Bayesian nonparametric inference in mechanistic models of complex biological systems

Parameter estimation in expensive computational models is a problem that commonly arises in science and engineering. With the increase in computational power, modellers started developing simulators of real life phenomena that are computationally intensive to evaluate. This, however, makes inference prohibitive due to the unit cost of a single function evaluation. This thesis focuses on computational models of biological and biomechanical processes such as the left-ventricular dynamics or the human pulmonary blood circulatory system. In the former model a single forward simulation is in the order of 11 minutes CPU time, while the latter takes approximately 23 seconds in our machines. Markov chain Monte Carlo methods or likelihood maximization using iterative algorithms would take days or weeks to provide a result. This makes them not suitable for clinical decision support systems, where a decision must be taken in a reasonable time frame. I discuss how to accelerate the inference by using the concept of emulation, i.e. by replacing a computationally expensive function with a statistical approximation based on a finite set of expensive training runs. The emulation target could be either the output-domain, representing the standard approach in the emulation literature, or the loss-domain, which is an alternative and different perspective. Then, I demonstrate how this approach can be used to estimate the parameters of expensive simulators. First I apply loss-emulation to a nonstandard variant of the Lotka-Volterra model of prey-predator interactions, in order to assess if the approach is approximately unbiased. Next, I present a comprehensive comparison between output-emulation and loss-emulation on a computational model of left ventricular dynamics, with the goal of inferring the constitutive law relating the myocardial stretch to its strain. This is especially relevant for assessing cardiac function post myocardial infarction. The results show how it is possible to estimate the stress-strain curve in just 15 minutes, compared to the one week required by the current best literature method. This means a reduction in the computational costs of 3 orders of magnitude. Next, I review Bayesian optimization (BO), an algorithm to optimize a computationally expensive function by adaptively improving the emulator. This method is especially useful in scenarios where the simulator is not considered to be a ``stable release''. For example, the simulator could still be undergoing further developments, bug fixing, and improvements. I develop a new framework based on BO to estimate the parameters of a partial differential equation (PDE) model of the human pulmonary blood circulation. The parameters, being related to the vessel structure and stiffness, represent important indicators of pulmonary hypertension risk, which need to be estimated as they can only be measured with invasive experiments. The results using simulated data show how it is possible to estimate a patient's vessel properties in a time frame suitable for clinical applications. I demonstrate a limitation of standard improvement-based acquisition functions for Bayesian optimization. The expected improvement (EI) policy recommends query points where the improvement is on average high. However, it does not account for the variance of the random variable Improvement. I define a new acquisition function, called ScaledEI, which recommends query points where the improvement on the incumbent minimum is expected to be high, with high confidence. This new BO algorithm is compared to acquisition functions from the literature on a large set of benchmark functions for global optimization, where it turns out to be a powerful default choice for Bayesian optimization. ScaledEI is then compared to standard non-Bayesian optimization solvers, to confirm that the policy still leads to a reduction in the number of forward simulations required to reach a given tolerance level on the function value. Finally, the new algorithm is applied to the problem of estimating the PDE parameters of the pulmonary circulation model previously discussed.