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Jurimetrics is the application of quantitative methods, and often especially probability and statistics, to law. [1] In the United States, the journal Jurimetrics is published by the American Bar Association and Arizona State University. [2] The Journal of Empirical Legal Studies is another publication that emphasizes the statistical analysis of law.


The term was coined in 1949 by Lee Loevinger in his article "Jurimetrics: The Next Step Forward". [1] [3] Showing the influence of Oliver Wendell Holmes Jr., Loevinger quoted [4] Holmes' celebrated phrase that:

“For the rational study of the law the blackletter man may be the man of the present, but the man of the future is the man of statistics and the master of economics.” [5]

The first work on this topic is attributed to Nicolaus I Bernoulli in his doctoral dissertation De Usu Artis Conjectandi in Jure, written in 1709.

Common methods


Gender quotas on corporate boards

In 2018, California's legislature passed Senate Bill 826, which requires all publicly held corporations based in the state to have a minimum number of women on their board of directors. [34] [35] Boards with five or fewer members must have at least two women, while boards with six or more members must have at least three women.

Using the binomial distribution, we may compute what the probability is of violating the rule laid out in Senate Bill 826 by the number of board members. The probability mass function for the binomial distribution is:

where is the probability of getting successes in trials, and is the binomial coefficient. For this computation, is the probability that a person qualified for board service is female, is the number of female board members, and is the number of board seats. We will assume that . Depending on the number of board members, we are trying compute the cumulative distribution function:

With these formulas, we are able to compute the probability of violating Senate Bill 826 by chance:

Probability of Violation by Chance (# of board members)

As Ilya Somin points out, [34] a significant percentage of firms - without any history of sex discrimination - could be in violation of the law.

In more male-dominated industries, such as technology, there could be an even greater imbalance. Suppose that instead of parity in general, the probability that a person who is qualified for board service is female is 40%; this is likely to be a high estimate, given the predominance of males in the technology industry. Then the probability of violating Senate Bill 826 by chance may be recomputed as:

Probability of Violation by Chance (# of board members)

Bayesian analysis of evidence

Bayes' theorem states that, for events and , the conditional probability of occurring, given that has occurred, is:

Using the law of total probability, we may expand the denominator as:

Then Bayes' theorem may be rewritten as:

This may be simplified further by defining the prior odds of event occurring and the likelihood ratio as:

Then the compact form of Bayes' theorem is:

Different values of the posterior probability, based on the prior odds and likelihood ratio, are computed in the following table:

with Prior Odds and Likelihood Ratio
Likelihood Ratio
Prior Odds123451015202550

If we take to be some criminal behavior and a criminal complaint or accusation, Bayes' theorem allows us to determine the conditional probability of a crime being committed. More sophisticated analyses of evidence can be undertaken with the use of Bayesian networks.

Screening of drug users, mass shooters, and terrorists

In recent years, there has been a growing interest in the use of screening tests to identify drug users on welfare, potential mass shooters, [36] and terrorists. [37] The efficacy of screening tests can be analyzed using Bayes' theorem.

Suppose that there is some binary screening procedure for an action that identifies a person as testing positive or negative for the action. Bayes' theorem tells us that the conditional probability of taking action , given a positive test result, is:

For any screening test, we must be cognizant of its sensitivity and specificity. The screening test has sensitivity and specificity . The sensitivity and specificity can be analyzed using concepts from the standard theory of statistical hypothesis testing:

Therefore, the form of Bayes' theorem that is pertinent to us is:

Suppose that we have developed a test with sensitivity and specificity of 99%, which is likely to be higher than most real-world tests. We can examine several scenarios to see how well this hypothetical test works:

With these base rates and the hypothetical values of sensitivity and specificity, we may calculate the posterior probability that a positive result indicates the individual will actually engage in each of the actions:

Posterior Probabilities
Drug UseMass Shooting

Even with very high sensitivity and specificity, the screening tests only return posterior probabilities of 60.1% and 0.98% respectively for each action. Under more realistic circumstances, it is likely that screening would prove even less useful than under these hypothetical conditions. The problem with any screening procedure for rare events is that it is very likely to be too imprecise, which will identify too many people of being at risk of engaging in some undesirable action.

Jurimetrics and law and economics

The difference between jurimetrics and law and economics is that jurimetrics investigates legal questions from a probabilistic/statistical point of view, while law and economics addresses legal questions using standard microeconomic analysis. A synthesis of these fields is possible through the use of econometrics (statistics for economic analysis) and other quantitative methods to answer relevant legal matters. As an example, the Columbia University scholar Edgardo Buscaglia published several peer-reviewed articles by using a joint jurimetrics and law and economics approach. [39] [40]

See also

Related Research Articles

In probability theory and statistics, Bayes' theorem, named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.

Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Fundamentally, Bayesian inference uses prior knowledge, in the form of a prior distribution in order to estimate posterior probabilities. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".

<span class="mw-page-title-main">Multivariate normal distribution</span> Generalization of the one-dimensional normal distribution to higher dimensions

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

<span class="mw-page-title-main">Student's t-distribution</span> Probability distribution

In probability and statistics, Student's t distribution is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

In probability theory and statistics, a Gaussian process is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

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<span class="mw-page-title-main">Laplace distribution</span> Probability distribution

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.

<span class="mw-page-title-main">Positive and negative predictive values</span> In biostatistics, proportion of true positive and true negative results

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Further reading