Bryan Caplan  

The Bayes Who Wasn't There

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From an early age, I've furrowed my brow at the claim that "Absence of evidence is not evidence of absence." Huh? Absence of evidence is not absolute proof of absence, but surely if you don't notice your friend in a room, that's evidence that he's not in the room, right?

That's why my inner child is delighted that Eliezer at Overcoming Bias has skewered the silly "absence of evidence" sophism:

[I]n probability theory, absence of evidence is always evidence of absence. If E is a binary event and P(H|E) > P(H), "seeing E increases the probability of H"; then P(H|~E) < P(H), "failure to observe E decreases the probability of H". P(H) is a weighted mix of P(H|E) and P(H|~E), and necessarily lies between the two...

Under the vast majority of real-life circumstances, a cause may not reliably produce signs of itself, but the absence of the cause is even less likely to produce the signs. The absence of an observation may be strong evidence of absence or very weak evidence of absence, depending on how likely the cause is to produce the observation... This is the fallacy of "gaps in the fossil record" - fossils form only rarely; it is futile to trumpet the absence of a weakly permitted observation when many strong positive observations have already been recorded.

Thanks, Eliezer. At last I have an intellectually satisfying response to my friends who continue to believe in Santa Claus. Yes - our failure to find his workshop at the North Pole is indeed evidence that Santa does not exist!


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COMMENTS (8 to date)
Caliban Darklock writes:

You're interpreting the phrase rather strangely. The absence of your friend in the room is evidence that your friend is not in this room, but it is quite definitely not evidence that your friend is not in any room - i.e. that he does not exist.

The key phrase in the quote you've found is "depending on how likely the cause is to produce the observation". For many years, we believed the coelocanth was extinct: absence of evidence was considered evidence of absence. Then someone caught one off the coast of South America. Turns out there are plenty of them. We just weren't looking where they were, looking instead where they used to be and saying "they're not here, they must not be anywhere". In retrospect, that's a pretty stupid line of reasoning.

In short, absence of evidence is only evidence of absence when the phenomenon for which you seek evidence is likely to produce the evidence you seek. The mathematical treatment of the question doesn't change that.

eric falkenstein writes:

This is a great example of tendentious rhetoric. Logically, they could get away with "proof", but instead use "evidence", which when you are reading fast, have similar meanings ('that which implies', roughly).

I think anyone who truly believes that, should tell their boss, investor, or lover, that their absence of performance does not at all reflect upon their ability to perform in the future.

Tobbic writes:

Correction: There's actually a Santa's workshop in the north pole in Finland ;). However, this Santa is just a hired guy, not the one with the flying sledge pulled by magical reindeers.

Lord writes:

The error is equating the absence of evidence to P(H|E). Naturally P(H|E) is evidence, the question is how significant.

Asim writes:

Given that the preceding comments have addressed why the verbal arguments you put forth are incorrect, I'd like to throw in my 2 cents by adding that the mathematics is garbage.

If E is a binary event and P(H|E) > P(H), "seeing E increases the probability of H"; then P(H|~E)

The author has misunderstood the definition of P(H) when it comes to Bayes' theorem. A quick refresher of simple high-school math ( http://en.wikipedia.org/wiki/Bayes'_theorem#Statement_of_Bayes.27_theorem) emphasizes that P(H) is the _total_ probability of H.

Meaning:

P(H) = P(H|E) + P(H|E')

(in english: the probability of event H = the probability of event H given event E or the probability of event H given not event E)

From the author's proof:

P(H|E) > P(H) = P(H|E) + P(H|E')

implies:

0 > P(H|E')

However, all probabilities, by definition, lie between 0 and 1 inclusive. The author is asserting a probability that is less than 0, hence the initial statement is contradictory, QED.

Ultimately there was a serious misunderstanding of what P(H) meant. Look at the original statement again:

If E is a binary event and P(H|E) > P(H), "seeing E increases the probability of H"

P(H|E) > P(H) is a logical absurdity, as the mathematics have shown. Perhaps the author really meant P(H|E) > P(H|E').

Asim writes:

Actually ignore my previous comment; I think my maths are even more incorrect.

Barkley Rosser writes:

Well, Eliezer is responding to the brilliant, but somewhat egomaniacal, Nessim Taleb and his Black Swan book, which trumpets this phrase repeatedly. Fine, but I don't think that it will necessarily convince a true Talebian as he is not clearly a Bayesian. Indeed, to carry out the Bayesian calculation, one must assume an underlying distribution on which to recalibrate. But that is one of Taleb's arguments: we do not know a priori, or even somewhat a posteriori, what the underlying distribution is, much less if there even is one or not.

I would remind that Bayes' Theorem has its limits as well, such as a continuous support and a finite dimensionality. Failure of the conditions can lead to convergence on cycles and other failures to get to the correct probability (Diaconis and Freeman, Annals of Statistics, 1988), even assuming that such exists in any meaningful sense, which for a hard-core subjectivist Bayesian is not true anyway.

mulp writes:

"Absence of evidence is not absolute proof of absence, but surely if you don't notice your friend in a room, that's evidence that he's not in the room, right?"

For the economist who views the environment as a subset of the economy, an externality, that is true, because you have if you have no demand for your friend, he will not exist.

But for the ecologist who sees the economy as a subset of the environment, your friend exists or not whether he is in your room or not. Only if there is demand for your friend in your room, your friend exists, and the environment cooperates, will your friend exist in your room.

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