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

• Bayesian inference
• Causal inference
  • Instrumental variables
• Design of experiments
  • Vital for epidemiological studies
• Generalized linear models
  • Ordinary least squares, logistic regression, Poisson regression
• Meta-analysis
• Probability distributions
  • Binomial distribution, hypergeometric distribution, normal distribution
• Survival analysis
  • Kaplan-Meier estimator, proportional hazards model, Weibull distribution

Applications

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:

$${\displaystyle \mathbb {P} (X=k)={n \choose {k))p^{k}(1-p)^{n-k))$$

${\displaystyle p}$

${\displaystyle k}$

${\displaystyle n}$

${\textstyle {n \choose {k))}$

${\displaystyle p}$

${\displaystyle k}$

${\displaystyle n}$

${\displaystyle p=0.5}$

Depending on the number of board members, we are trying compute the cumulative distribution function:

$${\displaystyle {\begin{cases}\mathbb {P} (X\leq 1)=(1-p)^{n}+np(1-p)^{n-1},\quad &n\leq 5\\\mathbb {P} (X\leq 2)=\mathbb {P} (X\leq 1)+{n(n-1) \over {2))p^{2}(1-p)^{n-2},\quad &n>5\end{cases))}$$

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:

Bayesian analysis of evidence

Bayes' theorem states that, for events ${\displaystyle A}$ and ${\displaystyle B}$, the conditional probability of ${\displaystyle A}$ occurring, given that ${\displaystyle B}$ has occurred, is:

$${\displaystyle \mathbb {P} (A|B)={\mathbb {P} (B|A)\mathbb {P} (A) \over {\mathbb {P} (B)))}$$

$${\displaystyle \mathbb {P} (B)=\mathbb {P} (B|A)\mathbb {P} (A)+\mathbb {P} (B|\sim A)[1-\mathbb {P} (A)]}$$

$${\displaystyle {\begin{aligned}\mathbb {P} (A|B)&={\mathbb {P} (B|A)\mathbb {P} (A) \over {\mathbb {P} (B|A)\mathbb {P} (A)+\mathbb {P} (B|\sim A)[1-\mathbb {P} (A)]))\\&={1 \over {1+{1-\mathbb {P} (A) \over {\mathbb {P} (A))){\mathbb {P} (B|\sim A) \over {\mathbb {P} (B|A)))))\end{aligned))}$$

${\displaystyle A}$

${\displaystyle \eta }$

${\displaystyle {\mathcal {L))}$

$${\displaystyle \eta ={\mathbb {P} (A) \over {1-\mathbb {P} (A))),\quad {\mathcal {L))={\mathbb {P} (B|A) \over {\mathbb {P} (B|\sim A)))}$$

$${\displaystyle \mathbb {P} (A|B)={1 \over {1+(\eta {\mathcal {L)))^{-1))))$$

${\displaystyle \mathbb {P} (A|B)}$ with Prior Odds and Likelihood Ratio

	Likelihood Ratio
Prior Odds	1	2	3	4	5	10	15	20	25	50
0.01	0.01	0.02	0.03	0.04	0.05	0.09	0.13	0.17	0.20	0.33
0.02	0.02	0.04	0.06	0.07	0.09	0.17	0.23	0.29	0.33	0.50
0.03	0.03	0.06	0.08	0.11	0.13	0.23	0.31	0.38	0.43	0.60
0.04	0.04	0.07	0.11	0.14	0.17	0.29	0.38	0.44	0.50	0.67
0.05	0.05	0.09	0.13	0.17	0.20	0.33	0.43	0.50	0.56	0.71
0.10	0.09	0.17	0.23	0.29	0.33	0.50	0.60	0.67	0.71	0.83
0.15	0.13	0.23	0.31	0.38	0.43	0.60	0.69	0.75	0.79	0.88
0.20	0.17	0.29	0.38	0.44	0.50	0.67	0.75	0.80	0.83	0.91
0.25	0.20	0.33	0.43	0.50	0.56	0.71	0.79	0.83	0.86	0.93
0.30	0.23	0.38	0.47	0.55	0.60	0.75	0.82	0.86	0.88	0.94

If we take ${\displaystyle A}$ to be some criminal behavior and ${\displaystyle B}$ 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 ${\displaystyle V}$ that identifies a person as testing positive ${\displaystyle +}$ or negative ${\displaystyle -}$ for the action. Bayes' theorem tells us that the conditional probability of taking action ${\displaystyle V}$, given a positive test result, is:

$${\displaystyle \mathbb {P} (V|+)={\mathbb {P} (+|V)\mathbb {P} (V) \over {\mathbb {P} (+|V)\mathbb {P} (V)+\mathbb {P} (+|\sim V)\left[1-\mathbb {P} (V)\right]))}$$

${\displaystyle \mathbb {P} (+|V)}$

${\textstyle \mathbb {P} (-|\sim V)=1-\mathbb {P} (+|\sim V)}$

• Sensitivity is equal to the statistical power ${\displaystyle 1-\beta }$, where ${\displaystyle \beta }$ is the type II error rate
• Specificity is equal to ${\displaystyle 1-\alpha }$, where ${\displaystyle \alpha }$ is the type I error rate

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

$${\displaystyle \mathbb {P} (V|+)={(1-\beta )\mathbb {P} (V) \over {(1-\beta )\mathbb {P} (V)+\alpha \left[1-\mathbb {P} (V)\right]))}$$

• We screen welfare recipients for cocaine use. The base rate in the population is approximately 1.5%,^[38] assuming no differences in use between welfare recipients and the general population.
• We screen men for the possibility of committing mass shootings or terrorist attacks. The base rate is assumed to be 0.01%.

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:

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]