## Concluding remarks

Frequentist and Bayesian methods of statistical analysis differ in how they treat the notion of probability. Bayesian methods use probabilities to assign degrees of belief to hypotheses or parameter values. In contrast, frequentist methods (null hypothesis testing and information theoretic methods) are confined to stating the frequency with which data would be collected given hypothetical replicate sampling and specified hypotheses being true. Given the disagreement about which approach to statistics is preferred, the relative merits of the different methods are clearly a matter of opinion. My preference is for Bayesian methods because I believe ecologists are usually attempting to assign degrees of belief in parameter values, models or hypotheses more generally (Table 2.2). Ecologists regard the truth as uncertain and attempt to use science to gain an improved understanding of the truth. Such an approach is consistent with Bayesian statistics.

Many of the criticisms of the different statistical methods are directed at the use of the methods, rather than their underlying basis. Null hypothesis significance testing is criticized because of its

Table 2.2. Benefits and limitations of Bayesian statistics (adapted from O'Hagan and Luce, 2003).

Benefits

Allows for intuitive interpretation

Uses prior information

Addresses a greater range of problems Allows complex models to be analysed easily

Accommodates decision making Use all the information transparently

### Limitations

Introduces an element of subjectivity (although treating it explicitly rather than ignoring it may be a benefit) There are difficulties in constructing priors

Bayesian methods are not commonly taught widespread misuse. However, it also has logical shortfalls because unobserved data influence the results, and acceptance of the alternative hypothesis does not depend on how well the evidence supports it. Frequentist methods in general are forced to ignore any relevant prior information. Additionally, they are not well-suited to decisions about individual cases, being restricted to assessing long-run frequencies obtained from hypothetical samples. Bayesian methods are criticized because it can be difficult to determine how prior information should be incorporated into analyses.

Despite the differences, Bayesian and frequentist methods often generate numerically similar answers, especially when estimating parameters and prior information is uninformative. In these circumstances, the best approach will be largely determined by which is most easily and successfully taught, learnt, and executed. Therefore, the success of the different methods lies firmly in the realm of cognitive psychology not just statistics. However, Bayesian methods have the distinct advantage that when the numerical results differ, the Bayesian methods are invariably correct (Jaynes, 1976).