prior probability

Note: This is the second blog post in a four part series.

As I mentioned in my previous post, Ross Douthat’s recent NY Times column on conspiracy thinking, “A Better Way to Think about Conspiracies,” formulates a four-part test for deciding which alleged conspiracies to keep an open mind about, or in Douthat’s own words, “a tool kit for discriminating among different fringe ideas.” Among other things, Douthat recommends: “Prefer simple theories to baroque ones.” This first criterion can thus be restated in Occam’s Razor terms as follows: prefer simpler conspiracy theories to more complex ones. Let’s call this principle “Ross’s Razor.”

In brief, Ross’s Razor tells us that when we are presented with competing explanations of the same event (e.g., Germany’s defeat in World War I; Trump’s loss in 2020 despite winning in Florida and Ohio), we should select the simplest explanation, the explanation with the fewest assumptions. As an aside, this preference for simplicity, though attributed to William of Ockham (1287?–1347), a Franciscan theologian and scholastic philosopher (see image below), may, in fact, go as far back as Aristotle’s treatise Physics, which states, “Nature operates in the shortest way possible.” As a further aside, whether we define simplicity in terms of the number of background assumptions or in terms of how nature or the world operates, I personally prefer to frame the simplicity/parsimony criterion in probabilistic terms, since one of the rationales for this preference for parsimony is a probabilistic one: the idea that the simplest explanation is most likely to be the correct one.

Either way, however, what does “simpler” mean in the domain of alternate realities or conspiracy theories? Does simplicity refer to the number of conspirators? The goal of the conspiracy? The number of steps necessary for the conspiracy to succeed? Worse yet, however we answer the foregoing questions, one of the supreme ironies of many conspiracy theories is that they pass Douthat’s parsimony test with flying colors, especially when it is the truth that is often ambiguous and messy! By way of illustration, consider the German “Stab-in-the-Back” Myth that I mentioned in my previous post. In many ways, this particular conspiracy theory provides a far simpler and parsimonious explanation of Germany’s defeat in World War I than the truth does.

Yes, the German Army was low on reserves, and yes the United States changed the course of the war after the Battle of Cantigny (28 May 1918), but how could the German public know these things at the time? Also, even if the number of German reserves and the number of U.S. troops were publicly-available information, what could be more simpler than to believe that Germany was stabbed-in-the-back by a visible group of traitors, the “November Criminals” who signed the armistice in November of 1918? Simply put (pun intended), it is this tempting yet misleading simplicity that is one of the main attractions of so many fringe conspiracy theories! That said, I will consider the remaining three factors in Douthat’s four-part test in my next few blog posts.

Happy St. Patrick’s Day! Ross Douthat, an influential columnist for The New York Times, recently wrote a fascinating essay titled “A Better Way to Think about Conspiracies.” As it happens, I have always been puzzled by one of the most famous conspiracy theories of all time, the “stab-in-the-back” myth that was popular in Germany during the ill-fated Weimar Republic era (1919 to 1933). How did the Imperial German Army–an army that was said to be “undefeated on the battlefield”–end up losing the First World War (WWI)? According to one popular conspiracy theory at the time, Germany lost WWI because she was “stabbed in the back” by a wide variety of left-wing politicians and intellectuals, who were collectively referred to as “the November Criminals” for agreeing to Germany’s surrender on 11 November 1918. In reality, however, Germany had lost the war because her army lacked sufficient reserves and because the USA had entered the war in full force in mid-1918. So, how did this dangerous myth persist for so long and win over so many hearts and minds?

Now, fast forward to the JFK assassination or, even more recently, to 2020? Did Harvey Lee Oswald act alone? Were the 2020 elections stolen from President Trump? If the JFK plot or Trump’s election fraud claims are just crazy conspiracy theories, why do so many people still believe in them? In short, where do we draw the line between plausible conspiracy theories and far-fetched ones? Here is why Douthat’s conspiracy theory essay is worth reading: he formulates a four-part test for deciding which alleged conspiracies to keep an open mind about, “a tool kit for discriminating among different fringe ideas.” In brief, Douthat’s conspiracy theory test consists of the following four criteria:

1. “Prefer simple theories to baroque ones.”
2. “Avoid theories that seem tailored to fit a predetermined conclusion.”
3. “Take fringe theories more seriously when the mainstream narrative has holes.”
4. “Just because you start to believe in one fringe theory, you don’t have to believe them all.”

Alas, Douthat’s four-part test is woefully inadequate for several reasons, which I shall discuss in detail in my next few posts. For now, it suffices to say that both the German “stab-in-the-back” myth as well as Trump’s stolen election story–indeed, most of the conspiracy theories mentioned in the chart below–would most likely pass Douthat’s four-part test with flying colors.

I shall close this series of Bayesian blog posts with a confession. Ex ante, before I began building my Bayesian model of the litigation process, I had taken a dim view of the legal game. Given the complexity and ambiguity of substantive as well as procedural rules, the indeterminate nature of most legal standards, and the high levels of strategic behavior by both litigants and judges, I expected my Bayesian model to confirm my negative view of the legal process. Ironically, however, the results of my Bayesian model of the litigation game were very surprising. In essence, my model shows that, regardless of the operative rules of procedure and substantive legal doctrine, a guilty verdict is nevertheless a highly reliable indicator of a defendant’s actual guilt. Specifically, my model demonstrates that when a defendant is found guilty of committing a wrongful act (civil or criminal), there is a high posterior probability that the defendant actually committed such a wrongful act, even when the underlying process of adjudication is random and even when the moving parties are risk-loving or less-than-virtuous!

Note: Because of two other research projects I am currently working on right now–both of which must be completed by April 16–as well as my regular spring semester teaching duties, I will be suspending my series of Bayesian blog posts for the time being. (After April 16, I will resume this series by showing how Bayesian methods can solve the blue bus problem and other evidentiary paradoxes.) In the meantime, I will switch gears, so to speak, and blog about my two ongoing research research projects in the days ahead …

Note: This is my fourteenth blog post in a month-long series on the basics of Bayesian probability and its application to law.

Happy Monday! Let’s now suppose that litigation is still a crapshoot but that plaintiffs and prosecutors are risk-loving or ‘less-than-virtuous’; that is, let’s assume that the moving parties are more willing to gamble than their virtuous colleagues. Specifically, I will assume that the litigation game is 50% sensitive and 50% specific and that plaintiffs and prosecutors are willing to play the litigation game even when they are only 60% certain that the named defendant has committed a wrongful act. Although these assumptions do not appear to be plausible, this permutation of my model, however implausible, may nevertheless provide an instructive counter-factual or hypothetical illustration of my Bayesian approach to litigation.

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Note: This is my thirteenth blog post in a month-long series on the basics of Bayesian probability and its application to law.

Suppose that litigation is a crapshoot (to quote my mentor and favorite law school professor John Langbein); that is, what if litigation outcomes are only 50% sensitive and 50% specific? In other words, what if litigation games are completely random? Under this scenario, the process of adjudication is no better than a coin toss. Although this assumption may appear fanciful, as I explained in a previous post (see “Bayes 10“), the randomness of adjudication might be a function of the level of the complexity or the level of ambiguity of the applicable legal doctrines (e.g., assumption of risk) or procedural rules (e.g., res judicata). Simply put (pun intended), the more complex or ambiguous the applicable law is, the more random or arbitrary the outcome of litigation will be.

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Note: This is my twelfth blog post in a month-long series on the basics of Bayesian probability and its application to law.

Suppose the litigation game is still highly sensitive and specific as before (i.e., 90% sensitive and 90% specific), but that plaintiffs and prosecutors are risk-loving or less-than-virtuous actors. Specifically, assume that the moving parties are willing to play the litigation game even when they are only 60% certain (instead of 90% certain, as we assumed earlier) that the named defendant has committed a wrongful act. (Such behavior is ‘less-than-virtuous’ in my model because the moving party is less concerned with the defendant’s actual guilt than a risk-averse or virtuous moving party is.) The intuition behind this revised assumption is that, in reality, the litigation game might be played by litigants who are engaged in rent-seeking and self-serving behavior. [1]

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Note: This is my eleventh blog post in a month-long series on the basics of Bayesian probability and its application to law.

Suppose the litigation game is 90% sensitive and 90% specific, that is, suppose the process of litigation is able to determine correctly, at least 90% of the time, when a defendant has committed a wrongful act, and suppose further that the process will also determine correctly, again at least 90% of the time, when a defendant has not, in fact, committed a wrongful act. The intuition behind this assumption (non-random adjudication) is that reliable legal procedures will tend to produce just and fair results. See, for example, Henry M. Hart and Albert M. Sacks, The Legal Process (William N. Eskridge and Philip P. Frickey eds, Foundation Press, 1994). [1] Simply put, such a litigation game appears to be a highly accurate one, since it will correctly determine with 90% probability, or nine times out of 10, whether the defendant has or has not committed a wrongful act, an essential precondition before liability may justly be imposed.

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Note: This is my tenth blog post in a month-long series on the basics of Bayesian probability and its application to law.

I will now consider four possible scenarios or types of litigation games:

1. non-random adjudication with risk-averse or ‘virtuous’ moving parties,
2. non-random adjudication with risk-loving or ‘less-than-virtuous’ moving parties,
3. random adjudication with risk-averse moving parties, and
4. random adjudication with risk-loving moving parties.

This schema may thus be depicted in tabular form as follows:

In summary, the adjudication variable in my model refers to the reliability or screening effectiveness of the process of adjudication. Specifically, ‘non-random adjudication’ refers to litigation games that are 90% sensitive and 90% specific, an assumption based on the classic and oft-repeated legal maxim ‘it is better that ten guilty men escape than that one innocent suffer’.

Random adjudication, in contrast to non-random adjudication, occurs when litigation games are only 50% sensitive and 50% specific and thus no more reliable than the toss of a coin. As an aside, it is worth asking, why would the process of adjudication ever produce a ‘random’ outcome in the real world? One possibility is that the level of randomness or unpredictability of adjudication might be a function of the level of complexity or ambiguity of legal rules. Consider, for example, the ‘reasonable man’ standard in tort law: the more complex or ‘open-textured’ the rules of substantive and procedural law are, the more random the litigation game will be. Also, before proceeding, notice that the adjudication variable can never be 100% sensitive nor 100% specific since errors are inevitable in any process of adjudication, regardless of the litigation procedures that are in place. 

In addition, the term ‘risk-averse’ or ‘virtuous’, as applied to moving parties, refers to plaintiffs and prosecutors who play the litigation game only when they are at least 90% certain that the named defendant has committed an unlawful wrongful act, while ‘risk-loving’ or ‘less-than-virtuous’ moving parties refers to plaintiffs and prosecutors who are willing to play the litigation game even when they are only 60% certain that the named defendant has committed a wrongful act. Stated colloquially, virtuous plaintiffs are civil plaintiffs who rarely file frivolous claims and criminal prosecutors who rarely abuse their discretion; by contrast, less-than-virtuous moving parties are more willing to gamble on litigation games than their more virtuous colleagues.

In my next post, I will focus on-random adjudication with risk-averse moving parties.

Note: This is my ninth blog post in a month-long series on the basics of Bayesian probability and its application to law.

I am now ready to present a stylized Bayesian model of litigation. To do so, I will have to make a number of simplifying assumptions about the litigation process. First, I will define ‘litigation’ broadly to include both criminal and civil cases. In essence, the litigation game (whether civil or criminal) is a contest in which the moving party, the plaintiff or the prosecutor, attempts to impose civil or criminal liability on the defendant for the commission of an unlawful or wrongful act (whether civil or criminal in nature). And likewise, seen from the defendant’s perspective, litigation is a contest in which defendants attempt to avoid the imposition of liability. My model thus presents litigation as a game with two possible outcomes: ‘positive’ (+) and ‘negative’ (–). Specifically, a ‘positive’ outcome occurs when the moving party successfully imposes civil or criminal liability on the defendant; a ‘negative’ one, when the defendant is able to avoid the imposition of liability. (Secondly, my simple Bayesian model of the litigation game ignores the temporal dimension of adjudication–‘time costs’ and the problem of delay. Instead, I will assume for simplicity that litigation is an instantaneous event, like a coin toss or the roll of a die.)

Having stated my simplifying assumptions, I will now proceed to apply Bayes’ theorem to the litigation process. Recall the statement of Bayes’ rule from one of my previous blog posts (“Bayes 6“):

Pr(A|B) = [Pr(B|A) × Pr(A)] ÷ Pr(B)

Translated into the legal language of litigation, my legal version of Bayes’ rule may now be restated as follows:

Pr(guilty|+) = [Pr(+|guilty) × Pr(guilty)] ÷ Pr(+)

In other words, we want to find the posterior probability, Pr(guilty|+), that a defendant will be found liable at trial, given that he or she has actually committed some wrongful act. (As the meme pictured below shows, Bayes’ rule is quite versatile; it can be applied to many other settings.) Ideally, of course, liability should be imposed only when a defendant has actually committed a wrongful act, and conversely, no liability should be imposed on innocent defendants. (In an ideal or perfect legal system, the value for Pr(guilty|+) should be equal to or close one, or stated formally: Pr(A|B) ≈ 1.) But in reality, false negatives and false positives will occur for a wide variety of reasons, such as heightened pleading standards and abuse of discovery in civil actions as well as prosecutorial discretion and prosecutorial misconduct in criminal cases. Stated colloquially, some guilty defendants will be able to avoid the imposition of liability, while some innocent ones will be punished. (See, for example, my “Bayes 3” blog post.)

To sum up, my Bayesian approach to litigation takes into account both (i) the possibility of a false positive (i.e., the imposition of liability when the defendant has not committed any wrongful act) as well as (ii) the possibility a false negative (no liability even though the defendant has, in fact, committed a wrongful act). The purpose of my stylized model, however, is not to explore the many systemic imperfections–procedural or practical or otherwise–in the existing legal system, imperfections contributing to the problem of false positives and negatives. This well-worn path has been explored by many others. (See, for example, Marc Galanter’s classic 1974 paper on ‘Why the “Haves” Come Out Ahead’, reprinted in David Kennedy and William W. Fisher, editors, The Canon of American Legal Thought, Princeton U Press (2006), pp. 495-545.) Instead, the goal of my model is to solve for Pr(guilty|+) and answer the following key question: How reliable is the litigation game, that is, how likely is it that a defendant who is found liable is, in fact, actually guilty of committing a wrongful act?

I will proceed to answer this question in my next post …