“The Bayesian approach can have a clarifying effect on one's thinking about evidence.” (Koehler and Saks, 1991, p 364)

Traditional training in statistical methods for those who go on to become practicing biological anthropologists has focused primarily on classical hypothesis testing. This is apparent in both textbooks geared toward anthropologists in general (Thomas, 1986; Madrigal, 1998; Bernard, 2011) and specialized texts for biological anthropologists (Slice, 2005; D'Aoãut and Vereecke, 2011). While Bayes' Theorem may be mentioned in passing in introductory statistics courses, this is typically restricted to examples of such limited interest that the student has little motivation to recall the theorem, and even less motivation to assume that there may be future value in having learned about Bayes' Theorem. In Bayesian terms, the prior probability that the student will retain Bayes' Theorem is quite low. In contrast, the student and eventual practitioner is likely to learn about confidence intervals, Type I and Type II errors in hypothesis testing, and *P*-values, and to blithely assume that what they have learned represents the near totality of what is available and useful within modern statistical practice. This represents an unfortunate omission of Bayesian methods and inference.

Bayesian methods and inference are particularly helpful for creating estimates and uncertainties about those estimates without asymptotic approximation, and for incorporating prior information with data to generate problem-specific distributions in a systematic and logical way. Such methods obey the likelihood principle (unlike classical inference), generate interpretable answers in terms of a probability distribution, readily accommodate missing data and complex parametric models, and allow comparison between models. This is not to say that Bayesian methods and inference are appropriate in all contexts: there is no single best practice for selecting prior distributions, and Bayesian methods often have high computational costs. However, these drawbacks do not explain why biological anthropologists in the Americas have largely chosen to ignore Bayesian methods, while these tools have become popular and useful elsewhere. Courgeau (2012) gives a very complete account of the use of both frequentist and Bayesian methods within the broader social sciences, and McGrayne's (2011) popular history of Bayes' Rule (another name for the theorem) gives insight into why Bayesian methods have only fairly recently come to the fore.

In large measure, the recent increase in Bayesian applications across diverse fields and around the world has occurred because of the development of computer simulation methods and related software (Geyer, 1992; Gilks et al., 1996; Gamerman, 1997; Lunn et al., 2009, 2000; Brooks et al., 2011) that remove the computational burden from the user. Our goals here are to explain Bayesian principles in a way that make their applicability understandable and straightforward, to provide concrete examples of computer simulations and statistics grounded in Bayes Theorem that address questions relevant to biological anthropology, and to simultaneously review how Bayesian methods and inference have been used in biological anthropology to date. We first review maximum likelihood estimation to establish some terminology, and then use a simple example of Bayes' postulate (Bayes' Theorem with a uniform prior) to examine: 1) the likelihood, prior and posterior, and 2) differences between highest posterior density (HPD) regions and confidence intervals. We move on to Bayes Theorem and how it can be used to 1) create new priors (sequential use), 2) generate predictive densities for new samples, 3) evaluate competing models (Bayes' factor), and 4) estimate normally distributed parameters. We then delve into computer simulation, Bayesian statistics, and freeware applications, reserving the “nuts and bolts” of different methods for simulating values out of various distributions for the Appendix.

The final sections of the article illustrate various Bayesian methods using published and practical “toy” examples from bioarchaeology and from forensic anthropology. The bioarchaeology examples involve modeling mortality and accounting for uncertainty in age estimates in paleodemography, and using full posterior density distributions to address disease prevalence, specificity, and sensitivity in paleopathology. The forensic anthropology examples use Bayesian methods to address the analysis of commingled remains and issues of identification in closed population mass disasters. The forensics section also includes a discussion of the potential problems that arise when conditional probabilities are transposed in evidentiary settings and when prior probabilities are misinterpreted. We then conclude with a brief review of the frequentist–Bayesian debate and texts that focus on Bayesian inference.