prior probability

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

Note: This post is part 8 of our review of Ron Allen & Mike Pardo’s paper “Relative plausibility and its critics.” If you are already familiar with the subjectivist ideas of Frank Ramsey (1931) and Bruno de Finetti (1974), you can skip today’s post.

We have spent a significant amount of time and mental energy comparing and contrasting our Bayesian or “subjective probability” view of juridical proof with Ron Allen and Mike Pardo’s alternative theory of relative plausibility, but we just realized that we haven’t yet defined what we mean by subjective probability. So, before we address Allen and Pardo’s final objection to subjective probability, we thought we would take a moment to explain subjective probability, especially for those of you who may not be familiar with the work of our intellectual heroes Frank Ramsey or Bruno de Finetti. (Because this post is somewhat technical, we include a formal bibliography below.) Continue reading →

Note: This post is part 7 of our extended review of Allen & Pardo’s new paper on juridical proof “Relative plausibility and its critics.”

We replied to the pesky conjunction problem in our previous post. Today, we will discuss the irony of Allen and Pardo’s two most serious objections to the Bayesian or subjective view of juridical proof. In short, Allen and Pardo assert that subjective probability is subjective, and that being purely subjective, this account bears “no relationship to advancing accurate outcomes” in legal trials (p. 10). Say what? By definition, it’s true that subjective probability is subjective. In fact, all methods of decision-making under uncertainty are subjective! As such, the irony of Allen and Pardo’s bald assertions does not escape us, since the same could be said of their pet theory of relative plausibility! Think about it: just because explanation A is more plausible or persuasive than a competing explanation (explanation B) does not, by itself, make explanation A true, especially if the individual probabilities of A and B are each below 0.5 or don’t add up to 1. The question, then, is not whether judgments about evidence are subjective or not–all are–; the question is which view of proof generates less inaccuracy. Continue reading →

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. Continue reading →

As the infographic below shows, there are various ways of using law to reduce a harm or “negative externality,” to use the lingo of economics. These methods include (1) flat out prohibition, (2) Pigovian taxes, (3) contracts, and (4) doing nothing, i.e letting the market determine what the optimal level of harm is. (In the case of plastic bags, it looks like method #4 is the dominant approach.) Which method is best?

hat tip: u/ohzemartins, via Reddit

That is the title of my review of Randy Kozel’s excellent book “Settled Versus Right: A Theory of Precedent.” My essay is now posted on SSRN, the first page of which is included below:

Page 1 of 7

We’ve been meaning to write up a scholarly critique of Richard Delgado’s law review article titled “A plea for narrative,” which was published in the pages of the Michigan Law Review, vol. 87 (1989), pp. 2411-2441. (Professor Delgado’s classic paper has been cited over 2600 times according to Google Scholar, and it was assigned reading for the first annual Margaret Montoya Legal Writing Retreat that we attended last month.) So, without further ado, here it goes: while we generally that lawyers are essentially storytellers and that stories are a forceful means of communicating ideas, there are three potential problem with storytelling. Professor Delgado identifies two of these perennial problems in footnote 87 of his paper (internal citations omitted):

There are dangers in storytelling …. The hearer of an unfamiliar [story] may reject it, precisely because the story unmasks hypocrisy and increases discomfort. *** Or, the hearer may consciously or unconsciously reinterpret the story, in light of the hearer’s own belief system and inventory of stock stories, so as to blunt, or even reverse [!], its meaning.

In other words, stories can actually strengthen the audience’s preexisting biases (the problem of confirmation bias) or produce an unintended polarizing effect (the problem of polarization). But these problems, in turn, are symptoms of a much deeper storytelling dilemma, one that Delgado does not address in his essay. I will dub this dilemma the “truth-value problem.” (This quandary is also referred to pejoratively as “just-so stories” in the philosophy of science literature. According to Wikipedia, a just-so story is “an unverifiable narrative explanation for a cultural practice, a biological trait, or behavior of humans or other animals.” As a further aside, Professor John Alcock writes in his 2003 book The Triumph of Sociobiology that the term just-so story is “one of the most successful derogatory labels ever invented.”) But as Professor Delgado correctly notes in his 1989 paper, storytelling is not just an indispensable feature of law practice; it is also a crucial part of human culture. So, how are we to respond to the “just-so story” label?

One response is to openly acknowledge the truth-value problem. Another response–one consistent with Professor Delgado’s scholarly project–is to tell better stories. (Exhibit A: Harper Lee’s “To Kill a Mockingbird.”) Yet another response is to change the incentive structure of scholarly work by rewarding “academic storytelling.” Most legal scholarship is either descriptive (telling us how what the law is; i.e. how the law plays out in the real world) or normative (telling us what the law ought to be; how the law should play out in an ideal world). There is no reason in principle why we can’t have a third category of “scholarly storytelling.” Even if most such stories in this third canon are unable to counteract or negate the polarization, confirmation bias, and truth-value problems we have identified in this post, all it takes is a few good stories to enrich our understanding of the world …

In case you’ve ever wondered what would happen if artificial intelligence were trained to perform image captions of Rorschach inkblots, this crazy website created by researchers at the MIT Media Lab is for you. Here are some sample Rorschach inkblots with AI generated captions:

Hat tip: kottke.