This thesis is concerned with the modelling of galaxy clusters, applying these models to real and simulated data using Bayesian inference, and the development of Bayesian inference algorithms applicable to a wide range of astrophysical problems.

I present a comparison of mass estimates for $54$ galaxy cluster candidates from the second Planck catalogue (PSZ2) of Sunyaev--Zel'dovich sources. I compare the mass values obtained with data taken from the Arcminute Microkelvin Imager (AMI) radio interferometer system and from the Planck satellite. The former of these uses a Bayesian analysis pipeline that parameterises a cluster in terms of its physical quantities, and models the dark matter & baryonic components of a cluster using Navarro-Frenk-White (NFW) and generalised-NFW profiles respectively. The mass estimates derived from Planck data are obtained from the results of the Bayesian detection algorithm PowellSnakes (PwS). I also analyse simulated AMI data with input values based on PwS mass estimates.

I then compare three cluster models using AMI data for the 54 cluster sample. The two observational models considered only model the gas content of the cluster. To compare the physical and observational models I consider their posterior parameter estimates, including the calculation of a metric defined between two probability distributions. The models' fit to the cluster data is evaluated by looking at the Bayesian evidence values.

Improvements to the physical modelling of galaxy clusters are then considered, either by relaxing some of the assumptions underlying the physical model, or by introducing a new profile for the dark matter component of clusters. The resultant models are compared with the physical model introduced previously.

The final part of the cluster analysis work focuses on Bayesian analysis using a joint likelihood function of data from both AMI and the Planck satellite simultaneously. The results of this joint analysis are compared with those obtained from the individual likelihood analyses using simulated data and with real data taken from the $54$ cluster sample.

Finally, a new Bayesian inference algorithm based on nested sampling is presented. The algorithm, named the "geometric nested sampler", is an adaption of the Metropolis-Hastings nested sampler and makes use of the geometrical interpretation of sets of parameters to sample from their domains efficiently. The geometric nested sampler is tested on several toy models as well as a model representing the emission of gravitational waves from binary black hole mergers. The results obtained using the geometric nested sampler are compared with those from popular nested sampling algorithms.