Bayesian experimental design for implicit models using mutual information
Scientists regularly face the challenging task of designing experiments in such a way that the collected data is informative and useful. The field of Bayesian experimental design formalises this task by phrasing it as an optimisation problem. Here, the goal is to maximise a utility function that describes the value of an experimental design according to the scientific aims of the experiment. The mutual information, which quantifies the expected information gain about variables of interest, is a principled choice of utility function that has seen extensive use in literature. However, computing the mutual information is intractable for all but the most simple computational models of nature. Indeed, as our scientific theories improve, scientists are increasingly devising models that have intractable likelihood functions, so-called implicit models. The increased realistic behaviour of implicit models comes at the cost of severely complicating the Bayesian design of experiments. The work presented in this thesis provides several solutions to Bayesian experimental design for implicit models using the mutual information utility function. Although a desirable quantity, mutual information is generally prohibitively expensive to compute because it involves posterior distributions, which are naturally intractable for implicit models. Therefore, existing literature has, mostly, either considered special settings where mutual information can be approximated, or utilised other simplified utility functions altogether. First, we present a method of approximating the mutual information using density ratio estimation techniques, where the only requirement is that we can sample data from the computational model, which is naturally satisfied for implicit models. This allows us to efficiently estimate the mutual information and then solve the Bayesian experimental design problem by maximising it by means of gradient-free optimisation techniques. Following this, we present an extension that concerns sequential Bayesian experimental design, where the aim is to find optimal designs and gather data in a sequential manner. We use the density ratios learned through the aforementioned approach to update our beliefs of the variable of interest at every iteration, which then repeatedly changes the optimisation landscape. Similar to before, we optimise the sequential mutual information at every iteration using gradient-free techniques. Next, we present a method where we construct a lower bound on the mutual information that is parametrised by a neural network. Neural networks provide great flexibility and, more importantly, allow us to back-propagate from the lower bound estimate to the experimental designs. We can therefore simultaneously tighten and maximise the mutual information lower bound using stochastic gradient-ascent. As opposed to previous gradient-free approaches, this results in greater scalability with respect to the number of experimental design dimensions. Following this, we provide a general framework that accommodates the use of (a) several lower bounds with different bias-variance trade-offs and (b) several important scientific tasks instead of only a single one (as is common in exist ing literature), such as parameter estimation, distinguishing between competing models and improving future predictions. Lastly, we present an application of this approach to cognitive science, where we design behavioural experiments with the aim of estimating parameters of and distinguishing between cognitive models. We showcase the advantages of our method by performing real-world experiments with human participants, demonstrating how scientists can use and profit from Bayesian experimental design methods in practice, even when likelihood functions are intractable.