In thesis, I aim to apply advanced methods in Bayesian statistical modelling on Type Ia Supernovae (SNIa) data to determine tighter constraints on the fiducial Lambda-Cold-Dark-Matter (LCDM) cosmology and improve the modelling of systematic uncertainties in the data. The body of work covered herein can be broadly classified into two main topics:

I re-examine the contentious question of constraints on anisotropic expansion from SNIa in the light of a novel determination of peculiar velocities, which are crucial to test isotropy with SNe, out to distances < 200/h Mpc.The Bayesian hierarchical model BAHAMAS is adopted to constrain a dipole in the distance modulus in the context of the LCDM model and the deceleration parameter in a phenomenological Cosmographic expansion. I find no evidence for anisotropic expansion, and place a tight upper bound on the amplitude of a dipole, in a LCDM setting, and the Cosmographic expansion approach. Using Bayesian model comparison, I obtain posterior odds in excess of 900:1 (640:1) against a constant-in-redshift dipole for LCDM (Cosmographic expansion).

One of the modern problems of Supernovae cosmology is accounting for selection effects caused by Malmquist bias in a principled way. Here, I present a complete formalism for handling selection effects in Type Ia supernova (SNIa) cosmology in the context of Bayesian Hierarchical Modeling. I demonstrate the method on simulated data sets where selection cuts are made on the apparent magnitude and show that previous results by Rubin et al, (2015) are incorrect and can lead to biased cosmological parameters reconstruction. I how this formalism is easily extended to include the Phillips corrections that are used to standardize SNe. The formalism presented exhibits better statistical properties in terms of bias and mean squared error relative to a traditional ad hoc style correction and the model of Rubin et al, (2015)