Dear Colleagues,

Many statistics textbooks treat frequentist statistics as the main subject and relegate Bayesian statistics to a single chapter, often near the end of the book. Students are often told that there are two "schools'' of statistics, the frequentist (or “classical”) and the Bayesian, and that the two are in opposition to each other. Students are implicitly, or even sometimes explicitly, asked to choose between them. The reality, of course, is much more complex. There are other views and tools—likelihoodism, decision theory, “objective Bayesianism,” and fiducial inference, for example—meant to sit between the supposed extremes of subjective Bayesianism and frequentism. Some use priors and posteriors that are not probability functions (do not integrate into unity). There are statisticians, the chief editor of this Special Issue being one of them, who are considered Bayesians, but who have used frequentist techniques or applied frequentist concepts in their work. It is worth remarking here that some of the main tools of Bayesian statistics used in the 21st Century are based on frequencies in simulations collectively known as Markov chain Monte Carlo. Bayesian methods have also been used by statisticians considered frequentists to solve problems that arise in frequentist statistics. Even Sir Ronald Fisher, who first proposed fiducial inference and is considered the “founding father” of frequentist inference, was in his later works moving toward some of the inductive arguments of Bayesian inference and emphasizing the likelihoods stronger. 

In the end, statistics comes down to trying to infer something about a larger population from a smaller sample, and any approach to such a task will have strengths and weaknesses, will surely have pathological cases it cannot resolve, and will be subject to valid criticisms. Therefore, it is not surprising that mixing the ideas of the different “schools” of inference has been a successful approach and has expanded and enriched the palette of tools available to researchers in every quantitative field of study.  

The idea of this Special Issue is to treat Bayesian statistics as the main topic, but that does not mean it is to be treated as superior to any other paradigm of inference.  Contributions from practitioners and theoreticians using and advancing Bayesian thoughts and methods are obviously welcome, but so are contributions from those who use other paradigms, including criticisms of aspects of Bayesian inference in comparison to the authors' preferred methods. Our idea is to provide a snapshot of how Bayesian inference is understood and how it contributes to scientific endeavor today, late in the first quarter of the 21st Century.

When we speak of Bayesianism, we are referring to a philosophical and statistical framework that involves the representation of degrees of belief or justification using probabilities. It is characterized by the idea that belief comes in degrees that can be formalized using the axioms of probability theory. Bayesianism involves the assessment of the rationality of degrees of belief based on a set of rules, and these beliefs can be updated using Bayes's theorem based on new information or evidence.

Prof. Dr. Carlos Alberto De Bragança Pereira
Prof. Dr. Paulo Canas Rodrigues
Dr. Mark Andrew Gannon
Guest Editors

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• prior distributions
• posterior probabilities or densities
• likelihood optimizations: weighted average or maximization