Reconceptualizing the burden of proof.

AuthorCheng, Edward K.

ESSAY CONTENTS INTRODUCTION I. COMPARISONS, NOT ABSOLUTES A. Explaining the 0.5 Standard B. Resolving the Conjunction Paradox C. Story Definition II. BAYESIAN HYPOTHESIS TESTING A. Resolving the Blue Bus and Gatecrasher Paradoxes B. The Puzzle of Epidemiology III. OPTIMALITY IV. AN EXTENSION TO CRIMINAL CASES CONCLUSION

INTRODUCTION As every first-year law student knows, the civil preponderance-of-the-evidence standard requires that a plaintiff establish the probability of her claim to greater than o.5. (1) By comparison, the criminal beyond-a-reasonable-doubt standard is akin to a probability greater than 0.9 or 0.95? Perhaps, as most courts have ruled, the prosecution is not allowed to quantify "reasonable doubt," (3) but that is only an odd quirk of the math-phobic legal system. We all know what is really going on with burdens of proof, especially with respect to 0.5.

But are these time-honored quantification moves actually correct? Is preponderance really p > 0.5 and beyond a reasonable doubt really p > 0.95? One need not dig too deeply to find immediate problems. Take, for example, the so-called Conjunction Paradox, which has long bedeviled legal scholars attempting to place the process of proof on probabilistic foundations. (4) Assume that a court is faced with a conventional negligence claim in which the plaintiff seeks to prove that: (A) the defendant was driving negligently; (B) the defendant's negligence caused him to crash into the plaintiff; and (C) the plaintiff suffered a soft-tissue neck injury as a result. Assume further that through the trial process, the plaintiff makes out each of these elements to a probability of 0.6. Should the plaintiff win? Each of the elements surely meets the preponderance standard; they all exceed 0.5. However, if all three elements are independent, their conjunction (ABC) has a probability of o.6 * 0.6 * o.6, or 0.216, suggesting that the plaintiff should lose. Even if the elements are not independent, their conjunction is always mathematically less than 0.6, so that with each additional element, the plaintiff finds it increasingly difficult to win. (5)

These types of problems present serious and fundamental impediments to scholars hoping to articulate a probabilistic theory of evidence. (6) They arguably even inhibit attempts to use probability and statistics to improve legal decisionmaking. After all, as it currently stands, the mathematics do not adequately model the legal system in operation. Along these lines, Ron Allen and Mike Pardo, among others, have argued that the legal system does not engage in this type of probabilistic reasoning at all, but instead proceeds through abductive reasoning, also known as inference to the best explanation. (7) Consistent with the story model of jury decisionmaking made famous by Nancy Pennington and Reid Hastie, (8) Allen and Pardo suggest that jurors choose the best explanation for the evidence with which they are presented. They do not accumulate evidence through conventional probability models.

But how could this state of affairs possibly be? On the one hand, probabilistic models of inference have been incredibly successful in science, leading to dramatic insights and findings into the way the world works. On the other hand, inference to the best explanation is compelling and intuitively correct to any lawyer. From law school on, lawyers learn that presenting a sagaciously chosen core theory (in appellate argument) or telling a compelling story (in trial argument) is critical to legal success. (9) Is legal factfinding simply different from scientific factfinding?

In this Essay, I argue that the answer to this question is in fact no. The use of probabilistic tools and the story model are not as antithetical as they may first appear. Indeed, the problem is neither in the use of probabilistic reasoning, nor in the use of a story model, but rather in the legal system's casual recharacterization of the burden of proof into p > 0.5 and p > 0.95. Indeed, once we recognize that mistake, we can construct a model of legal decisionmaking that is both compatible with the story model and potentially based on probabilities. As proponents of the story model have long argued, the legal system does not ask decisionmakers to determine whether litigants have established their cases to a particular level of certainty. Instead, decisionmakers compare the stories or theories put forward by the parties, and determine which story is more compelling in light of the evidence. However, far from calling into doubt the viability of probabilistic theories of evidence, this comparative procedure is found at the heart of standard methods of hypothesis testing in statistics. To make the two harmonize, evidence scholars need only let go of their love for p > 0.5.

The discussion proceeds as follows. In Part I, the Essay reconceptualizes the preponderance standard. It proposes viewing preponderance not as an absolute probability, such as 0.5, but rather as a ratio test that compares the probability of the narratives offered by the plaintiff and defendant. With a probability ratio test in hand later Sections show how the 0.5 standard came to be-essentially as an oversimplification-and how the ratio test avoids the Conjunction Paradox.

Part II pushes further on the reconceptualized preponderance standard by employing a Bayesian perspective. This Bayesian perspective offers a method of incorporating evidence into the decisionmaking process, and it provides an explanation for the Blue Bus problem famous in statistical proof circles. Part III critiques the reconceptualized preponderance standard on normative grounds, departing from the otherwise explanatory goals of the Essay. As it turns out, the preponderance test as implemented by the legal system neglects base rates, which may explain the base rate problem's frustrating persistence. Part IV tentatively extends the ideas from the preceding Parts into the criminal context, and a Conclusion follows.

  1. COMPARISONS, NOT ABSOLUTES

    Conventional legal thinking equates the preponderance standard in civil litigation with a requirement that the plaintiff prove her case to a probability greater than 0.5. This Part argues that this characterization is wrong. Because the adversarial structure of legal trials promotes jury comparisons of the parties' claims, preponderance is not an absolute probability. Rather, the preponderance standard is better characterized as a probability ratio, in which the probability of the plaintiffs story of the case is compared with the defendant's story of the case. Indeed, while one can technically derive the p > 0.5 standard from the ratio, it involves assumptions sharply at odds with current legal practice.

    Looking at the statistical world, we immediately see that characterizing any decision rule as a 0.5 probability threshold is odd. Statisticians rarely attempt to prove the truth of a proposition or hypothesis by using its absolute probability. Instead, hypothesis testing is usually comparative. (11) There is a null hypothesis and an alternative hypothesis, and one is rejected in favor of the other depending on the evidence observed and the consistency of that evidence with the two hypotheses. (12)

    If one were to model the preponderance standard statistically, the natural move would therefore not be a 0.5 probability threshold. Rather, following standard decision theory,, it might look something like this (13): We start with two competing views of the world-for example, that the average height of an adult male is either 5'9" or 6'2" or, more generally for our purposes, that either the defendant's story ([H.sub.[DELTA]]) or the plaintiff's story ([H.sub.[pi]]) is true. Depending on the actual state of the world, concluding that the world is either [H.sub.[DELTA]] or [H.sub.[pi]] has error costs, as depicted in Figure 1. If the conclusion matches the truth, then obviously there is no error cost. However, if we conclude [H.sub.[pi]] but the world is actually [H.sub.[DELTA]], we incur the costs of c,. For the reverse, we incur the costs of [c.sup.2]. (14)

    Our goal is to construct a decision rule that minimizes our expected error costs. Let [q.sub.[DELTA]] and [q.sub.[pi]] represent the probabilities at which the states of the world HA and [H.sub.[pi]] occur, respectively. We can then make some useful calculations about expected costs. For example, whenever we choose [H.sub.[DELTA]], our expected costs will be:

    o * [q.sub.[DELTA]] + [ c .sub.2] * [q.sub.[pi]]

    Similarly, if we choose [H.sub.[pi]]

    [c.sub.1] * [q.sub.[DELTA]] + o. [q.sub.[pi]]

    To minimize the expected costs, we will choose [H.sub.[pi]] if its expected costs are lower than those of [H.sub.[DELTA]], or in other words if:

    [c.sub.2] * [q.sub.[pi]] > [C.sub.1] * [q.sub.[DELTA]]

    or equivalently, if:

    Equation (1).

    [q.sub.r]/[q.sub.[DELTA]] > [C.sub.1] [C.sub.2]

    Now, how do we determine these values? To estimate the probabilities [q.sub.[pi]] and [q.sub.[DELTA]], we use all of the evidence: P([H.sub.[pi]] | E) and P([H.sub.[DELTA]] | E), respectively, where E represents all of the available (or presented) evidence. At the same time, in a civil trial, the legal system expresses no preference between finding erroneously for the plaintiff (false positives) and finding erroneously for the defendant (false negatives). The costs [c.sub.1], and [c.sub.2] are thus equal, resulting in the decision rule that plaintiff wins if and only if:

    Equation (2).

    P([H.sub.[pi]] | E)/ P([H.sub.[DELTA]] | E) > 1

    1. Explaining the 0.5 Standard

      If the preponderance standard is actually a probability ratio, then where does the 0.5 number come from? After all, 0.5 seems awfully intuitive, which is arguably why it is so popular among lawyers. As it turns out, 0.5 arises from an error in assumptions.

      Assume that the defendant's theory of the case is merely that the "plaintiff's theory is false." In the set of all possible stories of what happened...

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