As we move into the next era of cosmology, dominated by large extensive surveys in the name of precision. The question of how best to constrain our current models with the vast amount of data available to us now and in the future has become prominent. This thesis looks to answer that question in the field of weak lensing, specifically for redshift distributions, a typical weakness of the field.

To begin, this thesis will look at the cosmological parameters that describe our universe to the best of our ability and the probes available for constraining them. Chapter 2 looks at one of those probes in more detail, weak lensing, and how a light bending phenomenon can be used to infer statistical properties of the matter distribution of the universe and how its precision is hampered by weaknesses in redshift inference. Methods for achieving this is the focus of Chapter 3, photometric redshifts; using light collected from galaxies across a small number of broadband filters to infer its redshift based on known spectral features. We look into how such methods developed, how these spectral features exist, and the various approaches used. One approach requires a change to our statistical thinking, and thus Chapter 4 focuses on Bayesian statistics and ways it can be applied to problems. One such application is considered in Chapter 5 where we modify the work of Leistedt 2016 for weak lensing-like settings. In Chapter 6 we apply the work to Kilo Degree Survey, specifically KV450 and constrain cosmological parameters potentially lowering a current tension in $S_8$ values between weak lensing and CMB studies. Lastly, in Chapter 7, we gather preliminary results using our approach to KiDS-1000 data, considering improvements required for the levels of precision and accuracy of future surveys.