Information Scientist and Professor of Electrical Engineering and Law, University of Southern California; Author, Noise, Fuzzy Thinking
Negative Evidence

Negative evidence is a concept that deserves greater currency in the intellectual trades and popular culture. Negative evidence helps prove that something did not occur. University registrars routinely use negative evidence when they run a transcript check to prove that someone never got a degree at their university.

Negative evidence is the epistemic dual to positive evidence. It is just that evidence that tends to prove a negative. So it collides headfirst with the popular claim that you cannot prove a negative. A more sophisticated version of the same claim is that absence of evidence is not evidence of absence.

Both claims are false in general.

It may well be hard to prove a negative to a high degree. That does not diminish the probative value of doing so. It took the invasion and occupation of Iraq to prove that the country did not have weapons of mass destruction. The weapons may still turn up someday. But the probability of finding any has long since passed from unlikely to highly unlikely. The search has simply been too thorough.

Absence of evidence can likewise give some evidence of absence. A chest CAT scan can give good image-based negative evidence of the absence of lung cancer. The scan does not prove absence to a logical certainty. No factual test can do that. But the scan may well prove absence to a medical certainty. That depends on the accuracy of the scan and how well it searches the total volume of lung tissue.

The CAT-scan example shows the asymmetry between positive and negative evidence. The accurate scan of a single speck of malignant lung tissue is positive evidence of lung cancer. It may even be conclusive evidence. But it takes a much larger scan of tissue to count as good negative evidence for the absence of cancer. The same holds for finding just one weapon of mass destruction compared with searching for years and finding none.

Simple probability models tend to ignore this asymmetry. They have the same form whether the evidence supports the hypothesis or its negation.

The most common example is Bayes theorem. It computes the conditional probability of a converse in terms of a hypothetical condition and observed evidence. The probability that you have a blood clot given a high score on a D-dimer blood test differs from the probability that you would observe such a test score if you in fact had a blood clot. Studies have shown that physicians sometimes confuse these converses. Social psychologists have even called this the fallacy of the inverse or converse. Bayes theorem uses the second conditional probability (of observing the evidence given the hypothetical condition) to help compute the first probability (of having the hypothetical condition given the observed evidence). The simple ratio form of the Bayes calculation ensures that a symmetric ratio gives the probability that the hypothesis is false given the same observed evidence. The numerical values may differ but the ratio form does not.

The law is more discerning with negative evidence.

Courts often allow negative evidence if the proponent lays a proper foundation for it. The opponent must also not have shown that the lack of evidence involved something improper or untrustworthy. A convenience store’s videotapes of a robbery can give compelling negative evidence that a suspect did not take an active part in the robbery. But the video recordings would have to cover a sufficient portion of the store and its parking lot. The cameras must also have run continuously throughout the robbery.

Federal Rule of Evidence 803 uses a telling phrase to describe when a proponent can use a public record as negative evidence that a prior conviction or other event did not occur. The rule demands a “diligent search” of the public records.

Diligent search is the key to negative evidence.

It is easy to conduct a diligent search to prove the negative that there is not a five-carat diamond in your pocket. It takes far more effort to conduct a diligent search for the diamond in a room or building or in an entire city.

The strength of negative evidence tends to fall off quickly with the size of the search area. That is why we cannot yet technically rule out the existence in the deep seas of kraken-like creatures that can drag ships down to their doom. The ocean contains more than three hundred million cubic miles of water. High-resolution sonar has mapped only a fraction of it. That mapping itself has only been a snapshot and not an ongoing movie. It is far from diligent search of the whole volume.

Search size also justifies patience in the search for extra-terrestrial intelligence. Satellites have only recently mapped the surface of Mars. Radio telescopes have searched only a minuscule fraction of the expanding universe for some form of structured energy in interstellar radio signals.

Good negative evidence that we are alone could take thousands or millions of years of diligent search. Positive evidence to the contrary can come at any second.