prior probability

Shout out to the Russian Ballet of Orlando for their whimsical production of the French fairy tale “Beauty and the Beast.” I attended many of the rehearsals (my daughter had a small role in the production) and noticed that ballet is like most human activities: a mixture of rigid discipline and individual creativity. Below are my beautiful daughter Adys Ann with her instructor (and lead ballerina) Miss Jeanna:

1. Common law. On the one hand, we could apply traditional common law doctrines like fraudulent misrepresentation and breach of contract. The common law approach generally consists of “simple rules,” to borrow Richard Epstein’s term, but for better or worse, the common law can get messy, since these rules are developed piecemeal and “bottom-up,” as judges decide individual controversies on a case-by-case basis. For my part, I develop a common law approach to data fraud in my 2017 paper “Legal Liability for Data Fraud” (available here).

2. Regulation. On the other hand, we could apply a regulatory framework or a rational “top-down” approach to this problem, such as mandatory disclosure rules, flat-out prohibition, or some form of Pigovian taxes. By way of example, our colleague Omri Ben-Shahar (pictured below), a law professor at the University of Chicago, who argues for a regulatory approach, compares harmful “data emissions” to environmental pollution. (Check out his fascinating paper here.) Such an approach has the advantage of promoting uniformity but is always open to the risk of regulatory capture and the law of unintended consequences. In short, since neither the common law or regulatory approach is perfect, the real question is, which approach is “less bad”?

That is the title of my review of Nate Oman’s excellent book “The Dignity of Commerce.” My essay was originally published in The New Rambler; a slightly revised version is now posted on ssrn, the first page of which is included below:

Which precursor to the Stars & Stripes do you like the best?

Happy Flag Day! (source: popchartlabs.com)

We’ve just read Frederick Schauer’s review of Randy Kozel’s excellent book “Settled Versus Right: A Theory of Precedent,” Cambridge University Press, 2017. (Schauer’s paper is provocatively titled “On treating unalike cases alike” and is posted on SSRN here. Hat tip: Larry Solum. As an aside, our own review of Kozel is available via SSRN here.) So, before we conclude our review of Ron Allen and Mike Pardo’s relative plausibility paper, we want to highlight one of our favorite passages from Professor Schauer’s review:

… precedential constraint … is not about treating like cases as alike. On the contrary, precedential constraint is about treating cases that are somewhat alike and somewhat different as being alike for purposes of precedential constraint. It is about treating the similarities as relevant and the differences as irrelevant, and about deciding which similarities matter and which do not. Thus, identifying what is a precedent for what is about attributing or ascribing likeness; and it is not about discovering, locating, or unearthing likeness. Determining precedential similarity … entails the question of what a decision-maker in the instant case deems to be similar, and not about what is actually similar in some deeper ontological sense.

Professor Schauer’s powerful critique of Aristotle’s axiom (see image below) is also worth noting. In fact, Schauer begins his review of Kozel thus: “Perhaps we should blame Aristotle. In his enduring discussion of justice in the Nicomachean Ethics, Aristotle offered the now-ubiquitous maxim that like cases should be treated alike” (footnote omitted). Despite the intuitive appeal of Aristotle’s axiom, Schauer frankly acknowledges “the almost complete emptiness of the ‘treat like cases alike’ maxim.” Why is this celebrated maxim empty, a mere tautology? Because as Schauer correctly notes, “Given that any two items in the world share some but not all of the properties of the respective items, any two items can be deemed alike in some respects and unalike in others, thus making the mere idea of likeness or unlikeness singularly unhelpful.” In short, we need some independent criterion of likeness “to make the maxim anything other than a largely useless tautology.”

So, what criterion (or criteria) should judges use when deciding whether two unalike cases are sufficiently similar to be treated alike? That is the $64 question, and your guess or theory is as good mine!

Credit: Shawn Copeland

Note: This is our penultimate post on Allen & Pardo’s paper “Relative plausibility and its critics.”

We defined subjective probability in terms of “degrees of belief” in our previous post. Here, we will consider Ron Allen and Mike Pardo’s relative plausibility theory in light of the Dutch book theorem, but first, let’s explore the connection between degrees of belief and betting behavior, for this connection is what subjective probability is all about and shows why subjective probability is more logical and mathematically rigorous than relative plausibility. (Like our previous post, we include a bibliography below.)

On the subjective probability view of proof, a juror’s decision is like a bet on which party is telling the truth about the facts of the case. So, let Pr(∏) be the probability that the plaintiff is presenting the truth about the facts of case A, and let Â be the probability space containing all possible outcomes of case A, e.g. jury verdicts going 12-0, 11-1, 10-2, etc., or jury verdicts going 6-0, 5-1, 3-2, etc., depending on the composition of the jury. (Recall that most verdicts are collective decisions in Anglo-American trials with 12-man or 6-man juries.)

By a bet on ∏ ∈ Â we mean a contract between a bettor and a bookie whereby the bookie agrees to pay the bettor the amount $a if the plaintiff prevails and the bettor agrees to pay the bookie the amount $b if the defendant prevails. The stake of this bet is the sum $(a + b), and bettor’s odds is the ratio a ⁄ b. If Pr is the bettor’s subjective degree of belief function, the expected monetary value of the bet to him is [$a × Pr(∏)] − [$b × Pr(1 − ∏)].

The bet is a fair one if the expected value of the bet is zero. The bet is favorable (to the bettor) if the expected value of the bet is positive. And the bet is unfavorable to him if the expected value of the bet is negative. In algebraic notation, the condition for a fair bet, i.e. the bettor’s fair betting quotient, comes to Pr(∏) = b/(a + b ). If Pr(∏) = x is your degree of belief in ∏, then you should be willing to bet on ∏ as follows:

According to the Dutch book theorem, if your degrees of belief fail to conform to the axioms of probability (i.e.: Pr(∏) + Pr(1 − ∏) = 1), then your net will always be negative. (For a formal proof of the Dutch Book Theorem, see Paris, 2001; Kemeny, 1955; Earman, 1992, p. 39.)

Why is the Dutch Book Theorem relevant to the literature on juridical proof? Because subjective beliefs that are logically coherent–i.e. that conform to the axioms of probability–are more likely to be accurate. (See Joyce, 2009; Williams, 2012.) The same, however, cannot be said for beliefs held under Allen and Pardo’s less demanding relative plausibility theory. Why not? Because the relative plausibility approach is susceptible the unknown probability space problem. Allen and Pardo not only “reject the assumption that the plaintiff’s explanation [in civil cases] must fill more than half of the possible probability space” (Allen & Pardo, 2018, p. 26); by their own admission, their theory of relative plausibility does not require the competing explanations of the evidence to add up to 1. Consider Allen & Pardo’s own preferred example, “where a [juror] concludes that a plaintiff’s explanation is 0.4 likely and a defendant’s explanation is 0.2 likely….” (Ibid.) Last we checked, 0.4 + 0.2 = 0.6, so what about the remaining 0.4 probability? It’s getting Dutch booked by creative trial lawyers!

To sum up, although numerical values are not needed to operationalize a subjective probability account of proof, one of the advantages of the subjective approach is that it is more logical and mathematically rigorous than Allen and Pardo’s relative plausibility theory. In addition, as we have mentioned in a previous post, the other major problem with Allen & Pardo’s approach is that they neglect the collective nature of jury voting. In our next post, our final one in this series, we will explain why the collective nature of jury voting lends itself to a subjective or Bayesian view of proof.

Bibliography

John Earman, Bayes or bust? A critical examination of Bayesian confirmation theory, MIT Press (1992).

James M. Joyce, “Accuracy and coherence: prospects for an alethic epistemology of partial belief,” in Franz Huber and Christoph Schmidt-Pierre, editors, Degrees of belief, Springer (2009), pp. 263-297.

Daniel Kahneman & Amos Tversky, “Subjective probability: a judgment of representativeness,” Cognitive Psychology, Vol. 3, no. 3 (1972), pp. 430-454.

G. Kemeny, “Fair bets and inductive probabilities,” Journal of Symbolic Logic, Vol. 20 (1955), pp. 263-273.

B. Paris, “A note on the Dutch book method,” in The Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing (2001), pp. 301-306.

Abner Shimony, “Coherence and the axioms of confirmation,” Journal of Symbolic Logic, Vol. 20 (1955), pp. 1-28.

Robert G. Williams, “Money and truth: Dutch books and accuracy domination,” Journal of Philosophical Logic, Vol. 41, No. 5 (October 2012), pp. 811-840.

Credit: Frederica Russo