Bayes 9: the model

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 …