We present the Bayesian oxygen and nitrogen abundance determinations (bond) method. bond is a Bayesian code (available at: http://bond.ufsc.br) to simultaneously derive oxygen and nitrogen abundances in giant HII regions. It compares observed emission lines to a grid of photoionization models without assuming any relation between O/H and N/O. Our grid spans a wide range in O/H, N/O and ionization parameter U, and covers different starburst ages and nebular geometries. Varying starburst ages accounts for variations in the ionizing radiation field hardness, which arise due to the ageing of HII regions or the stochastic sampling of the initial mass function. All previous approaches assume a strict relation between the ionizing field and metallicity. The other novelty is extracting information on the nebular physics from semistrong emission lines. While strong lines ratios alone ([OIII]/H{beta}, [OII]/H{beta} and [NII]/H{beta}) lead to multiple O/H solutions, the simultaneous use of [ArIII]/[NeIII] allows one to decide whether an HII region is of high or low metallicity. Adding HeI/H{beta} pins down the hardness of the radiation field. We apply our method to HII regions and blue compact dwarf galaxies, and find that the resulting N/O versus O/H relation is as scattered as the one obtained from the temperature-based method. As in previous strong-line methods calibrated on photoionization models, the bond O/H values are generally higher than temperature-based ones, which might indicate the presence of temperature fluctuations or kappa distributions in real nebulae, or a too soft ionizing radiation field in the models.