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On a generalization of the Laplace approximation

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  • Nott, David J.
  • Fielding, Mark
  • Leonte, Daniela

Abstract

Laplace approximation is one commonly used approach to the calculation of difficult integrals arising in Bayesian inference and the analysis of random effects models. Here we outline a procedure which is an extension of the Laplace approximation and which attempts to find changes of variable for which the integrand becomes approximately a product of one-dimensional functions. When the integrand is a product of one-dimensional functions, an approximation to the integral can be obtained using one-dimensional quadrature. The approximation is exact for a broader class of functions than the ordinary Laplace approximation and can be applied when the integrand is not smooth at the mode. As an illustration of this last point we consider calculation of marginal likelihoods for smoothing parameter selection in the lasso.

Suggested Citation

  • Nott, David J. & Fielding, Mark & Leonte, Daniela, 2009. "On a generalization of the Laplace approximation," Statistics & Probability Letters, Elsevier, vol. 79(11), pages 1397-1403, June.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:11:p:1397-1403
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    References listed on IDEAS

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    1. Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
    2. Jones, M. C., 2002. "Marginal Replacement in Multivariate Densities, with Application to Skewing Spherically Symmetric Distributions," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 85-99, April.
    3. Park, Trevor & Casella, George, 2008. "The Bayesian Lasso," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 681-686, June.
    4. Scott Foster & Arūnas Verbyla & Wayne Pitchford, 2008. "A random model approach for the LASSO," Computational Statistics, Springer, vol. 23(2), pages 217-233, April.
    5. Evans, Michael & Swartz, Timothy, 2000. "Approximating Integrals via Monte Carlo and Deterministic Methods," OUP Catalogue, Oxford University Press, number 9780198502784.
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