## Absence of Evidence, Evidence of Absence, and Fossils

In this post I’ll summarize Elliott Sober’s probabilistic reasoning on the topic of fossils evidence.

Evolutionary biologists often take the observing of intermediate fossils to be evidence for common ancestry while the failing to observe an intermediate fossil isn’t evidence against—this is explained away by imperfections in the fossil record. Creationists often say the opposite: failing to observe an intermediate fossil is evidence against common ancestry while the observing of an intermediate fossil isn’t evidence for. Elliott Sober says both sides are wrong.

We typically say that absence of evidence is evidence of absence when we would expect to see the evidence given a particular hypothesis; and, absence of evidence isn’t evidence of absence when we wouldn’t expect to see the evidence given a particular hypothesis. Elliott Sober considers this idea when the expectation of evidence is probabilistic rather than determinate.  More specifically, when does absence of evidence provide negligible evidence for absence and when does it provide substantial evidence for absence?

To consider this, he uses the probabilistic tool of the Law of Likelihood. The Law of Likelihood has a qualitative component and quantitative component.

• Qualitative component: evidence E favors hypothesis H1 over hypothesis H2 when Pr(E│H1) > Pr(E│H2).
• Quantitative component: The degree to which E favors H1 over H2 is given by the ratio of likelihoods Pr(E│H1)/Pr(E│H2).

This is very much in line with Bayesianism, since on Bayesianism the ratio of posteriors would be equivalent to the ratio of priors unless the ratio of likelihoods is different than 1.

$\frac{Pr(H1|E)}{Pr(H2|E)} = \frac{Pr(E|H1)}{Pr(E|H2)} \times \frac{Pr(H1)}{Pr(H2)}$

Or, equivalently, ratio of posteriors = ratio of likelihoods x ratio of priors.

Given common ancestry (CA), the probability is 1 that an intermediate organism exists between two species X and Y (assuming gradualism). Given separate ancestry (SA), it’s possible but not necessary that an intermediate exists between two species X and Y. Since the hypothesis of separate ancestry isn’t committed to evolutionary stasis—where traits remain unchanged through generations—there is a chance q<1 that traits changed and an intermediate exists.

 Pr(intermediate organism exists|CA)=1 Pr(intermediate organism exists|SA)=q Pr(no intermediate organism|CA)=0 Pr(no intermediate organism|SA)=1-q

The quantitative component of the Law of Likelihood says that if an intermediate exists, then this evidence favors common ancestry over separate ancestry by 1/q, which will be greater than 1. If an intermediate doesn’t exist then this evidence favors separate ancestry by (1-q)/0 = ∞. The quantitative aspect shows that the non-existence on an intermediate is far more evidentially significant than the existence of an intermediate, since a non-existent intermediate means the common ancestry hypothesis is impossible.

Let us now consider whether evolutionary biologists are right when they appeal to the imperfections of the fossil record to say that the absence of observing intermediate fossils isn’t evidence against common ancestry. (Imperfections in the fossil record are due to various things like the rarity of fossil formation and the difficulty of finding fossils.) So far our likelihood calculations were based on the existence or non-existence of intermediate organisms. For this next task we need to consider not the existence of intermediate organism but our failing to observe an intermediate fossil. To do this, Sober introduces a new quantity:

a = Pr(we have observed an intermediate │CA & there exists an intermediate) = Pr(we have observed an intermediate │SA & there exists an intermediate).

(The assumption that the two quantities are equal unfairly boosts the SA hypothesis since it’s more likely that we should observe an intermediate on CA.  It won’t matter for our purposes since the constant a is stipulated to prove a point.)

 Pr(we have observed an intermediate|CA)=a Pr(we have observed an intermediate|SA)=qa Pr(we have not|CA)=1-a Pr(we have not|SA)=1-qa

Observing an intermediate fossil favors CA over SA by 1/q—the same result we had when we considered how the evidence favors CA over SA given the existence of intermediate organisms. But, if we don’t observe an intermediate fossil, SA is favored over CA by (1-qa)/(1-a), which will be greater than 1. What this means is that it’s wrong to say that absence of observed fossils doesn’t favor SA over CA.

This qualitative result should be unsurprising because if follows from the axioms of probability that Pr(Obs|CA) > Pr(Obs|SA) if and only if Pr(not Obs|CA) < Pr(not Obs|SA). Or, given the Law of Likelihood, if observing an intermediate favors CA over SA, then failing to observe an intermediate favors SA over CA. It should be noted that favors is a technical term for Likelihoodists.

The next question is quantitative: When does observing a fossil favor CA over SA more strongly than failing to observe a fossil favors SA over CA? In other words, when is 1/q > (1-qa)/(1-a)? It turns out, in most cases, observing a fossil favors CA over SA more strongly than failing to observe a fossil favors SA over CA. I’ll skip over the math as you can see Sober’s paper to see it in detail.

The grey area shows when observing an intermediate favors CA over SA more strongly than failing to observe an intermediate favors SA over CA. Recall q = Pr(intermediate organism exists|SA), and a = Pr(we have observed an intermediate │CA & there exists an intermediate) = Pr(we have observed an intermediate │SA & there exists an intermediate).

As you can see, when a is low–meaning the probability of observing an intermediate fossil is low–observing an intermediate favors CA over SA more strongly than failing to observe an intermediate favors SA over CA.  It’s the low probability of the observation that makes observing an intermediate favor CA over SA more strongly than failing to observe an intermediate favor SA over CA.

Conclusion: While failing to observe an intermediate fossil is evidence against common ancestry, it is negligible compared to the evidence for common ancestry when we do find an intermediate fossil, when the probability of observing a fossil is low.

Sober, Elliott. Absence of Evidence and Evidence of Absence: Evidential Transitivity in connection with Fossils, Fishing, Fine-Tuning, and Firing Squads.