Bayesian Approaches in Ecological Analysis and Modeling

Bayesian analysis provides a normative framework for use of uncertain information in decision making and inference. From a practical perspective, Bayes Theorem has a logical appeal in that it characterizes a process of knowledge updating that is based on pooling precision-weighted information. For years however, Bayesian inference was largely ignored or even discredited in favor of frequentist inference; among the reasons were computational difficulties and the formal use of subjective probabilities in applications of Bayes Theorem. In recent years, new computational approaches (e.g., Markov chain Monte Carlo) have greatly reduced the first problem, while the general recognition of the role of expert judgment in science has at least lessened resistance with respect to the second problem. Beyond that, Bayesian approaches facilitate certain analyses and interpretations that are often important to scientists. For example, the growing recognition of the value of combining information or " borrowing strength " in ecological studies, as new information is acquired to augment existing knowledge, is one of several reasons why interest in Bayesian inference continues to increase. Many currently used analytic techniques, such as random coefficients regression, multilevel models, data assimilation, and the Kalman filter are focused on this theme; all of these techniques reflect the basic framework of Bayes Theorem for pooling information. Most ecologists initially are taught that probabilities represent long-run frequencies: A consequence of this perspective is that probabilities have no meaning in a single unique or nonreplicated analysis. Scientists often ignore this constraint and interpret probabilities to suit the particular analysis. Related confusion sometimes arises in classical hypothesis testing and in the interpretation of p Bayesian Approaches 151 values. Bayesian inference provides appealing options in these situations. Collectively, these developments and perspectives have resulted in an increase in the application of Bayesian approaches in ecological studies, a number of which are noted here. Specific examples dealing with combining information, hypothesis testing, and Bayes-ian networks are discussed in more detail. In sum, it seems reasonable to make the judgmental forecast that Bayesian approaches will continue to increase in use in ecology.