  We present a Bayesian methodology for infinite as well as finite dimensional
parameter identification for partial differential equation models. The Bayesian
framework provides a rigorous mathematical framework for incorporating prior
knowledge on uncertainty in the observations and the parameters themselves,
resulting in an approximation of the full probability distribution for the
parameters, given the data. Although the numerical approximation of the full
probability distribution is computationally expensive, parallelised algorithms
can make many practically relevant problems computationally feasible. The
probability distribution not only provides estimates for the values of the
parameters, but also provides information about the inferability of parameters
and the sensitivity of the model. This information is crucial when a
mathematical model is used to study the outcome of real-world experiments.
Keeping in mind the applicability of our approach to tackle real-world
practical problems with data from experiments, in this initial proof of concept
work, we apply this theoretical and computational framework to parameter
identification for a well studied semilinear reaction-diffusion system with
activator-depleted reaction kinetics, posed on evolving and stationary domains.
