Bayes 4

Note: This is my fourth blog post in a month-long series on the basics of Bayesian probability theory.

I resume by Bayesian blog series by explaining why Bayesian methods are like a meta-test, a way of testing the reliability of one’s tests. How? By allowing us to update and test our priors of a given uncertain event–e.g., whether x has cancer; whether y is a spam message; whether z committed a crime or a tort. In the case of a legal trial, as my last example suggests, the Bayesian approach is a formal method for updating one’s prior beliefs about a defendant’s guilt or innocence. Put another way, Bayesian reasoning is a method for testing the reliability and accuracy of a legal trial: it is a way of testing legal tests.

But how is a legal trial like a test? People often assume that trials are, instead, a search for the truth. The problem with this truth-function interpretation of trials, however, is that truth itself is a probabilistic ideal, not an absolute one, especially when there are different versions or interpretations of the truth. Furthermore, from the perspective of a trial attorney and the litigating parties, a trial is just a risk-taking activity, a game whose main object is to “win,” regardless of truth. Metaphorically speaking, then, a legal trial is more like poker and less like science. Of course, it helps to have the truth on one’s side, but my point here is that having the truth on one’s side is neither a necessary nor sufficient condition for winning at trial–the ultimate goal of litigation from the perspective of the parties. To sum up, a legal trial is not a dispassionate or scientific search for truth; it is a bet or wager on which party’s story or version of the truth is more likely to be accepted as true.

More importantly, truth is a probabilistic concept, for there can be competing conceptions of the truth in any given case. That is why a legal trial can be compared to a test–a test of the evidence the parties offer in support of their competing versions of the truth. Thus, a legal trial is more like a medical test or a spam filter, but instead of testing for cancer, HIV, or spam, litigation simply tests the strength of the moving party’s case. In Anglo-American law, the moving party is either the government (e.g., the prosecution in a criminal case) or a private party (e.g., the plaintiff in a civil case), and when a prosecutor or plaintiff goes to trial, he is literally putting his allegations (his evidence) to the test. In either case, civil or criminal, the moving party must submit sufficient evidence to pass the relevant test. In civil cases the test is pass/fail–e.g., the preponderance of the evidence standard. In criminal ones the test is more demanding–e.g., the reasonable doubt standard. In either case, the question is the same: has the prosecution proven its case beyond a reasonable doubt? Or, has the plaintiff proven his case by a preponderance of the evidence? This, then, is what I mean when I describe a legal trial as a test.

With this background in mind, how can we use Bayesian methods to test our legal tests? I shall turn to this question in my next few blog posts.

Is “truth” binary or probabilistic?
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Sowell versus AOC

Alternative title: Style versus Substance

I admit that I am a huge fan of AOC’s style and of her general anti-establishment rebelliousness, but in this short video clip the great Thomas Sowell points out some devastating problems and logical fallacies in the substance of AOC’s rhetoric and policies.

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Social media self-cancellation update

Farewell, Twitter and Facebook!!! I am interrupting my series of Bayesian blog posts to let people know that I have decided to “self-cancel” my social media presence by finally closing my Facebook and Twitter accounts for good. (I already closed my short-lived Instagram last year.) I took this step because I was spending way too much time on those websites and receiving little of value in return, with the possible exception of a few chess sites that I was following. In other words, the ratio of worthy-to-inane posts on my Facebook and Twitter feeds was just too low. I will, however, remain here on WordPress for the foreseeable future …

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On Bayes, part 3: all tests are imperfect

Note: This is my third blog post in a month-long series on the basics of Bayesian probability theory.

Tests are imperfect; that is, they sometimes detect things that do not exist (“false alarms” or false positives) and miss things that do exist (“misses” or false negatives). Consider spam filters, which are designed to detect unwanted email messages and reroute them into a separate spam or junk email folder. The problem is that no spam filter is perfect. Sometimes the spam net is cast too wide and some non-spam email messages fall into the spam filter and into one’s junk email folder; sometimes the spam net is not wide enough and spam messages get past the filter and into your regular inbox. (As an aside, I blogged about these so-called Type I and Type II errors, or the problem of false alarms and misses, four years ago here.)

By analogy, the same problem occurs in law and litigation. [1] Specifically, “frivolous” civil cases and criminal cases in which the prosecutor has “overcharged” the defendant can be compared to spam. In a perfect legal system, judges and juries would be able to detect and distinguish frivolous civil claims from valid ones as well as superfluous or trumped-up criminal charges from substantial ones. But judges and juries make mistakes. We all do. Sometimes, judges will dismiss valid claims (misses) and allow frivolous claims to go to trial (false alarms). Likewise, juries will occasionally convict innocent men (false alarms) or allow the guilty to go free (misses). I will further elaborate on this theme–on how legal trials are imperfects tests–in my next post.

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Image credit: @TheUSMLE

[1] See, e.g., Christoph Engel & Gerd Gigerenzer, “Law and Heuristics: An Interdisciplinary Venture,” in Engel & Gigerenzer, eds., Hueristics and the Law (2006), pp. 1-16, available here.

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What Bayes is: a test

Note: This is my second blog post in a month-long series on the basics of Bayesian probability theory.

In my previous post, I explained that Bayesian reasoning “is not just a method of drawing inferences from observations or evidence presented; it is also a method of testing the strength or weakness of such inferences.” This testing function of Bayesian methods is crucial: When we are designing, taking, or using a medical test, or a spam-filter test, or any other kind of test, we must cognitively distinguish the test from the underlying hypothesis or event being tested. Put another way, “tests are not the event.” [1] A spam filter tests for spam, but the spam test itself is separate from the event of actually having a spam message, and by the same token, a test for breast cancer, for example, is separate from the event of actually having such a cancer. Or, in the words of one Bayesian scholar:

Even a useful mammography test does not actually change [the underlying reality] whether or not a woman has cancer. She either has cancer or she doesn’t. Reality is not uncertain about whether or not the woman has cancer. We are uncertain about whether or not she has cancer. It is our information, our judgment, that is uncertain, not reality itself.” [2]

Likewise, a legal trial in many ways operates as a “test”–a test of guilt or innocence–and so is separate from the underlying condition being tested, i.e. a defendant’s guilt or innocence. Again, the defendant either has or has not committed a wrongful act (e.g., a crime, a tort, a breach of contract, etc.). [3] Reality itself is not uncertain about the defendant’s conduct. It is we (the jury or the trier or fact) who are uncertain about the defendant’s guilt or innocence. [4] As an aside, this point helps explain the logic of my Bayesian approach to adjudication. A Bayesian or any probabilistic method for trying cases is not designed to tell us with 100% certainty whether a given defendant is guilty or innocent. No method or test could produce such perfect outcomes–a point I will further discuss in my next post–instead, the Bayesian approach is just an alternative method for testing the strength of the moving party’s case, that is, for testing how likely the plaintiff or prosecutor has proven his case.

a branching charge of percentages showing false positives
“Of 10,000 women aged over fifty tested, 36 will be correctly identified as positive whereas 996 will be told they are positive despite not having cancer.” Source: Kit Yates (Science Friday).

[1] See Luke Muehlhauser, An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem (Dec. 18, 2010). See also Eliezer S. Yudkowsky, An Intuitive Explanation of Bayesian Reasoning (June 4, 2006).

[2] Muehlhauser op. cit.

[3] As Krista McCormack has pointed out to me, the legal elements of a wrongful act–that is, what
constitutes a particular crime or tort–are human-constructed elements, since we ultimately define
what elements constitute a tort or crime, such as “battery,” for example. As a result, the underlying
metaphysical reality (e.g., “did this defendant commit a battery?”) might not congruently align with
the human-defined elements of the wrongful act. Nevertheless, there is no reason in principle why we could not apply Bayesian methods to the task of interpreting what behavior constitutes a wrongful act in the first place. That is, we could apply Bayesian methods to predict what interpretation of law is most likely to prevail in a given close case.

[4] Even the practice of science, like litigation, is just a series of tests. There is a test for a given
phenomenon, and there is the event of the phenomenon itself. See Kalid Azad, An Intuitive (and Short) Explanation of Bayes’ Theorem (May 7, 2007).

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Bayes is not an oracle

Note: This is my first blog post in a month-long series on the basics of Bayesian probability theory.

It’s the “1st of tha month,” so as promised let’s begin my series of Bayesian blog posts, and let me start by stating “what Bayes is not.” It is not a magic conjurer, Delphic oracle, or “veritable sorcerer.” [1] That is, it is not a method for eliminating uncertainty. Instead, Bayesian probability is a method for measuring our level of uncertainty. [2] Put another way, the Bayesian approach to legal and moral judgements is not just a method of drawing inferences from observations or evidence presented; it is also a method of testing the strength or weakness of such inferences. This insight is extremely relevant to law and legal processes. Although Bayesian reasoning cannot replace human intuition or judgement or decide for us the ultimate guilt or innocence of a defendant, we can nevertheless use Bayesian methods to measure or evaluate the strength of a party’s evidence, whether it be evidence of guilt or evidence of innocence. As a result, Bayesian methods may not only be used offensively by a moving party (plaintiff or prosecutor) to measure the strength of his case; such methods can also be used defensively by a defendant to test or challenge the strength of the moving party’s case. (In my next post, I will show how legal trials are like bets.)

Delphic Oracle | Ancient greek art, Oracle art, Greek art
What Bayes is not.

[1] People v. Collins, 438 P.2d 33, 33 (1968).

[2] See, e.g., Colin Howson &Peter Urbach, Bayesian Reasoning in Science, 350 Nature 371, 372 (1991) (Bayesian reasoning is a method of “characterizing a scientific conclusion about a hypothesis as a statement of its probability”); Stephen Fienberg & Mark Schervish, The Relevance of Bayesian Inference for the Presentation of Statistical Evidence and for Legal Decisionmaking, 66 Boston University Law Review 771, 773 (1986) (“Bayesian probability theory … provide[s] both a framework forquantifying uncertainty and methods for revising uncertainty measures in the light of acquired evidence”).

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Back to Bayesian Basics

This blog is called “prior probability,” which refers to a special idea in the world of Bayesian probability theory, the idea of a “prior”: one’s personal or subjective belief/probability estimate of an event, before any data is collected or observed. I have long been fascinated with this concept and with Bayesian probability generally. Specifically, where do we get our priors from, and why are most people so reluctant to update their priors? Also, why is Bayesian reasoning so special, and why are Bayesian methods worth studying? These last two questions are especially poignant and relevant in my case. After all, what do Bayes or priors have to do with law, let alone ethics, my two areas of study? As it happens, I have written a number of scholarly essays, book reviews, blog posts, and law journal articles, just to name a few, on this subject:

  1. A Bayesian Model of the Litigation Game, European Journal of Legal Studies (2011).
  2. The Turing Test and the Legal Process, Information & Communications Technology Law (2012).
  3. Visualizing Probabilistic Proof, Washington University Jurisprudence Review (2014).
  4. Why Don’t Juries Try ‘Range Voting’?, Criminal Law Bulletin (2015).
  5. Bayesian Manipulation of Litigation Outcomes, unpublished manuscript (2016).
  6. The Case for Bayesian Judges, Journal of Legal Metrics (2019).
  7. Bayesian Verdicts, Journal of Brief Ideas (2020).
  8. Weyl Versus Ramsey: A Bayesian Voting Primer, unpublished manuscript (2020).
  9. Frank Ramsey’s Contributions to Probability and Legal Theory, work in progress.

But truth be told, only a handful of individuals have ever read any of my scholarly papers. Also, hardly anyone, especially in such non-mathematical fields like law and ethics, wants to take the time to figure out the meaning of technical formulas or solve equations, so beginning on Monday, March 1st–and for the entire month of March–I will be returning to my Bayesian roots, so to speak, and will be blogging about first Bayesian principles and explaining their relevance to law and ethics. (I will, however, in honor of my hero, the good Reverend Thomas Bayes, be taking Sundays off.)

Image Credit: Paul Epps
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True or false?

Background: The geometry problem pictured below, which popped up in my Twitter feed, was originally assigned on a seven-year-old’s math homework. For my part, I thought at first that the answer had to be “false,” but after seeing some of the replies in this fascinating thread, a thread which contains many deep mathematical insights, I have changed my answer to “true”!

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PSA: vaccine distribution should be FCFS

I agree with the sentiment of my colleague Andrew Fleischman (@ASFleischman), who tweeted the following rhetorical question: “… I wonder how vaccine rollout would have gone if it was just first come, first serve[?]” The answer, of course, is pretty clear: the people who really wanted the vaccines would have received them first. Most of the replies to Fleischman’s tweet, however, are clueless. (At the very least, young ladies won’t have to pretend to be grannies to get a vaccine! See here.)

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Lego Colosseum

The Lego image pictured below is from Dr Abigail Graham (@abby_fecit), an archaeology-minded historian and Romanophile. Her article is titled: “Reconstructing the Past: Lego Colosseum & 8 reasons why Lego is great training for an archaeologist.” Source:

Hat tip: @pickover
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