ABSTRACT
Classical, or frequentist, statistics is based on the basic principle that the parameters of a distribution are constants, and a statis tical model is used, on which we base our analysis and inference. Usually, it is the normal model, and the most popular topics are hy pothesis testing, estimation and analysis of variance. The meaning of the probability of an event is never clearly stated, but it could be a mixture of Laplace symmetry principle and Von Mises limit proportion. In the 20th century, other meanings of Probability have been suggested, in particular, subjective notions associated with bet ting. Works by Savage,and especially by De Finetti on exchangeable sequences, have provided a solid basis to the subjective approach to probability. There should be at least 3 kinds of Probability: physical, logical and subjective (Good, 1965), leading to 3 areas of Bayesian statistics: the Empirical Bayes (developed by Robbins), the Logi cal Bayes (Keynes, Carnap) and the Subjectivist Bayes (Ramsey, Savage, de Finetti). The interested reader can consult the thorough treatise on the foundations of different probability theories by Fine (1973), who concluded that 11 of all the theories considered, subjec tive probability holds the best position with respect to the value of probability conclusions, however arrived at ” (page 240). Subjective probability concepts constitute the main engine that permitted the development of the Bayesian approach to statistics, which is based on Reverend Thomas Bayes (1702-1761) simple formula:
when considering single events A and B, and
for the set of exhaustive events {i4j}^=1. In the formal Bayesian paradigm, the parameter 0 under consideration is a random variable with a prior distribution g(0), defined in Cl C TV1. In the statistical model f(x\6), a value of 0 serves to determine the distribution, from which a sample X = ( X i , . . . , Xn) will be taken. The likelihood func-
The prior is an indispensable component of the Bayesian paradigm and strictly speaking, it should be based on subjective probability only. However, in practice, its derivation can come from personal be liefs, historical data, or other means of information, or a combination of these sources. At the present time, although there is much debate on the nature and properties of the prior, there is still no concensus on how to derive it, not even general accepted guidelines to follow for that purpose. It can be fairly said that establishing the prior distri bution is the “weakest link” in the Bayesian chain, a situation that, in many cases, has seriously prevented the application of Bayesian statistical methods. Concerning this problem, Fine (1973, page 240) wrote: “The measurement problem in subjective probability is sizable and conceivably insurmountable”.
