This thesis seeks to advance modelling in cognitive science in three ways. The first is by introducing the field to MultiNest, an exemplar of the overlooked nested sampling approach to performing Bayesian inference borrowed from advanced physics; it will be demonstrated that, in applications involving cognitive models and data, MultiNest offers a number of advantages over the prominently proposed thermodynamic integration and steppingstone methods when calculating the crucial Bayes factor.

Second, the thesis puts forward new and more expansive models of simple reaction time task (SRT) performance. Despite the importance of empirically well-established foreperiod effects in simple responding, extant mathematical models of SRT performance confine themselves to describing the marginal response time distribution only; the new models presented here, developed by extending evidence accumulator models of decision making to possess foreperiod-modulated time-dependent decision thresholds, simultaneously account for both foreperiod effects and response time distribution.

Finally, it is argued that modelling work in cognitive science, pace current tendencies towards simplifying assumptions, should focus on providing full accounts of complex cognitive phenomena. Bayesian model comparison – undertaken with MultiNest and SRT data from numerous subjects – shows that the new SRT models, despite their increased complexity, receive far greater support from the data across subjects relative to their predecessors. This result is interpreted to mean that the additional complexity in the new models echoes complex patterns inherent in the data that must be preserved in any satisfactory theory explaining the data: the data, it will be said, possess irreducible complexity. If this is true when considering a task as simple as the SRT, it is likely true across a host of other cognitive domains; thus, it is concluded, cognitive modelling would do well to embrace the goal of developing models which give full accounts of the irreducible complexity of cognitive data.