A probabilistic formulation for empirical population synthesis:: sampling methods and tests

We revisit the classical problem of synthesizing spectral properties of a galaxy by using a base of star clusters, approaching it from a probabilistic perspective. The problem consists of estimating the population vector x, composed by the contributions of n(u), different base elements to the integrated spectrum of a galaxy, and the extinction Av, given a set of absorption line equivalent widths and continuum colours. The formalism is applied to the base of 12 elements defined by Schmidt et al. as corresponding to the principal components of the original base employed by Bica, and subsequently used in several studies of the stellar populations of galaxies. The exploration of the 13D parameter space is carried out with a Markov chain Monte Carlo sampling scheme, based on the Metropolis algorithm. This produces a smoother and more efficient mapping of the P(x,A(v)) probability distribution than the traditionally employed uniform-grid sampling. This new version of empirical population synthesis is used to investigate the ability to recover the detailed history of star formation and chemical evolution using this spectral base. This is studied as a function of (i) the magnitude of the measurement errors and (ii) the set of observables used in the synthesis. Extensive simulations with test galaxies are used for this purpose. The emphasis is put on the comparison of input parameters and the mean x and Av associated with the P(x,A(v)) distribution. It is found that only for extremely low errors [signal-to-noise ratio (S/N) > 300 at 5870 Angstrom] all 12 base proportions can be accurately recovered, though the observables are recovered very precisely for any S/N. Furthermore, the individual (x) over bar components are biased in the sense that components which carry a large fraction of the light tend to share their contribution preferably among components of same age. Old, metal-poor components can also be confused with younger, metal-rich components because of the age-metallicity degeneracy. These compensation effects are linked to noise-induced linear dependences in the base, which redistribute the likelihood in x-space very effectively. The age distribution, however, can be satisfactorily recovered for realistic S/N (similar to 30). We also find that synthesizing equivalent widths and colours produces better focused results that those obtained synthesizing only equivalent widths, despite the inclusion of the extinction as an extra parameter.