# Boole's Probability of Judgments

**7/25/18**

Over at William Briggs' blog, he posted Is Presuming Innocence A Bayesian Prior? In it, he issued a challenge to the frequentist definition of probability (or rather his strawman version of it):

"Jury trials are perfect at showing frequentism fails as a definition of probability."

The "probability of judgments" was actually a hot topic some generations ago in the time of Poisson, Cournot, Boole, and some others around the mid to late 1800s. The popularity of the topic, however, declined drastically after that time, relative to other areas being studied with probability and statistics, probably having to do with it being too speculative, an aspect which Boole notes.

In Boole's __Laws of Thought__ (LOT), towards the end, in a "Probability of Judgments" section, he talks about using probability to study trials. What is amazing, is that from logic and relative frequency data, he finally derives the mean probability of a correct judgement as well as the general probability k of guilt of an accused person.

Here are some of the things Boole looks at

- P(accused person is guilty) = k
- P(juryman forms the correct opinion) = x
- P(accused person will be condemned) = X
- P(accused party is guilty and juryman judges him to be guilty) = kx
- P(accused person is innocent and juryman judges him guilty) = (1-k)(1-x)
- for 1 juryman, X = kx + (1-k)(1-x)
- If there are n jurymen whose separate probability of judgment is x
_{1}, x_{2}, ..., x_{n}, then X = k*x_{1}*x_{2}*...*x_{n}+ (1-k)(1-x_{1})(1-x_{2})...(1-x_{n}) - Suppose x
_{1}, x_{2}, ... , x_{n}are equal, then X = kx^{n}+ (1-k)(1-x)^{n} - In this case, P(accused person is guilty) = kx
^{n}/(kx^{n}+ (1-k)(1-x)^{n}) - P(i voices out of jury of n declares guilty)
- P(condemnation by a majority "a") (this is just the above probability with i = (n+a)/2)
- P(condemnation by a majority of at least m out of n jurors)
- All the above equations but without k (ie. where it is not a jury setting, but an assembly of n people)
- He also speculates on values of x and k. He notes Laplace's assumptions are not that good.
- He uses estimates from real trial data that Poisson collected. That is, he is using relative frequencies. He uses the estimates condemned/accused = 5,286/11,016 = .4782 from 1825-1830, and 743/2,046 = .3631 from 1831. He uses these frequencies to derive values for k and x.
- He assumes the accused is more likely to be guilty than not.
- He assumes a juryman is more likely to make a correct decision than not.
*and many other expressions*

Re-reading Boole never ceases to amaze me. Boole always starts super simple and gets super complex towards the end. Some great books by, and on, Boole, are and

There is still some great research being done on the "probability of judgments" in modern times. For example, There is the Journal of Empirical Legal Studies. See "Estimating the Accuracy of Jury Verdicts" for an interesting paper. Yes, frequentism was and is used in this area with good results. Why would one expect otherwise, when many, many trials are done all over the world all of the time.

Also, Understanding Random Forests: From Theory to Practice, by Louppe states

Condorcet's jury theorem Majority voting [as defined in Equation 4.20], finds its origins in the Condorcet's jury theorem from the field of political science. Let's consider a group of M voters that wishes to reach a decision by majority vote. The theorem states that if each voter has an independent probability p > 1/2 of voting for the correct decision, then adding more voters increases the probability of the majority decision to be correct. When M -> oo, the probability that the decision taken by the group is correct approaches 1. Conversely, if p < 1/2, then each voter is more likely to vote incorrectly and increasing M makes things worse.

Now, just where is Briggs' favored definition of probability being used to study jury trials?

Thanks for reading.

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