Bayesian posterior classification of planetary nebulae according to the Peimbert types

In this paper we present a re-analysis of the criteria used to characterize the Peimbert classes I, IIa, IIb, III and IV, through a statistical study of a large sample of planetary nebulae previously classified according to these groups. In the original classification, it is usual to find planetary nebulae that cannot be associated with a single type; these most likely have dubious classifications into two or three types. Statistical methods can greatly contribute in providing a better characterization of planetary nebulae groups. We use the Bayes Theorem to calculate the posterior probabilities for an object to be member of each of the types I, IIa, IIb, III and IV. This calculation is particularly important for planetary nebulae that are ambiguously classified in the traditional method. The posterior probabilities are defined from the probability density function of classificatory parameters of a well-defined sample, composed only by planetary nebulae unambiguously fitted into the Peimbert types. Because the probabilities depend on the available observational data, they are conditional probabilities, and, as new observational data are added to the sample, the classification of the nebula can be improved, to take into account this new information. This method differs from the original classificatory scheme, because it provides a quantitative result of the representativity of the object within its group. Also, through the use of marginal distributions it is possible to extend the Peimbert classification even to those objects for which only a few classificatory parameters are known.