## Bayes 6: the logic of Bayesian reasoning

Note: This is my sixth blog post in a month-long series on the basics of Bayesian probability theory.

In this post I present the logic of my Bayesian model of litigation outcomes in algebraic terms as follows:

Pr(A|B) = ([Pr(B|A)] × [Pr(A)]) ÷ Pr(B)

At first glance, this formidable and intimidating formula looks like an impenetrable set of terms, but it can be explained in words and broken down into the following five parts:

(i) The term on the left-hand side of the equation, Pr(A|B), refers to the conditional probability (or posterior probability) of event A, given the occurrence of event B.

(ii) The right-hand side of the equation is a fraction: the numerator contains two parts, Pr(B|A) × Pr(A), while the denominator consists of one term, Pr(B).

(iii) The first term in the numerator, Pr(B|A), refers to the conditional probability of event B, given the occurrence of event A.

(iv) The second term in numerator, Pr(A), refers to the prior probability (or unconditional probability) of event A, that is, the probability of A in the absence of any information about event B.

(v) Lastly, the denominator, Pr(B), is the prior probability (or unconditional probability) of event B in the absence of any information about event A.

In plain words, B or “+” is the probability of a positive litigation outcome from the perspective of the moving party in the litigation game, the plaintiff (in a civil trial) or the prosecutor (in a criminal trial). In other words, the main idea here is that the moving party—the plaintiff or prosecutor, as the case may be—obtains a favorable or positive outcome, which is denoted by the symbol +, when the defendant is found civilly or criminally liable at trial. Our Bayesian model of the litigation game thus poses the following fundamental question: what is the posterior probability that a defendant in a civil or criminal trial will be found liable, given that the defendant has not, in fact, committed any wrongful act? [1]

That is the question we will answer in my remaining blog posts. In the meantime, I will equate the term ‘guilty’ (or the letter ‘A’) with the event that the defendant in a particular litigation game has committed a wrongful or unlawful act, that is, an act for which he should be civilly or criminally liable. [2] In addition, I will equate the term the symbol + (or the letter ‘B’) with the event that the defendant is actually found liable at trial for the commission of a civil or criminal wrongful act. [3]

[1] The term Pr(A) or Pr(guilty) (in contrast to the terms ‘A’ or ‘guilty’) refers to the prior probability in the absence of additional information that this event (i.e., the imposition of civil or criminal liability) has in fact occurred.

[2] In other words, the symbol + and the term ‘positive litigation outcome’ is not meant to convey a pro-plaintiff or pro-prosecutor bias; instead, we use it to indicate a litigation outcome in which civil or criminal liability is imposed on the defendant.

[3] Like the term ‘litigation’, I define ‘wrongful act’ broadly to include both civil wrongs, such as torts and breaches of contract, as well as criminal wrongs, such as homicide and theft.