## The conjunction problem is not a problem for Bayesians

Note: This post is part 6 of our ongoing review of Allen & Pardo’s explanatory account of juridical proof.

We have been reviewing Ron Allen and Mike Pardo’s excellent paper, “Relative plausibility and its critics.” We agree with their critique of objective probability in law, so in the next few posts, we will restate and answer, point by point, their objections to subjective or Bayesian probability. (In the process, we hope to show why our Bayesian view of proof provides a better explanation of juridical proof than Allen and Pardo’s relative plausibility theory.) Here, we will turn our attention to the so-called conjunction problem.

Allen and Pardo restate this proof paradox on p. 11 of their paper (emphasis added; footnote omitted): “in a two-element claim …, if the plaintiff proves each to 0.6, then the plaintiff wins [under the 0.5 preponderance standard]. However, if these two elements are independent of each other, then the probability of plaintiff’s claim is 0.36, not greater than 0.5.” (The conjunction problem is also described in Friedman, 1997, p. 280.) In particular, the problem here is this: should the standards of proof used in legal trials (e.g., preponderance; clear and convincing; reasonable doubt) be applied to the individual elements of the moving party’s claims or to each claim as a whole (i.e. to the conjunction or totality of all the elements of each claim)?

Before proceeding, however, it’s worth noting that the conjunction problem is not the only “proof paradox” discussed in the evidence literature. Perennial puzzles like the “paradox of the gatecrasher” and the “blue bus hypothetical” are supposed to debunk a purely probabilistic view of proof. But as we have explained in a previous post and in our paper “Visualizing Probabilistic Proof,” published in The Washington University Jurisprudence Review, Vol. 7 (2014), pp. 39-75, these anomalies or puzzles are relatively easy to solve using Bayesian methods. We won’t go into the details of the stylized gatecrasher and blue bus scenarios here, however, in order to focus on the conjunction problem.

In fact, it turns out that subjective Bayesians need not lose any sleep over the conjunction problem either. (To follow along with our argument, see Richard D. Friedman’s paper, “Answering the bayesioskeptical challenge,” published in The International Journal of Evidence and Proof, Vol. 1 (1997), pp. 276-291.) To begin with, the problem only arises if the elements of a case are independent events. (From a probabilistic perspective, two events are independent of each other when the probability that one event occurs in no way affects the probability of the other event occurring. Say, for example, you roll a die and then flip a coin. Those are two independent events.) But this condition will rarely occur in legal trials, since the same set of facts will be relevant to most or all of the elements of a claim or defense.

Furthermore, even if the elements of a given claim were totally independent events, jurors will consider these elements in a holistic or cumulative manner, i.e. in light of the evidence as a whole and in light of the stories and explanations presented by the parties at trial. Indeed, Allen and Pardo themselves concede that jurors reason with evidence holistically; we would only add that such holistic or cumulative reasoning will occur regardless of whether jurors are assessing the “relative plausibility” of the parties’ stories or whether they are updating their subjective degrees of belief in the truth of these stories. The main question, then, is whether a Bayesian theory of proof is superior to Allen and Pardo’s relative plausibility or explanatory account? (We will return to this key question in a moment.) To illustrate the cumulative or holistic nature of fact-finding in law (and to see why the elements of a case are rarely independent), consider the following thought experiment in Friedman, 1997, p. 284 n. 17: a sequential trial in which one legal element (element A) of the plaintiff’s case is tried before the others (elements B, C, and D). In this element-by-element scenario, the plaintiff will present his evidence and tell his side of the story at the trial of element A. If the fact-finder finds for the plaintiff on element A, then the plaintiff will most likely represent the same pieces of evidence and tell the same story again to prove the next required element at the trial of element B, and so on.

Now, let’s return to the conjunction problem. For Bayesians, this proof paradox is a chimera. As Professor Richard Friedman has shown (Friedman, 1997, pp. 279-284), jurors are fully capable of assigning their subjective probabilities to the entire body of evidence presented at trial, just as they are fully capable of determining which explanation of the evidence is the most plausible, independent of the number of elements in each claim or defense. The real question, then, is this: which of the two theories of juridical proof–i.e. subjective probability or relative plausibility–makes more sense, either as a descriptive or normative manner? As we have shown in many previous posts, and as Allen and Pardo themselves concede in their paper, “relative plausibility” is just a poor man’s probability theory. Moreover, even if Allen and Pardo’s explanatory account were true, the process of fact-finding and of weighing the relative plausibility, i.e. relative probability, of each side’s explanations is itself a subjective process!!! (Cf. Friedman, 1997, p. 277.) Rest assured, we will have a lot more to say about the subjective nature of juridical proof next week …

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