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Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming-Based Approach

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  • Belmiro P. M. Duarte
  • Weng Kee Wong

Abstract

type="main" xml:id="insr12073-abs-0001"> This paper uses semidefinite programming (SDP) to construct Bayesian optimal design for nonlinear regression models. The setup here extends the formulation of the optimal designs problem as an SDP problem from linear to nonlinear models. Gaussian quadrature formulas (GQF) are used to compute the expectation in the Bayesian design criterion, such as D-, A- or E-optimality. As an illustrative example, we demonstrate the approach using the power-logistic model and compare results in the literature. Additionally, we investigate how the optimal design is impacted by different discretising schemes for the design space, different amounts of uncertainty in the parameter values, different choices of GQF and different prior distributions for the vector of model parameters, including normal priors with and without correlated components. Further applications to find Bayesian D-optimal designs with two regressors for a logistic model and a two-variable generalised linear model with a gamma distributed response are discussed, and some limitations of our approach are noted.

Suggested Citation

  • Belmiro P. M. Duarte & Weng Kee Wong, 2015. "Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming-Based Approach," International Statistical Review, International Statistical Institute, vol. 83(2), pages 239-262, August.
    • Handle: RePEc:bla:istatr:v:83:y:2015:i:2:p:239-262
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    File URL: http://hdl.handle.net/10.1111/insr.12073
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    References listed on IDEAS

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    1. Dette, Holger & Melas, Viatcheslav B. & Wong, Weng Kee, 2005. "Optimal Design for Goodness-of-Fit of the MichaelisMenten Enzyme Kinetic Function," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1370-1381, December.
    2. Fu-Chuen Chang & Hung-Ming Lin, 2007. "On Minimally-supported D-optimal Designs for Polynomial Regression with Log-concave Weight Function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(2), pages 227-233, February.
    3. R. R. Sitter & C. F. J. Wu, 1999. "Two-Stage Design of Quanta1 Response Studies," Biometrics, The International Biometric Society, vol. 55(2), pages 396-402, June.
    4. Joy King & Weng-Kee Wong, 2000. "Minimax D-Optimal Designs for the Logistic Model," Biometrics, The International Biometric Society, vol. 56(4), pages 1263-1267, December.
    5. Harman, Radoslav & Pronzato, Luc, 2007. "Improvements on removing nonoptimal support points in D-optimum design algorithms," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 90-94, January.
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    Cited by:

    1. Duarte, Belmiro P.M. & Sagnol, Guillaume & Wong, Weng Kee, 2018. "An algorithm based on semidefinite programming for finding minimax optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 99-117.
    2. Belmiro P. M. Duarte, 2023. "Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
    3. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2019. "Optimal design of experiments for liquid–liquid equilibria characterization via semidefinite programming," LSE Research Online Documents on Economics 102500, London School of Economics and Political Science, LSE Library.
    4. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2022. "Optimal design of experiments for implicit models," LSE Research Online Documents on Economics 107584, London School of Economics and Political Science, LSE Library.
    5. Belmiro P. M. Duarte & Guillaume Sagnol, 2020. "Approximate and exact optimal designs for $$2^k$$ 2 k factorial experiments for generalized linear models via second order cone programming," Statistical Papers, Springer, vol. 61(6), pages 2737-2767, December.
    6. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2021. "A model-based framework assisting the design of vapor-liquid equilibrium experimental plans," LSE Research Online Documents on Economics 107448, London School of Economics and Political Science, LSE Library.
    7. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Oliveira, Nuno M.C., 2024. "Using hierarchical information-theoretic criteria to optimize subsampling of extensive datasets," LSE Research Online Documents on Economics 121641, London School of Economics and Political Science, LSE Library.

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