Bayesian Inference and Boolean Logic

Conceptual Overview of Bayesian Inference

Bayesian inference is an approach to the interpretation of statistical evidence inspired by the contributions of 18th-century statistical theorist Thomas Bayes. It is based on the concept of conditional probability, in which one recalculates the likelihood of an event based on new evidence, as codified in Bayes's theorem. Bayesian inference introduces the concept of the “degree of belief,” emerging through an iterative process, in which successive exercises generate new understandings of reality. It is to be distinguished from traditional hypothesis testing with null hypotheses, Type I and Type II errors, confidence intervals, and critical values.

Bayesian inference produces a conception of the “state of knowledge” at a given moment. The researcher posits prior probabilities, perhaps based on his or her own intuition or on a broader consensus, and these are taken into account as an experiment or study produces posterior probabilities. Posterior probabilities represent revised assessments as determined by subsequent analysis.

Following the Bayesian theorem, posterior probabilities can be calculated according to the following formula:

P(HIE) is the posterior probability that Hypothesis H is true given that Evidence E has emerged. P(H) is the prior probability that H is true, as assessed before the arrival of Evidence E. P(EIH) is the probability Evidence E would arise given the truth of Hypothesis H. P(E) is the probability of E occurring regardless of the correct hypothesis.

Many critics regard Bayesian analysis as necessarily subjectivist, in that inquiry begins with the researcher's initial judgments (which may be characterized as “biased”) and continues to be guided by the researcher's belief. On the other hand, subsequent iterations of analysis may suppress initial biases and introduce grounds for broad credibility. A minority of critics find Bayesian inference consistent with objectivity and Aristotelian logic, and interest in Bayesian approaches is increasing.

Bayesian inference is quite useful in combination with case study analysis. That is, the researcher may consider statistical evidence, make an inference along with an assessment of degree of belief, and then incorporate case analysis, which either reinforces or weakens earlier judgments. Bayesian analysis provides a justification for the integration of multiple methodologies in a single project.

Consider, for example, the effective use of polling in political research. The point estimate of support for candidate or party is of limited value in and of itself. A Bayesian approach builds an analysis conditioned by the estimate but refined through the analysis of rally size, polling place lines, and other factors. Projections of United States presidential elections necessarily combine state-level predictions and the consideration of electoral college scenarios ensuing from them.

Conceptual Overview of Boolean Logic

Boolean logic is a system for performing logical operations in the analysis of sets, or groups, of numbers, objects, or events. It is the foundation of database search engines and the basis of computer information processing. It incorporates the logical operators AND and OR, which are equivalent to union and intersection in symbolic logic. These operators render large databases more manageable; for example, multiple search terms can identify relevant articles from an extensive library of journals.

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