Deterministic approximations

Content

  • Bernstein–von Mises theorem
  • Laplace approximations
  • integrated nested Laplace approximation (INLA)

Learning objectives

At the end of the chapter, students should be able to

  • derive the Laplace approximation for simple models to compute marginal likelihood, posterior expectation or approximation to density.
  • describe the difference between deterministic and stochastic approximations.

Readings

Complementary readings

  • Laplace approximation use in statistics: Tierney & Kadane (1986)
  • skewed Bernstein–von Mises: Durante et al. (2024)
  • Laplace approximation for Bayes factor: Raftery (1995)
  • integrated nested Laplace approximation: Rue et al. (2009), Wood (2019)

Slides

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Code

  • Laplace approximations for exponential-gamma model R script
  • INLA package R script

References

Durante, D., Pozza, F., & Szabo, B. (2024). Skewed Bernstein–von Mises theorem and skew-modal approximations. The Annals of Statistics, 52(6), 2714–2737. https://doi.org/10.1214/24-aos2429
Raftery, A. E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–163. https://doi.org/10.2307/271063
Rue, H., Martino, S., & Chopin, N. (2009). Approximate bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(2), 319–392. https://doi.org/10.1111/j.1467-9868.2008.00700.x
Tierney, L., & Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393), 82–86. https://doi.org/10.1080/01621459.1986.10478240
Wood, S. N. (2019). Simplified integrated nested Laplace approximation. Biometrika, 107(1), 223–230. https://doi.org/10.1093/biomet/asz044