Cumulative Evidence

As a measure of evidential strength, the Bayes factor helps us to understand the cumulative weight of multiple lines of evidence. It’s tempting to suppose that the combined strength of several pieces of evidence is just the sum of their individual strengths. We tend to think, intuitively, of “adding up” the weights of various pieces of evidence, and this naïve intuition even shows up in common expressions of the English language. Try searching Google for the phrase “add up the evidence,” and it will return hundreds of thousands of results!I ran a search for that phrase on February 22, 2025. Google returned approximately 788,000 results. That intuition, however, is mistaken. When two or more evidential propositions are probabilistically independent, the Bayes factor of the combined evidence is the product—not the sum—of their individual Bayes factors. For example, if two independent pieces of evidence E₁ and E₂ have Bayes factors of 5 and 20, respectively (relative to some hypothesis H), then the conjunction (E₁•E₂) has a Bayes factor of 100. In other words, the strength of the combined evidence is 5 times 20, not 5 plus 20.

The same applies in cases of mixed evidence—that is, cases where we have some evidence for a hypothesis H and some evidence against it. Recall that a proposition E provides evidence for H if its Bayes factor relative to H is greater than 1, and it provides evidence against H if its Bayes factor is less than 1. (For instance, a Bayes factor of 10 means E is evidence for H, but a Bayes factor of 1/10^th would be equally strong evidence against H.) Provided each piece of evidence is independent of the others, we simply multiply their Bayes factors together to calculate the total strength of the evidence.

Suppose propositions A, B, and C are probabilistically independent. Proposition A provides weak evidence for hypothesis H, with a Bayes factor of 5. Proposition B provides equally weak evidence against H, with a Bayes factor of ⅕. Proposition C provides relatively strong evidence for H, with a Bayes factor of 100. What is the combined strength of all three evidential propositions taken together? The Bayes factor of the combined evidence is:

5 × ⅕ × 100 = 100

Now, what if we gain one additional piece of independent evidence D, with a Bayes factor of only 2? We might be tempted to dismiss this additional bit of evidence as unimportant, since it is extremely weak compared to the other evidence. That would be a huge mistake! If D is independent of A, B, and C, it will increase the total strength of the evidence from a Bayes factor of 100 to a Bayes factor of 200:

5 × ⅕ × 100 × 2 = 200

The seemingly insignificant Bayes factor of 2 actually doubles the total strength of the evidence!

As illustrated in the preceding example, even weak evidence makes a massive difference if it’s independent of other evidence you have. Independent evidence with a Bayes factor of merely 2, for instance, means that the strength of your total evidence is double what it would be otherwise!

Because the cumulative effect of independent evidence is multiplicative (rather than additive) in this way, several pieces of weak evidence can constitute extremely strong evidence when combined. For instance, if nine pieces of independent evidence each have a Bayes factor of 10, their combined strength is 10⁹, that is, a billion! Thus, the combined strength of independent evidence increases exponentially with each new piece of evidence. This is the crucial insight behind so-called cumulative case arguments in philosophy, where numerous lines of relatively weak evidence are aggregated to support a conclusion. Multiple kinds of weak evidence can accumulate to yield a powerful case, provided each bit of evidence is probabilistically independent of the others.

The situation is more complicated when different lines of evidence are probabilistically dependent—that is, when one evidential proposition affects the probability of another. When pieces of evidence affect each other’s probabilities, their combined strength can’t be calculated simply by multiplying their individual Bayes factors together. In such cases, the total strength of the evidence they provide for any given hypothesis depends on what other hypotheses are plausible. Regardless of whether several pieces of evidence are independent or not, their conjunction may strongly support one hypothesis by ruling out alternative hypotheses. The following example illustrates this possibility.

Let’s modify the police detective example discussed in the previous chapter. As before, you are considering three plausible hypotheses: the mysterious death was due to either an accident, suicide, or murder. This time, however, you discover a suicide note on the premises. This effectively rules out the accident hypothesis, reducing its probability nearly to zero.As stated in the original example, you consider a note extremely unlikely to turn up if the death was an accident—so unlikely, in fact, that you can safely disregard the possibility. Thus, the “accident and note” conjunction (A•N) occupies a negligibly tiny sliver of your prior Bayes bar. In contrast, the (S•N) and (M•N) conjunctions each have significant prior probabilities. So, conditionalizing on N will raise the probabilities of those other hypotheses at the expense of the accident hypothesis, leaving the accident hypothesis with almost no probability. However, you suspect the note may have been forged by a cunning murderer to mislead your investigation. You take the note to a handwriting expert, who confirms your suspicion: the handwriting on the note doesn’t match the victim’s.

Obviously, this second bit of evidence is not probabilistically independent of the first. The proposition about the note’s handwriting clearly depends on the fact that there was a note in the first place! Nevertheless, this new line of evidence greatly strengthens your confidence in the murder hypothesis, by reducing the probability of suicide nearly to zero. Although the two pieces of evidence were not probabilistically independent, their conjunction nonetheless provides much stronger evidence than the first piece of evidence (the existence of the note) alone. The reason is that each successive piece of evidence eliminates a different rival hypothesis. First, the discovery of the note ruled out the accident hypothesis. Then, the discovery of the mismatched handwriting ruled out the suicide hypothesis. Thus, even though these two lines of evidence were not independent, their conjunction still yields extremely strong support for the murder hypothesis.

This last example illustrates the close relationship between Bayesian confirmation theory and inference to the best explanation (IBE). As we saw in the previous chapter,On the page titled Surprising Implications of Bayesianism, I explained how “Bayesianism reveals that evaluating the predictions of a single hypothesis or theory, in isolation, is the wrong way to think about confirmation. Even if a hypothesis makes some observation unlikely, it does not follow that the observation disconfirms the hypothesis. It might, in fact, support the hypothesis over its rivals. Conversely, even if a theory makes a successful prediction, it doesn’t follow that this success confirms the theory. It may even disconfirm the theory by supporting alternative theories. Bayesianism reveals that the strength of evidence for a hypothesis H doesn’t depend merely on what H predicts, or what we expect if H is true. It also depends, crucially, on what we expect for each of the alternative hypotheses we consider plausible.

The connection between Bayesian reasoning and IBE illuminates the importance of evidential diversity. A diverse body of evidence—consisting of many different types of evidence—typically provides stronger support for a hypothesis than repeated instances of the same type of evidence. This is a key insight behind the methodology of robustness analysis frequently employed in experimental science. An experimental result is said to be robust if it can be replicated under a variety of different experimental conditions, indicating that it is probably not an artifact of a particular experimental setup or specific means of detection. Robust results provide stronger evidence for the hypothesis being tested, because they rule out numerous rival hypotheses, such as the hypothesis that the result is an artifact of malfunctioning equipment.For further discussion of evidential diversity, robustness analysis, and way in which Bayesianism sheds light on these concepts, see Jonah N. Schupbach, Bayesianism and Scientific Reasoning (Cambridge: Cambridge University Press, 2022), Chapter 3, Section 3.3 (pp. 84-89).

Since Bayesian reasoning typically involves comparisons between rival hypotheses, Bayesian confirmation theory can be seen as a sophisticated framework for making inferences to the best explanation. The mathematical tools of Bayesianism help to clarify the reasoning involved in IBE. These tools even provide a way to evaluate the cumulative strength of the evidence for each of the candidate explanations.