We’ve seen the Drake equation in many forms, none of them very impressive. They all share the same failing, as we’ll see.

Two fellows, Frank and Sullivan, have another go. The lite version is (appropriately) in the *New York Times* with the title “Yes, There Have Been Aliens“. The full version is the *peer-reviewed* paper “A New Empirical Constraint on the Prevalence of Technological Species in the Universe” in *Astrobiology*. From the *Times*:

Instead of asking how many civilizations currently exist, we asked what the probability is that ours is the only technological civilization that has ever appeared. By asking this question, we could bypass the factor about the average lifetime of a civilization. This left us with only three unknown factors, which we combined into one “biotechnical” probability: the likelihood of the creation of life, intelligent life and technological capacity.

You might assume this probability is low, and thus the chances remain small that another technological civilization arose. But what our calculation revealed is that even if this probability is assumed to be extremely low, the odds that we are not the first technological civilization are actually high. Specifically, unless the probability for evolving a civilization on a habitable-zone planet is

less than one in 10 billion trillion, then we are not the first.

You can already see that abuses of probability are coming up. From the paper:

We define the “A-form” of the Drake equation, which describes the total number of technological species that have ever evolved anywhere in the currently observable Universe:

A = [N

_{*}f_{p}n_{p}][f_{l}f_{i}f_{t}]…where N

_{*}is the total number of stars, f_{p}is the fraction of those stars that form planets, n_{p}is the average number of planets in the habitable zone of a star with planets, f_{l}is the probability that a habitable zone planet develops life, f_{i}is the probability that a planet with life develops intelligence, and f_{t}is the probability that a planet with intelligent life develops technology (of the “energy intensive” kind such as that of our own civilization).

After this comes manipulations of the equation which aren’t especially interesting. There are no insurmountable problems in the leading elements of this or the modified equation. But there is a universe of trouble in the second terms in the brackets, [f_{l} f_{i} f_{t}].

All of these elements are said to be probabilities. Skip whether we can discover unique numbers for each probability and instead focus on probability’s Golden Rule: *All probabilities are conditional.* From that simple and honest truth flows everything, including the proof that the Drake equation, modified or no, is meaningless.

To prove that, pick the element f_{t}, “the probability that a planet with intelligent life develops technology (of the ‘energy intensive’ kind such as that of our own civilization).” That probability *does not exist*—no probability does—without premises, assumptions, givens, or conditions. And what might these premises be?

The point of the Drake equation is to count on or, in the modified form, to put a probability to the proposition “Rational creatures on other planets exist”. Yet a main element, f_{i}, is very nearly that same probability. You might argue that it isn’t precisely the same, in an attempt to save Drake from circularity, but f_{i} surely smacks of assuming what it set out to prove.

Even if circularity is missing, f_{i} has no meaning. None. No probability in that equation is sensible. All probabilities need evidence, and these have none. The *best* we can do is infer the propositions of f_{i} and so on are contingent, and therefore have any number between 0 and 1, which means, thus the final result (in the modified Drake) is itself between 0 and 1, and which is nothing more than knowledge that “Creatures exist” is itself contingent. Nothing has been gained.

In order to calculate f_{i}, we need a list of accepted premises that unambiguously lead to a unique number, or at least to a tight interval. None exist: none that are acceptable and agreed to by all, I mean. Plentiful premises exist that might be used, of course. You might say, “7 out of 10 planets with life develop rational creatures” and thus f_{i} = 0.7. But who would agree to these premises?

Since nobody has any idea of how life began (on Earth) nor do any know of rationality arose, premises which can fix f_{i} just don’t exist. And that same is true for the other probabilities. The Drake equation leads nowhere.

New Drake factor: fd — the likelihood peer-reviewed nonsense gets published in elite journals.

Stop the world… I want to get off.

BTW — I just pre-ordered your book.

Best regards.

Peer-reviewed!! (Good comment, John Z)

“Instead of asking how many civilizations currently exist, we asked what the probability is that ours is the only technological civilization that has ever appeared. “ Doesn’t this indicate too much time and money on one’s hands? This used to be the realm of scifi and it was quite fine there.

Is there an equation like the Drake equation to indicate the probability my mouse trap will catch the mouse before it eats through the sugar package in the pantry? The equation seems like it could be used for most anything with a couple of modifications. You know, like “m” is the number of mice, “m1” is the probability he is trapped, “m2” the chance he finds the sugar before the trap, “m3” that the trap misses him, etc. Of course, in the case of the mouse, there might be actual data to support the “m” values, but we’ll pretend we have no time to gather it and are just estimating (now a true scientific process in peer-reviewed journals—saves on data collection expense, I guess).

The Donald Duck equation is x = a*b*c where a, b, & c are anything that results in x being peer reviewable.

You are correct in that there is no way to assign real probabilities to f(l) f(i) or f(t)

However, there is another way to look at the Drake equation.

N is empirically quantifiable for the observable universe and it is a very large number.

f(p) is also empirically quantifiable and is a very large number.

n(p) is more difficult to quantify empirically but not impossible.

You can look at f(l), f(i), and f(t) from the other direction, how small do they have to get before the very large numbers in the first three terms stop causing the probability that there is or has been other intelligent life in the universe to = 1.

As Fermi put it, “where are they?”

Chick Pea: whaddya mean? I’ve been here all along.

Meanwhile, we should should all put our heads together and try to come up with a tight interval for

Pr(G|C) where

C=God cretted other rational animals, and

G=he woulda guv them their own damn universe.

Again, Sci-fi, provides the definitive word. A middle-aged anthropologist and his wife go exploring the universe to find life. Their lives are extended via time-dilation at speeds approaching c. The wife dies, and after some years with a pet cat (who also dies), the anthropologist has a revelation: God did create life on one planet. I’m sorry, I can’t recall the title and a brief search doesn’t reveal it.

Briggs, look in the mirror. There is your evidence. Really, try it. You will actually observe an intelligent creature living on a planet.

JMJ

They left out a key probability: the probability of milking another peer-reviewed paper out of this.

A good reviewer would certainly have called for some supporting calculations to be done, preferably with large scale simulations on clouds of multiple processors. They left out some key “probabilities.” One would be the distribution of the time required from planetary formation to 1) the occurrence of life, 2) the development of intelligence, 3) development of technology sufficient to escape the stellar system. Then one needs to convolve that with the distribution of times between 1) stellar deaths, 2) cataclysmic extinction events, 3) cataclysmic confrontations with other civilizations 4) natural extinction by disease or other biological phenomenon, e.g. interspecies competition, etc. Then one needs to convolve those with the times of paths between points inside one causal horizon and others, the probability of surviving transiting such paths, etc. etc. and they could just churn out paper after paper. It would be even better if the follow-on papers referred to some Russians, say Kolmogorov and Markov and involved elegant saddlepoint analyses.

The work opens up vistas of sophisticated computer based color graphics and tables and tables of output. They have only scratched the surface here.

Or they could just invoke the famous logical argument: reductio ad yberra –

1. If the world were perfect, it wouldn’t be.

And one of its corollaries

2. You’ve got to be very careful if you don’t know where you are going, because you might not get there.

The Drake’s Cake equation:

(For the topping: 8 tablespoons unsalted butter + ? cup granulated sugar + ? cup dark brown sugar + ¾ teaspoon ground cinnamon + ¼ teaspoon salt + 1¾ cups cake flour) + (For the cake: 1¼ cups cake flour + ½ cup granulated sugar + ¼ teaspoon baking soda + ¼ teaspoon table salt + 6 tablespoons unsalted butter + 1 large egg at room temperature + 2 tablespoons plain Greek yogurt + 1 teaspoon vanilla extract + ? cup buttermilk).

Like all intelligent, technological civilizations, how it’s constructed makes a difference. Failure is a possibility.

From empirical observations, there is a very low probability that someone will dislike properly constructed results.

Evidently some fraction symbols don’t translate.

(For the topping: 8 tablespoons unsalted butter + 1/3 cup granulated sugar + 1/3 cup dark brown sugar + 3/4 teaspoon ground cinnamon + 1/4 teaspoon salt + 1-3/4 cups cake flour) + (For the cake: 1-1/4 cups cake flour + 1/2 cup granulated sugar + 1/4 teaspoon baking soda + 1/4 teaspoon table salt + 6 tablespoons unsalted butter + 1 large egg at room temperature + 2 tablespoons plain Greek yogurt + 1 teaspoon vanilla extract + 1/3 cup buttermilk).

I like the Lex Luther Equation, which lays out the probability that any sufficiently advanced civilization will produce a billionaire genius psychopathic murderer with the means and motive to exterminate said advanced civilization. Unlike the Drake Equation, it relies on measurable, confirmable facts: there are, in fact, billionaires, geniuses and and psychopathic murderers. Since there are none-zero values for the occurrences of billionaire genius sociopaths, it is only a matter of time before a Lex Luther arises and destroys whatever species generated him.

The Drake Equation rests on Uniformitarianism writ large: whatever we see on earth MUST be what WILL happen everywhere the same conditions (a relentlessly poorly-defined concept in this case) prevail. Well, billionaires, geniuses and mass-murderers all arose here BY NATURE, dammit! and thus we can rest assured that one will arise eventually anywhere intelligent life arises. Thus, we can also rest assured that such an one will then wipe out his world using whatever weapon or poison or doomsday devise his genius can dream up and his billions pay for. (Superman being as adverse to being discovered as space aliens.)

And that, Fermi, is where everybody is.

JMJ: Huh?

fah: Great points. The color graphics are essential, of course, 😉

All: I suspected there were other such equations. Thanks for the examples!

I have to take a stand in defense of the Drake equation, after all Fermi clearly believed in its value and he was thoroughly familiar with an equation from nuclear reactor theory that was of similar form and character: the six-factor formula.

The fact that probabilities are conditional is in no way damning, as each successive term is conditional on the previous (i.e. fp is short for P(fp|N), etc.) along with its priors. This is true of all of probability theory. Just because we don’t have statistics of other occurrences doesn’t mean we have no knowledge that applies.

Furthermore, to say that a probability /does not exist/ is simply sloppy thinking. After all, one assigns probabilities to things to describe one’s state of knowledge, and thereby all of probability theory is inherently /subjective/. A literal interpretation of your statement is that you believe the OP is lying about assigning a probability to it, which is clearly absurd. Do you instead mean it as emphasis on how strongly you feel the probability OP gave it is wrong? If so simply say that instead.

As for where the Drake equation leads… we will have to wait and see, but it certainly has value. The value Fermi no doubt saw in it is well understood by looking at the six-factor formula, which Fermi likely developed. The six-factor formula describes the neutron multiplication factor, k, as a function of a bunch of conditional probabilities (just like the Drake equation). However, it is useless as a means of obtaining the value of k, because you can’t get accurate enough values for those probabilities to get a useful result (just like the Drake equation). Despite that it is extremely useful as a means of understanding the behavior of the system (just like the Drake equation).

On a side note, for anyone who enjoys mathematics and physics, nuclear reactor theory is a blast to read. It is largely a self-contained topic and requires only a solid basis in math.

I began this as a comment on the Gambler’s Fallacy but decided it could run over into this post.

From Briggs in the Gambler’s fallacy:

[quote]And not just those people, but anybody who believes in physical probability embraces the Gambler’s Fallacy; frequentists are just their most visible representatives. Physical probability must be causative to make observed frequencies work out in balance. But since probability is only a state of mind, unless one is staunch Idealist, probability-as-cause makes no sense.[/quote]

This post is a little puzzling.

I’ve never seen a Roulette Wheel except as a whirling something in a picture show where the hero scoups a series of winnings that endears him to the most beautiful girl in the picture show. Anything can be made to happen in a picture show.

However, an astute gambler will always know that no matter how many times the ball lands in Red the likelihood of it being red or black next time is exactly 50:50 unless the wheel is contaminated or “loaded”.

I presume that our ole friend Briggs is trying to create an impression that if possibilities are infinite then improbabilities are possible and, given enough time, possibilities become certainties.

Anyhow, let’s stick with the Roulette wheel and the ludicrous assumptions that that render Drake and all mathemagical systems that render statistics and probabilities entirely irrelevant to the appearance of Life.

Assumption: Life spontaneously arises out of a fortuitous accretion of atoms and molecules.

The Wheel: Let’s imagine that the wheel has on it the numerous elements that must accrete in a specified order to form the “molecule of life”. Presumably, if the wheel is spun often enough then the required sequence would occur.

However, the wheel is “loaded” with entropy… that is, in most of the pockets the ball would have to go into are “uphill”, against entropy, so that even if the ball was flung into the “needed” pocket it would roll straight out again. So, even if “the chances are” that it could happen it will never happen because the physics is against it.

There are many simple, pragmatic observations about organic life that preclude any possibility that organic life is just a matter of chemistry.

Briggs: [quote]

In order to calculate fi, we need a list of accepted premises that unambiguously lead to a unique number, or at least to a tight interval. None exist: none that are acceptable and agreed to by all, I mean. Plentiful premises exist that might be used, of course. You might say, “7 out of 10 planets with life develop rational creatures” and thus fi = 0.7. But who would agree to these premises?

Since nobody has any idea of how life began (on Earth) nor do any know of rationality arose, premises which can fix fi just don’t exist. And that same is true for the other probabilities. The Drake equation leads nowhere.

[/quote]

There is a “unique number”… it is 0.

If there is life “out there” it was put there by the same Life that put it “in here”.

I doubt very much that the above will affect the thinking of those ideologically committed to rationalising de Chardin’s insanely egomaniacal Materialism but, with a bit of luck, it may make someone not committed think about it.

What we have is a bunch of coots flogging a dead horse with the promise that a liberal dose of Mathemagical Snake Oil will make this corpse into the artist’s impression of the greatest galloper that insanity could ever conceive.

The mathemagicians don’t have it on their own, though, there’s a whole cartel of scientismagicians and philosophists producing and rationalising the illusion.