Stochastic gradient descent for optimal design in ordinary differential equations

Background:

Many experiments are dependent on time through an underlying model for which the parameters are unknown. Here we will focus on models defined by ordinary differential equations. In order to gain information for the true value of such parameters, observations need to be made at various time points. It is intuitive that the more observations, the more information we gain, however this is not always feasible due to constraints on budget and resources available. This highlights the problem of when to take the limited number of observations with the aim of attaining the most information about the model parameters. Realisations from the model are noisy, making choosing the best design points difficult to identify. Optimal design is the exploration of a solution to this problem.

The statistical models we shall be using have parameters which are unknown. We aim to learn as much about these parameters as possible through inference methods such as importance sampling or Markov Chain Monte Carlo. To gain a quantitative summary on how good the inference results are in comparison to the true values, we use a utility function. There are various choices of utility function but initially we shall use the negative mean square error. Our aim is to choose a design which optimises expected utility i.e. utility averaged appropriately over all possible true values of the parameters and resulting observations.

In practice this is incredibly difficult as only noisy estimates of expected utility can be made, meaning it is difficult to identify the optimal times. In order to find the maximum of the expected utility we will use a stochastic gradient descent algorithm. This algorithm estimates the gradient of the utility at given times, using this to iteratively get closer to the optimal times.

Project:

The project will initially look at a variation of the pharmacokinetic study model defined by Ryan, Drovandi and Pettitt. I shall use an importance sampler with prior proposal to obtain draws from the posterior distribution of the parameters. In order to distinguish which design maximises the utility a simple stochastic gradient descent algorithm will be implemented. This will be compared to a basic grid search.

After this simple example has been explored, I could then progress to research better ways of finding the optimal design. These could include:
* using similar approach to alternative models such as the SIR model
* looking at various utility functions and how they perform
* not only is it important to identify when to take observations, we should identify how many observations should be made
* are there more efficient ways of implementing an algorithm to find the times which maximises the utility function