We present a new method for inferring the metallicity (Z) and ionization parameter (q) of HII regions and star-forming galaxies using strong nebular emission lines (SEL). We use Bayesian inference to derive the joint and marginalized posterior probability density functions for Z and q given a set of observed line fluxes and an input photo-ionization model. Our approach allows the use of arbitrary sets of SELs and the inclusion of flux upper limits. The method provides a self-consistent way of determining the physical conditions of ionized nebulae that is not tied to the arbitrary choice of a particular SEL diagnostic and uses all the available information. Unlike theoretically calibrated SEL diagnostics the method is flexible and not tied to a particular photo-ionization model. We describe our algorithm, validate it against other methods, and present a tool that implements it called IZI. Using a sample of nearby extra-galactic HII regions we assess the performance of commonly used SEL abundance diagnostics. We also use a sample of 22 local HII regions having both direct and recombination line (RL) oxygen abundance measurements in the literature to study discrepancies in the abundance scale between different methods. We find that oxygen abundances derived through Bayesian inference using currently available photo-ionization models in the literature can be in good (~30%) agreement with RL abundances, although some models perform significantly better than others. We also confirm that abundances measured using the direct method are typically 0.2 dex lower than both RL and photo-ionization model based abundances.
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