Bayesian Statistical Inference for Psychological Research '

Bayesian statistics, a currently controversial viewpoint concerning statistical inference, is based on a definition of probability as a particular measure of the opinions of ideally consistent people. Statistical inference is modification of these opinions in the light of evidence, and Bayes' theorem specifies how such modifications should be made. The tools of Bayesian statistics include the theory of specific distributions and the principle of stable estimation, which specifies when actual prior opinions may be satisfactorily approximated by a uniform distribution. A common feature of many classical significance tests is that a sharp null hypothesis is compared with a diffuse alternative hypothesis. Often evidence which, for a Bayesian statistician, strikingly supports the null hypothesis leads to rejection of that hypothesis by standard classical procedures. The likelihood principle emphasized in Bayesian statistics implies, among other things, that the rules governing when data collection stops are irrelevant to data interpretation. It is entirely appropriate to collect data until a point has been proven or disproven, or until the data collector runs out of time, money, or patience. The main purpose of this paper is to introduce psychologists to the Bayesian outlook in statistics, a new fabric with some very old threads. Although this purpose demands much repetition of ideas published else-ford for their comments on earlier versions. where, even Bayesian specialists will find some remarks and derivations hitherto unpublished and perhaps quite new. The empirical scientist more interested in the ideas and implications of Bayesian statistics than in the mathematical details can safely skip almost all the equations; detours and parallel verbal explanations are provided. The textbook that would make all the Bayesian procedures mentioned in this paper readily available to experimenting psychologists does not yet exist, and perhaps it cannot exist soon; Bayesian statistics as a coherent body of thought is still too new and incomplete. Bayes' theorem is a simple and fundamental fact about probability that seems to have been clear to Thomas Bayes when he wrote his famous article published in 1763 (recently reprinted), though he did not state it there explicitly. Bayesian statistics is so named for the rather inadequate reason that it has many more occasions to apply Bayes' theorem than classical statistics has. Thus, from a very broad point of view, Bayesian statistics dates back at least to 1763. From a stricter point of view, Bayesian statistics might properly be said to have begun in 1959 with …