Operaciones y logística Infraestructura (O)

In this paper we have introduced a method for solving Bayesian design problems which require a large number of designs points to be found, through incorporation of lower dimensional parameterisations into existing stochastic optimisation algorithms. These lower dimensional parameterisations consisted of a few design variables, which were then input into various functions to generate multiple design points. This approach was found to have substantial computational savings since one simply has to search over a few design variables, rather than a large number of design variables. Also, it was much easier to obtain the multivariate mode for a few design variables than for a large number of design variables. However, it should be stressed that the designs presented in this paper are not optimal but near optimal, which is a compromise of the computational savings achieved through these methods.

Our approach is useful for design variables (e.g., sampling times) that require multiple measures to be taken at specific points that are separated from one another in the design space (although as shown in a toy example, the lower dimensional parameterisations can produce simple replicated designs). This approach does not overcome the problem of having a large number of different design variables (e.g., temperatures, pressures), and further research should be conducted for solving this design problem.

The functions that were used in this paper to generate the designs were purely illustrative for the examples chosen for this paper. There are many other functions and transforma- tions available that one could choose to generate their design points and choice of an appropriate function would depend on the application of interest. To determine which function is most appropriate for a particular application, one could run several different parameterisations in parallel on different CPUs and choose the function which gives rise to the design with the highest utility.

We found that the beta proposal scheme, where the designs come from the (evenly-spaced) percentiles of a beta distribution, gave quite flexible designs, and so this function may be appropriate in many situations to generate a large number of design points. One could also extend the beta proposal scheme to propose from a generalised beta distribution (e.g., Sepanski and Kong (2007)), which may offer further flexibility in constructing the designs. One could also include another design variable in the parameterisation of the beta proposal scheme that determines the optimal percentiles of the beta distribution to use, e.g., percentile = 100( _ni
α

) where α is an additional design variable to search over. In our PK example, the designs which were generated by the lower dimensional parame- terisations gave higher utility values than the replicate designs. This is useful since it may not be feasible to take replicate designs for studies in which one is interested in determin- ing the optimal sampling times. However, this may not always be the case and replicate designs may be preferred in some applications. If replicate designs are practically feasible for the experiment of interest, then one may wish to instead generate a large number of design points based on the estimated weights of a continuous or approximate design, which also falls into our framework of lower dimensional parameterisations. We will be investigating this in future work.

A fixed number of sampling times were assumed for the examples used in this study, so that we could demonstrate our methodology for generating many design points. The number of sampling times used in this study may not be optimal and future studies may wish to investigate the optimal number of design points for their applications. For experiments where cost is of little importance and the utility function does not contain a cost function, in general, the more design points there are the better.

Importance sampling did not perform particularly well (low effective sample sizes were often obtained) when it was used to estimate the Bayesian utility functions (especially for the KLD) for nonlinear models with a large number of observations. A large number of samples (10000-20000) was required to ensure that the utility was well estimated. To estimate a utility well, a certain number of samples is required. This number of samples

may vary across the utilities. KLD is harder to estimate in general since it involves the estimation of the evidence. The mean response variance utility may require less importance samples since the utility is estimated by bringing in observations one-at-a- time. For another paper we are investigating alternative methods for estimating Bayesian utility functions, such as Laplace approximations.

Acknowledgments

E.G. Ryan was supported by an APA(I) Scholarship which came from an ARC Linkage Grant with Genentech (LP0991602). The work of C.C. Drovandi and A.N. Pettitt was also supported by an ARC Discovery Project (DP110100159).

The authors would like to thank Dr James McGree from Queensland University of Tech- nology for his helpful comments throughout the process of writing this paper. The authors would also like to thank the referees and the associate editor for their insightful comments and suggestions throughout the review process of this paper.

Statement of Authorship for Chapter 4

This chapter has been written as a journal article. The authors listed below have certified that:

x They meet the criteria for authorship in that they have participated in the concep- tion, execution, or interpretation, of at least that part of the publication in their field of expertise;

x They take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

x There are no other authors of the publication according to these criteria;

x Potential conflicts of interest have been disclosed to (a) granting bodies, (b) the edi- tor or publisher of journals or other publications, and (c) the head of the responsible academic unit, and

x They agree to the use of the publication in the students thesis and its publica- tion on the QUT ePrints database consistent with any limitations set by publisher requirements.

In the case of this chapter, the reference for the associated publication is: Ryan, E.G., Drovandi, C.C., and Pettitt, A.N. (2014). Fully Bayesian Designs for Nonlinear Mixed Effects Models. Submitted to Computational Statistics and Data Analysis.

Contributor Statement of contribution

Elizabeth Ryan Wrote all Matlab code required for the manuscript,

performed all computations in the manuscript, inter- preted and reported the results, constructed all figures presented in the manuscript, wrote the manuscript, and acted as the corresponding author

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Christopher Drovandi Assissted in the writing of the Matlab code, directed the research and proofread the manuscript.

Tony Pettitt Directed the research and proofread the manuscript.

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I have sighted email or other correspondence from all co-authors confirming their certifying authorship.