Laplace approximations in inference

Posterior updates without likelihood



Second mode? I see no second mode.

Approximating probability distributions by a Gaussian with the same mode. Thanks to limit theorems this is not always a terrible idea.

References

Breslow, N. E., and D. G. Clayton. 1993. “Approximate Inference in Generalized Linear Mixed Models.” Journal of the American Statistical Association 88 (421): 9–25. https://doi.org/10.2307/2290687.
Gorad, Ajinkya, Zheng Zhao, and Simo särkkä. 2020. “Parameter Estimation in Non-Linear State-Space Models by Automatic Differentiation of Non-Linear Kalman Filters.” In, 6.
Huggins, Jonathan H., Trevor Campbell, Mikołaj Kasprzak, and Tamara Broderick. 2018. “Practical Bounds on the Error of Bayesian Posterior Approximations: A Nonasymptotic Approach.” arXiv:1809.09505 [Cs, Math, Stat], September. http://arxiv.org/abs/1809.09505.
Ingebrigtsen, Rikke, Finn Lindgren, and Ingelin Steinsland. 2014. “Spatial Models with Explanatory Variables in the Dependence Structure.” Spatial Statistics, Spatial Statistics Miami, 8 (May): 20–38. https://doi.org/10.1016/j.spasta.2013.06.002.
Kristiadi, Agustinus, Matthias Hein, and Philipp Hennig. 2021. “Learnable Uncertainty Under Laplace Approximations.” In Uncertainty in Artificial Intelligence. http://arxiv.org/abs/2010.02720.
Opitz, Thomas, Raphaël Huser, Haakon Bakka, and Håvard Rue. 2018. “INLA Goes Extreme: Bayesian Tail Regression for the Estimation of High Spatio-Temporal Quantiles.” Extremes 21 (3): 441–62. https://doi.org/10.1007/s10687-018-0324-x.
Rue, Håvard. n.d. “Bayesian Spatial Modelling with R-INLA.” Accessed January 14, 2021. https://core.ac.uk/reader/161914450.
Rue, Håvard, Andrea Riebler, Sigrunn H. Sørbye, Janine B. Illian, Daniel P. Simpson, and Finn K. Lindgren. 2016. “Bayesian Computing with INLA: A Review.” arXiv:1604.00860 [Stat], September. http://arxiv.org/abs/1604.00860.
Saumard, Adrien, and Jon A. Wellner. 2014. “Log-Concavity and Strong Log-Concavity: A Review.” arXiv:1404.5886 [Math, Stat], April. http://arxiv.org/abs/1404.5886.
Tang, Yanbo, and Nancy Reid. 2021. “Laplace and Saddlepoint Approximations in High Dimensions.” arXiv:2107.10885 [Math, Stat], July. http://arxiv.org/abs/2107.10885.

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