Gaussian likelihoods for ’intractable’ situations

Various modelling situations – including chaotic dynamics, stochastic differential equations, random patterns such as produced by the Turing reaction-diffusion systems, or the Cahn-Hilliard equation – share the analogy that a  fixed  model parameter corresponds to a family of solutions rather than a fixed deterministic one. This may be due to extreme sensitivity with respect to the initial values,  randomized or unknown initial values, or the  explicit stochasticity of the system. As a result, standard methods based on directly measuring the distance between model output and data are no more available. We discuss an approach that allows a unified construction of likelihoods for such ‘intractable’ situations. The starting point is the Donsker theorem stating that the cumulative distribution function of i.i.d scalar samples tends to a Gaussian distribution. But the approach can be extended also to weakly dependent, and vector-valued data. Several cases from the above list are presented as examples.
 

invited by Jana de Wiljes