more water and friendlier atmospheres. But the elements of life – such as carbon and oxygen – are among the most common elements in the universe. Earth isn’t uniquely water-abundant – five known moons, plus Pluto, have more water than Earth. While the odds are low of a planet having Earth-like chemistry, they aren’t one-in-a-million low. I doubt they’re even one-in-a-thousand low. (For example, Titan is believed to resemble early Earth, although it’s much colder.)


No matter how you calculate, it seems unlikely that there’s fewer than 100,000 potentially habitable planets in the Milky Way. That’s being extremely cautious; the number could be in the billions.

Even if all the conditions are right, how likely is life to develop? We have no idea. Again: our sample size is 1. Via the principle of indifference, a 0.1% chance is just as extreme a guess as a 99.9% chance. For Earth to be the only life-bearing planet in the galaxy, the chances of life developing must be infinitesimally small. We currently have no data to support such an extreme claim.

What about intelligent (human-like) life? There, we do have some data: the fact that we haven’t found any.


Here’s where Fermi’s paradox comes in.

A species spreading at just 0.25% the speed of light would colonize the galaxy within 50 million years. And that’s a conservative guess – at humanity’s rate of technological advancement, we might reach every star in just a few million years. Cosmically, that’s no time.

So why hasn’t someone else done it yet? There’s no sign of aliens or alien artifacts within the Solar System. (Which ought to be a popular target – the Sun outshines 90% of the galaxy.) Nor is anyone blasting radio signals our way. If intelligent life is common, at least one species should have contacted us. So, as Fermi asked, “where is everybody?”

Perhaps nobody got past the Great Filter. (There’s a great explanation of Fermi’s paradox and the Great Filter here.)

There are many steps between the development of life and galactic colonization, each of them hard to get past. To get to humanity, life on Earth had to evolve from simple, prokaryotic bacteria into complex, eukaryotic cells; sentient animals; and finally, intelligent life. Each of these steps have happened only once in evolution’s 4 billion year history, so they must be pretty rare.great-filter1-1024x727

That’s assuming that the Great Filter is behind us. It could be yet to come. Perhaps crossing stars is simply too difficult. Perhaps intelligent life is more likely to destroy itself than expand. The Great Filter could be just about anywhere.

If it exists, that is. Extraterrestrial life may have already visited Earth – we just didn’t notice. I like to think of this as another aspect of the Filter – something is preventing us from seeing aliens, whether they exist or not.

In that case, time is the greatest filter of all. Think about it: if aliens visited 1 million years ago, would anyone know? What about 1,000 years ago? If they came 1 billion years ago, they might have set up a colony on Earth and we’d have no idea. Mass communication is very new – 100-500 years old, depending on how you count. Recorded history goes back only a few thousand years. Cosmically, that’s less than no time.

There are many reasons why these aliens might not have stayed. Maybe the Solar System held nothing of value to them. Perhaps they came too early, jotted down “mostly harmless,” and left for someplace more interesting. They might also have practical, ethical, or legal reasons to leave us alone. They don’t know if we’d react hostilely, for one thing. Intelligent races might have some contract to leave primitive life alone until it’s more evolved. In any of these cases, they’d be wise to monitor us, perhaps with probes – which, perhaps, we occasionally catch a glimpse of. This is the “Zoo Hypothesis,” proposed by Fermi himself.

(Side note: I’m focusing exclusively on the Milky Way. If intergalactic travel is possible, the odds of an alien visitor exponentially rise. The Andromeda Galaxy, with an estimated trillion stars, is “only” 2.5 million light years away; perhaps an extremely advanced species could cross that gap. However, I want to avoid getting too speculative.)


Okay, time to use Bayes’ Theorem.

Take two events: A and B. Event A is the existence of aliens, or alien artifacts, within the Solar System. Event B is our failure to find evidence of such aliens.

The simplest form of Bayes’ Theorem I’ve seen comes from Nate Silver’s The Signal and the Noise. To find the posterior probability – the odds of A, given B – we’d use this equation:

xy/[xy + z(1-x)]

x is the prior – our initial odds for A. That is, the odds we’d give to aliens’ presence prior to knowing that we wouldn’t find them. This should be pretty high, for reasons explained above. Humans seem on track to colonize the galaxy; it’d be amazing if, over billions of years, we’d be the first. See also the mediocrity principle. I’ll put this at 50% (which seems low).

y is the probability of B, given that A is true. In other words: if you knew that aliens had visited, what odds would you give for us not finding them? This depends on how likely you find the Zoo Hypothesis, or some similar explanation, to be. I find it reasonable. Humans send out probes all the time; in general, we want to gather information before we act. If we found intelligent aliens, we’d be fools to contact them immediately without studying them first. There’s also the tiny amount of time we’ve been looking – far too small a sample to draw any conclusions. On the other hand, I admit, I’d expect us to find something by now. If space empires are common, you’d think one of them would have an outpost nearby. We’ve photographed all of the planets and large moons, looked everywhere for radio signals, and come up with nothing. I’ll put y at just 5%.

z is the probability of B, given that A is false. This is basically 100%.

Do the math, and you get a 5% posterior probability of alien presence.

That, however, is with no evidence. We do have some evidence, although nothing definitive.


Not all UFO sightings are equal. Mass sightings are more noteworthy than individual sightings; sightings that defy known science are more noteworthy than flashing lights. According to Wikipedia, “Between 5% and 20% of reported sightings are not explained”; Wikipedia adds that “the null hypothesis cannot be excluded that these reports are simply other more prosaic phenomena.” However, Bayesian inference has no concept of a null hypothesis; all hypotheses are treated as equal.

Instead, let’s ask: what would we expect to find, given the presence of aliens? Well, we wouldn’t expect nothing. Human civilization has spread so wide that a few encounters would be inevitable, even if the aliens are trying to hide. These encounters shouldn’t be limited in time or place: we should find them all over the world and throughout history.

Indeed, that’s what we find. Several Greek and Roman historians reported UFO sightings in antiquity. Plutarch wrote of a “wine-jar” shaped object that fell from the sky in Asia Minor, reported by two opposing armies. Pliny the Elder wrote of a “spark” falling from the sky, then rising back up. In 776, Frankish annals report two objects like “large flaming shields” floating overhead.

In the 1560s, central Europe experienced a spate of UFO sightings. This includes my favorite: a 1561 “battle” over Nuremberg, whose residents witnessed “hundreds of spheres, cylinders and other odd-shaped objects that moved erratically overhead.”


Illustration from a Nuremberg newspaper, printed in April 1561

Granted, people back then “saw” all sorts of omens. Still, you’d expect stories like these if alien ships flew over ancient skies.

The number of UFO sightings is less important than each one’s credibility. Some do seem fairly credible. In 1987, three Japanese pilots witnessed objects which defied known laws of aviation, flying “as if there was no such thing as gravity,” according to one pilot. A similarly bizarre object was seen by many O’Hare employees in 2006; the disc hovered over the airport, then rapidly shot up into the sky. The 1997 Phoenix lights, one of the most famous UFO events, was seen by thousands – including the sitting governor at the time, Fife Symington III (great name). Symington later said: “As a pilot and a former Air Force Officer, I can definitively say that this craft did not resemble any man made object I’d ever seen.” Elsewhere, he said: “I know just about every machine that flies. It was bigger than anything that I’ve ever seen. It remains a great mystery.” He also revealed that no one in the Federal government or the military had any explanation at the time.

For more sightings, see these Wikipedia and NatGeo articles. Common themes include discs (“flying saucers”) and massive, v-shaped aircraft (as seen over Phoenix). You’d expect such commonalities if any of these sightings are real.

To be clear, I don’t believe any of these sightings, individually, have even a 1% chance of alien origin. However, the question isn’t how many of these sightings are of alien ships, but whether even just one of them is.

Another piece of evidence comes from KIC 8462852, a star whose seemingly random fluctuations have scientists totally stumped. It’s also dimming far faster than any similar star. There is no theoretical model that explains this star’s behavior. It’s probably caused by some unknown natural phenomenon. However, it’s the kind of anomaly you’d expect to find if hyper-advanced alien life existed in our galaxy.

Would we expect all this evidence if alien life isn’t here?

Probably, yes. The huge number of disproven UFO sightings should alter our priors toward skepticism. There are also many sightings of Bigfoot, ghosts, the Virgin Mary, and other things for which my priors are much lower.

In fact, the history of folklore is rife with such sightings. People want to see, and people want to believe; sometimes, seeing and believing become interchangeable. Elves and witches were seen all the time back when people believed in them. That said, some UFO reports are a bit more credible than any elf sighting.

Overall, the evidence is enough to raise my prior, but only a little – from 5% to, say, 6%. Indeed, those are about the odds I’d have given aliens’ presence before the NYT reports.

How much should those reports change my prior?


Event A is still the presence of aliens or alien artifacts. My prior for this (x) is 6%. Event B, now, is everything written in those NYT reports.

What’s the probability of B, given that A is true? Not too low, I’d say. The UFOs in question acted skittish, yet inquisitive – about how I’d expect alien probes to act. They might want to avoid detection, but without knowing much about humans, this would be difficult. Still, you might expect them to be better at hiding, or so bad at it as to produce better evidence. Or maybe they wouldn’t even try to hide. Nonetheless, the incidents are well within the realm of what I’d expect if aliens are watching us. I’ll set y at 20%. This seems low, but I’m trying to be careful.

What’s the probability of B, given that A is false?

Now, these weren’t hack articles: two of the three authors were Pulitzer Prize-winning journalists. All of the sources were credible, and the Department of Defense did not dispute anything. The articles have been criticized because some of the information was already known within the UFO community. However, the reports have put this information in a context that makes it nearly impossible to deny.

So the 2004 incident isn’t a hoax. But what else could it be? Terrestrial theories are, if anything, more shocking than extraterrestrial ones. If the “white tic tac” is something natural, we’re looking at a new epoch in science. If it’s a secret government aircraft, we’re looking at a new epoch in politics and aviation. If it’s a mistake… then we’re looking at a new epoch in politics, aviation, and journalism, because a blunder of this scale is without precedent.

I can’t come up with an explanation that isn’t world-changing.

No matter what, this seems like a one-in-a-million story. However, I’m putting z at 1%. That seems very high, but again – caution.

So x=0.06, y=0.2, and z=0.01. Plug this into xy/[xy + z(1-x)], and we get…

0.56. A 56% posterior probability that aliens have visited the Solar System. And that is using extremely conservative assumptions.

Granted, these numbers are all subjective. Maybe your priors are lower. My point is that these revelations should substantially raise them. I’m seeing a lot of dismissive comments such as these Tweets and the Popsci article above. I’m a skeptic by nature, but this derisiveness seems misplaced.

The purpose of this article isn’t to convince you that aliens are watching. I’m not so sure of that, myself. However, I was driven to write this by the snide tone of some of these dismissive reactions. A lot of people hear “UFO” and immediately tune out. Many reasonable people seem to disagree, and I think they deserve to be heard. Each piece of evidence alters our priors, if only a little; it’s important to keep our eyes and ears open so that our judgments have a proper basis.


One thought on “The Bayesian Case for UFOs

  1. Eric Hyphenthenumberone

    If we were being visited by aliens, then that means that my new prior would be that the universe is simply awash in civilization. It’s everywhere and frequently traveling between stars. And almost certainly there would be millions or billions of civilizations old enough to have started moving stars around. If that’s the case, what are the odds that billions of active civilizations can hide their existence from *any* observation? But when they get here, despite being able to hide all evidence of stellar engineering, they then fly around with their running lights on, in radar-visible craft but only in locations where a handful of witnesses can see them.


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