EVIDENCE, PROBABILITY, AND THE BURDEN OF PROOF Ronald J. Allen* and Alex Stein** This Article analyzes the probabilistic and epistemological underpinnings of the burden of proof doctrine. We show that this doctrine is best understood as instructing factfinders to determine which of the parties’ conflicting stories makes most sense in terms of coherence, consilience, causality, and evidential coverage. By applying this method, factfinders should try—and will often succeed—to establish the truth, rather than a statistical surrogate of the truth, while securing the appropriate allocation of the risk of error. Descriptively, we argue that this understanding of the doctrine—the “relative plausibility theory”—corresponds to our courts’ practice. Prescriptively, we argue that the relative-plausibility method is operationally superior to factfinding that relies on mathematical probability. This method aligns with people’s natural reasoning and common sense, avoids paradoxes engendered by mathematical probability, and seamlessly integrates with the rules of substantive law that guide individuals’ primary conduct and determine liabilities and entitlements. We substantiate this claim by juxtaposing the extant doctrine against two recent contributions to evidence theory: Professor Louis Kaplow’s proposal that the burden of proof should be modified to track the statistical distributions of harms and benefits associated with relevant primary activities; and Professor Edward Cheng’s model that calls on factfinders to make their decisions by using numbers instead of words. Specifically, we demonstrate that both models suffer from serious conceptual problems and are not feasible operationally. The extant burden of proof doctrine, we conclude, works well and requires no far-reaching reforms. * John Henry Wigmore Professor of Law, Northwestern University School of Law. ** Professor of Law, Benjamin N. Cardozo School of Law, Yeshiva University. We thank Gideon Parchomovsky, Mike Pardo, and Richard Posner for helpful comments and suggestions. 558 ARIZONA LAW REVIEW [VOL. 55:557 TABLE OF CONTENTS INTRODUCTION ..................................................................................................... 558 I. THE NATURE OF THE BURDEN OF PROOF .......................................................... 565 A. Adjudicative Factfinding as Inference to the Best Explanation ................. 567 B. Justifying the Conventional Burden of Proof ............................................. 571 1. Two Modes of Factfinding ..................................................................... 571 2. Naturalism .............................................................................................. 575 3. Empirical Truth ...................................................................................... 577 II. EVIDENCE THRESHOLDS ................................................................................... 579 A. Do Evidence Thresholds Work?................................................................. 580 B. Evidence Thresholds and Bayes' Theorem ................................................. 584 C. Substantive Law and the Burden of Proof .................................................. 588 III. COMPARATIVE PROBABILITY .......................................................................... 594 A. Tinkering with Conjunctions ...................................................................... 594 B. Law, Science, and Probability .................................................................... 599 CONCLUSION ........................................................................................................ 602 INTRODUCTION Legal factfinding, like most real life decision-making, involves decision under uncertainty.1 Consequently, the legal system has adopted a set of decision rules to instruct judges and jurors how to decide cases in the face of uncertainty. These rules are collectively known as the burden of proof.2 They include the well- known requirement that all accusations against the defendant in criminal cases be proven “beyond a reasonable doubt.”3 For defenses that an otherwise guilty defendant may raise, the rules often require proof by a “preponderance of the evidence”4 or proof by “clear and convincing evidence.”5 In civil litigation, the burden of proof tends to treat plaintiffs and defendants as equals, normally requiring each party to prove her allegations—the plaintiff’s cause of action and the defendant’s affirmative defenses—by a “preponderance of the evidence.”6 For allegations of crime and fraud in civil cases, the proof burden is often set to “clear and convincing evidence”—a special proof requirement that also applies in proceedings that might deny a person certain civil rights, such as deportation, 1. See ALEX STEIN, FOUNDATIONS OF EVIDENCE LAW 34–36 (2005) (underscoring the inevitable presence of uncertainty in adjudicative factfinding). 2. See generally CHRISTOPHER B. MUELLER & LAIRD C. KIRKPATRICK, EVIDENCE §§ 3.1–3.3, 3.11–3.12, at 103–12, 134–42 (5th ed. 2012) (discussing civil and criminal burdens of proof). 3. Id. §§ 3.11–3.12 at 134–42. 4. Id. §§ 3.12 at 136–42. 5. See, e.g., 18 U.S.C. § 17(b) (2006) (“The defendant has the burden of proving the defense of insanity by clear and convincing evidence.”). 6. See MUELLER & KIRKPATRICK, supra note 2, § 3.3, at 111. 2013] BURDEN OF PROOF 559 denaturalization, involuntary confinement to a mental institution, and removal of parental rights.7 Some of these rules are entrenched in the Constitution;8 most are a matter of state policy. A defendant’s right to be acquitted when one or more elements of the crime are not proven beyond a reasonable doubt is part of his entitlement to “due process of law” under the Fifth and Fourteenth Amendments.9 The Due Process Clause also includes the “clear and convincing evidence” requirement for allegations that may lead to a denial of civil rights.10 The Ex Post Facto Clause does not allow the burden of proof—in criminal cases and with regard to statutory prohibitions that are not explicitly criminal but have a punitive intent—to be altered retroactively.11 Finally, the Erie doctrine (widely considered “quasi- constitutional”) gives the states precedence over Congress in setting up burdens of proof for diversity suits.12 Legal scholars have long recognized the centrality of the burden of proof and its effects on individuals’ entitlements and primary activities.13 This recognition led scholars to investigate the conceptual foundations of the burden of proof, as well as how it integrates into the factfinding process as a whole. Economically minded scholars have investigated the connections between the burden of proof, risk of error, primary behavior, and cost of litigation.14 Moral theorists, beginning with Immanuel Kant, have tried to identify the evidentiary 7. Id. § 3.3, at 112. 8. See Alex Stein, Constitutional Evidence Law, 61 VAND. L. REV. 65, 79–82 (2008) (attesting that the “proof beyond a reasonable doubt” requirement for criminal convictions and the “clear and convincing evidence” standard for allegations that justify deprivations of civil rights and liberties are mandated by due process). 9. Id. at 79–80. 10. Id. at 81–82. 11. Id. at 99–101. 12. Id. at 98–99. 13. See, e.g., Symposium on Presumptions and Burdens of Proof, 17 HARV. J. L. & PUB. POL’Y 613 (1994). 14. See Bruce L. Hay & Kathryn E. Spier, Burdens of Proof in Civil Litigation: An Economic Perspective, 26 J. LEGAL STUD. 413, 418–21 (1997) (analyzing burden of proof as an instrument for reducing the cost of litigation); Gideon Parchomovsky & Alex Stein, The Distortionary Effect of Evidence on Primary Behavior, 124 HARV. L. REV. 518, 530–42 (2010) (explaining people’s primary behavior as motivated by the burdens of proof and other evidentiary requirements); Richard A. Posner, An Economic Approach to the Law of Evidence, 51 STAN. L. REV. 1477, 1502–07 (1999) (unfolding economic analysis of the burden of proof as a tool for reducing the cost of errors and error-avoidance as a total sum); David Rosenberg, The Causal Connection in Mass Exposure Cases: A “Public Law” Vision of the Tort System, 97 HARV. L. REV. 849, 861–67 (1984) (carrying out economic analysis of the burden of proof and identifying the limits of the “preponderance” standard in tort cases with uncertain causation); Chris W. Sanchirico, Games, Information and Evidence Production: With Application to English Legal History, 2 AM. L. & ECON. REV. 342, 343–44 (2000) (unfolding an account of proof burdens that uses evidence production as a proxy for determining the harmfulness of primary behavior); Chris W. Sanchirico, Relying on the Information of Interested—and Potentially Dishonest—Parties, 3 AM. L. & ECON. REV. 320 (2001) (analyzing the proof burdens’ effect on primary behavior). 560 ARIZONA LAW REVIEW [VOL. 55:557 minimum that could justify an imposition of punishment or other deprivation on a person who may not have committed the alleged wrong.15 The body of literature produced by these scholars is rich, insightful, and multifaceted. This Article investigates the relationship between evidence, probability, and the burden of proof. We examine what factfinders do when they decide cases by applying the controlling proof burden. We demonstrate that factfinders decide cases predominantly by applying the relative plausibility criterion guided by inference to the best explanation, rather than by using mathematical probability.16 Indeed, we show that our courts apply mathematical probability only to a small number of well-defined categories of cases.17 We then evaluate this practice and commend it on the grounds of both pragmatism and principle. We show that the relative plausibility approach outperforms mathematical probability operationally and normatively. Application of mathematical probability in the courts of law engenders paradoxes and anomalies that are not easy to avoid or explain away. Relative plausibility, on the other hand, faces no such predicaments. A further advantage is its alignment with the natural reasoning of ordinary people, which reduces the cost of adjudication and helps the legal system guide individuals’ behavior. Last, but not least, relative plausibility is the best available tool to get factfinders to the actual facts of the case they are asked to resolve. Mathematical probability, on the other hand, abstracts away from those facts. As a substitute, it prods factfinders to derive their decisions from the general frequencies of events. We combine this discussion with our critique of the two most recent contributions to the burden of proof literature: Louis Kaplow’s radical proposal to revamp the burden of proof doctrine18 and Edward Cheng’s introduction of a new mathematical tool for factfinders’ use.19 Kaplow proposes a complete overhaul of the burden of proof doctrine, which he criticizes for having “almost nothing to do with what matters for society.”20 His analysis starts from the fundamental premise that, because certainty in factfinding is not within the legal system’s reach, the system should strive to achieve a socially optimal distribution of adjudicative errors: mistaken impositions 15. See Ernest J. Weinrib, Private Law and Public Right, 61 U. TORONTO L.J. 191, 210 (2011) (explaining Kant’s rationalization of the burden of proof as “an aspect of the defendant’s innate right to be considered beyond reproach in the absence of an act that wrongs another”). 16. For foundational articles on this subject, see Ronald J. Allen, A Reconceptualization of Civil Trials, 66 B.U. L. REV. 401, 403 (1986); Ronald J. Allen, Factual Ambiguity and a Theory of Evidence, 88 NW. U. L. REV. 604 (1994) [hereinafter Allen, Factual Ambiguity]; Ronald J. Allen, The Nature of Juridical Proof, 13 CARDOZO L. REV. 373 (1991). 17. See infra note 108 and accompanying text. 18. Louis Kaplow, Burden of Proof, 121 YALE L.J. 738 (2012). 19. Edward K. Cheng, Reconceptualizing the Burden of Proof, 122 YALE L.J. 1254, 1258–59 (2013). 20. Kaplow, supra note 18, at 789. 2013] BURDEN OF PROOF 561 of legal liability (“errors of commission”) and mistaken failures to impose legal liability (“errors of omission”). According to Kaplow, optimal distribution of those errors does not correlate with the extent to which courts’ decisions are accurate. As established in Kaplow’s previous work, accuracy ex post has no value in and of itself.21 Distribution of adjudicative errors—regardless of the accuracy rate it produces over a run of cases—thus ought to promote a different goal: It ought to incentivize ex ante socially optimal primary behavior. Consistent with this vision, Kaplow criticizes the burdens of persuasion that function as proof requirements under extant law: “preponderance,” “beyond a reasonable doubt,” and “clear and convincing evidence.”22 These probability standards, Kaplow argues, work to achieve accuracy ex post—an economically inefficient goal that our legal system ought to abandon.23 They ought to be replaced by a different legal mechanism that incentivizes socially desirable conduct ex ante. To implement his idea, Kaplow argues for the creation of what he calls “evidence thresholds.”24 This novel mechanism is the core insight of Kaplow’s normative theory. Evidence that goes into Kaplow’s thresholds informs courts about the effects of the relevant activity—harmful and socially useful, or “benign”25—across a series of cases. This evidence will associate different activities with different concentrations of harm and benefit. Some of those concentrations yield a negative tradeoff; others do not. Policymakers consequently will desire to suppress activities associated with the undesirable concentrations of harm versus benefit, while allowing other activities to take place. Policymakers can achieve this result by setting up rules that sanction the undesirable concentrations of harm versus benefit. Sanctions will follow according to a sliding scale of the probability in which the higher the predominance of harm in the mix, the lower the probability needed for liability; and conversely, the lower the risk of harm, the higher the probability needed. According to Kaplow, this myriad of rules should replace the conventional burden of proof doctrine.26 Edward Cheng recasts the burden of proof doctrine in terms of standard mathematical probability.27 Kaplow’s theory presupposes that the extant proof requirements—“preponderance,” “beyond a reasonable doubt,” and “clear and convincing”—have numerical equivalents on the probability scale between 0 and 1 and that courts associate these requirements with mathematical probability. Cheng does not accept this presupposition, and for a good reason: Courts generally do not 21. See Louis Kaplow, The Value of Accuracy in Adjudication: An Economic Analysis, 23 J. LEGAL STUD. 307 (1994). 22. Kaplow, supra note 18, at 742–44. 23. Id. at 784–89. 24. Id. at 756–62. 25. Kaplow uses the term “benign” and the awkward term “benignancy,” for which we substitute the more straightforward term “benefit” and its derivatives. 26. Kaplow, supra note 18, at 755–72. 27. Cheng, supra note 19, at 1259–65. 562 ARIZONA LAW REVIEW [VOL. 55:557 use mathematical probability in applying the burden of proof doctrine.28 Importantly, the prevalent academic opinion approves this practice: Most evidence scholars believe that adjudicative factfinding is fundamentally incompatible with mathematical probability.29 Mathematical probability sometimes allows policymakers to evaluate the overall performance of a rule or a set of rules and macromanage the legal system as a whole.30 Carrying this tool to the process of determining individual facts is broadly considered a bad idea.31 Cheng’s article undertakes to overturn this widely accepted “incompatibility thesis.”32 To discharge this task, Cheng develops a mathematical method that removes the problems that make “trial by mathematics” operationally nonfeasible and normatively unattractive.33 One of those problems—the most difficult one, in the eyes of many—is the “conjunction paradox.”34 Consider a breach-of-contract suit that needs to be proven by a preponderance of the evidence, denoted as a mathematical probability greater than 0.5. Assume that the plaintiff makes two mutually independent allegations: (1) The defendant and she contracted for delivery of goods and (2) the defendant breached the contract by not delivering the goods that he undertook to deliver. Assume further that the evidence the parties adduce indicates that each of these allegations has a 0.7 probability. The conventional understanding of the burden of proof doctrine holds that the court 28. See STEIN, supra note 1, at 238–39. 29. See William L. Twining & Alex Stein, Introduction to EVIDENCE AND PROOF in VOL. XI OF INTERNATIONAL LIBRARY OF ESSAYS IN LAW AND LEGAL THEORY xxi–xxiv (William L. Twining & Alex Stein, eds. 1992) (discussing the probability debate and underscoring the mismatch between mathematical probability and adjudicative factfinding); Symposium, BAYESIANISM AND JURIDICAL PROOF, in 1 INT. J. EVIDENCE & PROOF 253, 254– 360 (Ron Allen & Mike Redmayne eds., 1997) (debating the applicability of mathematical probability to adjudicative factfinding). 30. See, e.g., Alex Stein, Inefficient Evidence 1 (Benjamin N. Cardozo School of Law, Cardozo Legal Studies Faculty Research Paper No. 380, 2013), available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2199601 (using mathematical probability to explain and guide the legal system’s macromanagement of evidence). 31. See, e.g., L. JONATHAN COHEN, THE PROBABLE AND THE PROVABLE (1977) (unfolding a broad philosophical theory that identifies a fundamental misfit between mathematical probability and adjudicative factfinding); Ronald J. Allen, Rationality, Algorithms, and Juridical Proof: A Preliminary Inquiry, 1 INT. J. EVIDENCE & PROOF 254, 275 (1997) (specifying incompatibilities between mathematical probability and juridical proof, while underscoring the virtues of natural reasoning to the best explanation); Craig R. Callen, Notes on a Grand Illusion: Some Limits on the Use of Bayesian Theory in Evidence Law, 57 IND. L.J. 1, 2–3 (1982) (demonstrating that application of mathematical probability in courts of law requires factfinders to carry out unbearably complex calculus); Alex Stein, Judicial Fact-Finding and the Bayesian Method: The Case for Deeper Scepticism about their Combination, 1 INT’L. J. EVIDENCE & PROOF 25, 41 (1996) (demonstrating that the Bayesian approach to adjudicative factfinding that employs subjective probabilities is tautological). 32. See Cheng, supra note 19, at 1259–65. 33. Id. at 1258–62. 34. Id. at 1263–65. See also COHEN, supra note 31, at 58–61 (original statement of the conjunction paradox in evidence law). 2013] BURDEN OF PROOF 563 should rule in favor of the plaintiff, whose case is much stronger than the defendant’s. Under the conventional understanding of probability, however, the plaintiff’s case is actually weaker than the defendant’s. The combined probability of the plaintiff’s allegations against the defendant is 0.49 (0.7  0.7)—just below the preponderance threshold. The probability of the defendant’s claim “I made no contract with the plaintiff or, alternatively, committed no breach” is 0.51 (0.3 + 0.3 – 0.32). Hence, the defendant should prevail. This mathematical outcome contradicts legal doctrine and common sense, which is why it received the name “the conjunction paradox.”35 Evidence scholars, including us, have tried to resolve this paradox or somehow explain it away.36 Cheng’s article makes an important addition to these efforts by developing a novel method to avoid the paradox. This method shifts away from a categorical assessment of probability to a comparative assessment. If successful, it would refute the incompatibility thesis and vindicate trial by mathematics. Cheng argues that the preponderance requirement (along with all other probability thresholds incorporated in the burdens of proof) should be understood in comparative, rather than categorical, terms.37 Courts should compare the individual probabilities attaching to the plaintiff’s factual allegations and to the defendant’s story. This comparison will determine whose case is stronger. As Cheng explains, courts should proceed in the same way in which scientists choose between competing hypotheses.38 This decision-making framework will not allow the defendant in our breach of contract case to rely on the probability of the disjunctive scenario “I made no contract with the plaintiff, and if I did make it somehow, I did not breach it.” This scenario is counterfactual and hence does not form a hypothesis comparable with the plaintiff’s allegations of fact. The probabilities of the parties’ comparable factual allegations thus show 0.7 on the plaintiff’s side and 0.3 on the defendant’s side. This mathematical outcome aligns with the decision that factfinders would reach by applying the relative plausibility method.39 Cheng’s probabilistic account thus connects a mathematical approach to factfinding to the best present understanding of burdens of persuasion.40 Our critique of Kaplow’s theory is threefold. First, we show that his proposal cannot be adopted because of its enormous (essentially, infinite) informational costs. Second, Kaplow’s evidence thresholds are direct analogues of 35. See STEIN, supra note 1, at 49–50. 36. See Id. at 49–56; Ronald J. Allen & Sarah A. Jehl, Burdens of Persuasion in Civil Cases: Algorithms v. Explanations, 2003 MICH. ST. L. REV. 893, 944; Alex Stein, An Essay on Uncertainty and Fact-Finding in Civil Litigation, with Special Reference to Contract Cases, 48 U. TORONTO L.J. 299, 311–12 (1998) [hereinafter Stein, Uncertainty and Fact-Finding]; Alex Stein, Of Two Wrongs that Make a Right: Two Paradoxes of the Evidence Law and their Combined Economic Justification, 79 TEX. L. REV. 1199, 1199– 2000 (2001) [hereinafter Stein, Two Wrongs]; Ronald J. Allen, Book Review: Laudan, Stein, and the Limits of Theorizing About Juridical Proof, 29 L. & PHIL. 195, 225–26 (2010). 37. Cheng, supra note 19, at 1259–61. 38. Id. at 1257, 1276–77. 39. Id. at 1259–62. 40. Id. at 1259–65. 564 ARIZONA LAW REVIEW [VOL. 55:557 Bayesian likelihood ratios.41 Bayes’ Theorem shows that basing decisions upon likelihood ratios instead of the posterior probabilities that account for all relevant information is a mistake.42 Under the Bayesian framework, the optimal proof burden in any given context will derive from the desired ratio of false positives (“errors of commission”) and false negatives (“errors of omission”), although the formulation of that ratio is, as we discuss, complicated—much more so than Kaplow seems to realize. This formulation of the burden of proof explains and to a significant extent justifies the conventional view. Last, the conventional proof burdens track the substantive definitions of tort and criminal liability that require courts to base liability decisions on the actor’s ex ante information, but Kaplow paid no attention to those definitions. This omission has two implications. First, substantive definitions of liability—both civil and criminal—go far toward aligning courts’ applications of the conventional burdens of proof with the ex ante distributions of harm versus benefit. We show that taking this factor into consideration substantially vindicates the conventional approach to the burden of proof. The conventional burden of proof doctrine is more sophisticated and better aligned with efficiency than Kaplow believes it to be. Similarly, Kaplow’s theory abstractly categorizes individuals’ activities as harmful and beneficial without regard to the specific nature of the primary behavior. As a result, the theory does not distinguish between accidents, contract breaches, and crimes. The theory’s failure to address these harms separately misses an important—indeed pivotal—characteristic of our legal system. The system prescribes separate combinations of proof burdens and other rules for accidents, breaches of contract, and crimes. For liability flowing from accidents, the system constructs evidentiary rules that motivate prospective wrongdoers to base their conduct on the ex ante probability of causing harm. These rules include liability presumptions driven by regulatory statutes and probability-based recovery of tort compensation. Accident law thus may not need Kaplow’s evidence thresholds. Contracts plainly require no such thresholds either, as parties are generally best situated to design their own evidentiary mechanisms for resolving allegations of breach,43 which both substantive and procedural laws unequivocally permit.44 The conventional burden of proof functions in contract law as a mere default,45 which Kaplow does not (and cannot) criticize. 41. See infra Part II.B. Kaplow’s illustrations of how evidence thresholds are supposed to work strengthen this association. Kaplow, supra note 18, at 785–86. 42. For an explanation of Bayes’ Theorem, see Alex Stein, The Flawed Probabilistic Foundation of Law and Economics, 105 NW. U. L. REV. 199, 211–12 (2011), and sources cited therein. 43. See Robert E. Scott & George G. Triantis, Anticipating Litigation in Contract Design, 115 YALE L.J. 814, 814 (2006). 44. See Robert G. Bone, Party Rulemaking: Making Procedural Rules Through Party Choice, 90 TEX. L. REV. 1329, 1330 (2012); John W. Strong, Consensual Modifications of the Rules of Evidence: The Limits of Party Autonomy in an Adversary System, 80 NEB. L. REV. 159, 160 (2001). 45. See Stein, Uncertainty and Fact-Finding, supra note 36, at 341–44. 2013] BURDEN OF PROOF 565 Another example of the consequences of failing to attend to different forms of liability involves the criminal law. Criminal law aspires to optimal deterrence by adjusting applicable penalties while requiring the prosecution’s evidence to establish a very high posterior probability of guilt.46 This is a very sensible way to reduce crime while protecting innocents from wrongful conviction. As Gary Becker demonstrated long ago, penalty adjustments can achieve optimal deterrence more expediently and cost-effectively than adjustments of law enforcement.47 The reason is obvious. Enforcement efforts that require information are expensive: Indeed, Kaplow acknowledges that setting up his evidentiary thresholds is a costly exercise.48 Criminal penalties, on the other hand, can be set with a strike of a pen. Our critique of Cheng’s theory is straightforward. Cheng’s theory succeeds in developing a mathematical conceptualization of the burden of proof that avoids the conjunction paradox as it is presently understood. But that is all that it does. Critically, it ignores the consequences of the systematic suppression of the probabilities of opposite scenarios. Cheng also fails to explain why it would be good for society if our courts were to use his conceptualization instead of the conventional one. We address that very question in the pages ahead. We show that the conventional proof burden, conceptualized as inference to the best explanation, does a better job in promoting the fairness and efficiency of our legal system. Unlike the mathematical understanding of the proof burden, this conceptualization gives rise to no anomalies and paradoxes. Structurally, this Article unfolds as follows. In Part I, we explain how the conventional burden of proof doctrine works and how it promotes efficiency and fairness. In Parts II and III, respectively, we analyze and criticize Kaplow’s and Cheng’s theories of the burden of proof. A short Conclusion ensues. I. THE NATURE OF THE BURDEN OF PROOF Burdens of proof easily bear a probabilistic interpretation. In civil cases, the standard instruction tells jurors that each element of a claim and of an affirmative defense must be established by a preponderance of the evidence, where “preponderance” means more likely than not.49 This formulation of the proof burden leads directly to the probabilistic interpretation of greater than a 0.5 probability.50 In criminal cases, the “beyond a reasonable doubt” instruction decidedly avoids asking jurors to quantify their doubts concerning the defendant’s guilt. Asking jurors to do so is tantamount to asking them to sacrifice a number of innocents in order to allow the criminal justice system to convict and punish a 46. See, e.g., Richard A. Bierschbach & Alex Stein, Mediating Rules in Criminal Law, 93 VA. L. REV. 1197, 1210–12 (2007) (explaining and citing literature as to what optimal deterrence in criminal law requires). 47. See Gary S. Becker, Crime and Punishment: An Economic Approach, 76 J. POL. ECON. 169, 180–84 (1968). 48. See Kaplow, supra note 18, at 771, 786–89. 49. See MUELLER & KIRKPATRICK, supra note 2, § 3.3, at 111–12. 50. See, e.g., Allen & Jehl, supra note 36, at 894–95. 566 ARIZONA LAW REVIEW [VOL. 55:557 sufficient number of guilty offenders.51 Despite this operational difficulty, mathematical probability can give meaning to the criminal proof burden as well; the same is true for “clear and convincing” evidence.52 The significant questions here are whether any of these reconceptualizations are empirically accurate or normatively attractive as a potential improvement of our legal system. Our answer to both questions is no. Scholars’ attempts at mathematizing the burden of proof follow a frequentist interpretation of probability,53 and for good reason. Other interpretations of the concept of “probability”—logical, propensity, and subjective beliefs54—make no sense at all in the juridical context.55 The frequentist account of probability, however, does not do much better. Courts resort to frequentist probability in some very specific contexts.56 Outside these contexts, frequentist probability is of no use. Pragmatism and substance drive our courts’ general rejection of this probability.57 Courts have no information about the relative 51. See generally Alexander Volokh, n Guilty Men, 146 U. PA. L. REV. 173, 198 (1997). 52. Probability thresholds for these burdens can be set at any appropriate level, for example: 0.95 (“beyond a reasonable doubt”) and 0.75 (“clear and convincing evidence”). 53. Frequentist probability is a system of reasoning that associates an event’s chances of occurring with instantial multiplicity. Under this system, an event’s chances of occurring are favorable when it falls into the majority of the observed events. Conversely, an event’s chances of occurring are not favorable when it falls into the minority of the observed events. An event’s probability consequently equals the number of cases in which it occurred divided by the totality of relevant cases. See L. JONATHAN COHEN, AN INTRODUCTION TO THE PHILOSOPHY OF INDUCTION AND PROBABILITY 47–48 (1989); see also STEIN, supra note 1, at 143–48 (discussing mathematical approaches to the burden of proof and their uniform reliance on frequentist probability). 54. See generally DONALD GILLIES, PHILOSOPHICAL THEORIES OF PROBABILITY 1 (2000) (explaining different versions of probability); COHEN, supra note 53, at 53–80 (analyzing logical, propensity-based, and subjectivist interpretations of “probability” and explaining their limitations). 55. See Michael S. Pardo & Ronald J. Allen, Juridical Proof and the Best Explanation, 27 L. & PHIL. 223, 227–38 (2008); Alex Stein, Bayesioskepticism Justified, 1 INT. J. EVIDENCE & PROOF 339, 342 (1997) (formal demonstration of circularity and self- reference that plague the subjectivist version of probability as applied in juridical context); Stein, supra note 31, at 41 (rejecting the subjective-belief version of probability as tautological). 56. See infra note 108 and accompanying text. 57. See United States v. Shonubi, 998 F.2d 84 (2d Cir. 1993) (reversing a lower court decision in United States v. Shonubi, 802 F. Supp. 859, 860–64 (E.D.N.Y. 1992), that used mathematical probability to determine a fact aggravating the defendant’s crime and sentence); Stein, Two Wrongs, supra note 36, at 1204 n.6, 1205 (citing different jury instructions that run contrary to mathematical probability); see also Ronald J. Allen & Michael S. Pardo, The Problematic Value of Mathematical Models of Evidence, 36 J. LEGAL STUD. 107, 130–35 (2007) (rationalizing the Second Circuit’s reversal of the trial judge’s decision in Shonubi by the judge’s failure to carve out the relevant reference class). Cf. United States v. Veysey, 334 F.3d 600, 604–06 (7th Cir. 2003), cert. denied 540 U.S. 1129 (2004) (approving defendant’s arson conviction based on actuarial testimony estimating that