We present a new Bayesian hierarchical model (BHM) named Steve for performing type Ia supernova (SNIa) cosmology fits. This advances previous works by including an improved treatment of Malmquist bias, accounting for additional sources of systematic uncertainty, and increasing numerical efficiency. Given light curve fit parameters, redshifts, and host-galaxy masses, we fit Steve simultaneously for parameters describing cosmology, SNIa populations, and systematic uncertainties. Selection effects are characterised using Monte-Carlo simulations. We demonstrate its implementation by fitting realisations of SNIa datasets where the SNIa model closely follows that used in Steve. Next, we validate on more realistic SNANA simulations of SNIa samples from the Dark Energy Survey and low-redshift surveys. These simulated datasets contain more than $60,000$ SNeIa, which we use to evaluate biases in the recovery of cosmological parameters, specifically the equation-of-state of dark energy, $w$. This is the most rigorous test of a BHM method applied to SNIa cosmology fitting, and reveals small $w$-biases that depend on the simulated SNIa properties, in particular the intrinsic SNIa scatter model. This $w$-bias is less than $0.03$ on average, less than half the statistical uncertainty on $w$.These simulation test results are a concern for BHM cosmology fitting applications on large upcoming surveys, and therefore future development will focus on minimising the sensitivity of Steve to the SNIa intrinsic scatter model.
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