Confidence limits of evolutionary synthesis models IV.: Moving forward to a probabilistic formulation

Context. Synthesis models predict the integrated properties of stellar populations. Several problems exist in this field, mostly related to the fact that integrated properties are distributed. To date, this aspect has been either ignored ( as in standard synthesis models, which are inherently deterministic) or interpreted phenomenologically ( as in Monte Carlo simulations, which describe distributed properties rather than explain them). Aims. This paper presents a method of population synthesis that accounts for the distributed nature of stellar properties. Methods. We approach population synthesis as a problem in probability theory, in which stellar luminosities are random variables extracted from the stellar luminosity distribution function ( sLDF). Results. With standard distribution theory, we derive the population LDF ( pLDF) for clusters of any size from the sLDF, obtaining the scale relations that link the sLDF to the pLDF. We recover the predictions of standard synthesis models, which are shown to compute the mean of the luminosity function. We provide diagnostic diagrams and a simplified recipe for testing the statistical richness of observed clusters, thereby assessing whether standard synthesis models can be safely used or a statistical treatment is mandatory. We also recover the predictions of Monte Carlo simulations, with the additional bonus of being able to interpret them in mathematical and physical terms. We give examples of problems that can be addressed through our probabilistic formalism: calibrating the SBF method, determining the luminosity function of globular clusters, comparing different isochrone sets, tracing the sLDF by means of resolved data, including fuzzy stellar properties in population synthesis, among others. Additionally, the algorithmic nature of our method makes it suitable for developing analysis tools for the Virtual Observatory. Conclusions. Though still under development, ours is a powerful approach to population synthesis. In an era of resolved observations and pipelined analyses of large surveys, this paper is offered as a signpost in the field of stellar populations.