Florian Hartig

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An extended empirical saddlepoint approximation for intractable likelihoods

Matteo Fasiolo, Simon N. Wood, Florian Hartig & Mark V. Bravington

Electronic Journal of Statistics, 12(1), 1544–1578 (2018)
Cite this
@article{fasiolo2018extended,
  author = {Matteo Fasiolo and Simon N. Wood and Florian Hartig and Mark V. Bravington},
  title = {An extended empirical saddlepoint approximation for intractable likelihoods},
  journal = {Electronic Journal of Statistics},
  volume = {12},
  number = {1},
  pages = {1544--1578},
  year = {2018},
  doi = {10.1214/18-EJS1433},
}

DOI: 10.1214/18-EJS1433
Citation statistics: 48 (Google Scholar) · 7 (OpenAlex), as of 29 October 2026

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Funding

This study was supported through funding by EPSRC (Engineering and Physical Sciences Research Council) (EP/I000917/1; EP/K005251/1).

View article (DOI)

Abstract

The challenges posed by complex stochastic models used in computational ecology, biology and genetics have stimulated the development of approximate approaches to statistical inference. Here we focus on Synthetic Likelihood (SL), a procedure that reduces the observed and simulated data to a set of summary statistics, and quantifies the discrepancy between them through a synthetic likelihood function. SL requires little tuning, but it relies on the approximate normality of the summary statistics. We relax this assumption by proposing a novel, more flexible, density estimator: the Extended Empirical Saddlepoint approximation. In addition to proving the consistency of SL, under either the new or the Gaussian density estimator, we illustrate the method using three examples. One of these is a complex individual-based forest model for which SL offers one of the few practical possibilities for statistical inference. The examples show that the new density estimator is able to capture large departures from normality, while being scalable to high dimensions, and this in turn leads to more accurate parameter estimates, relative to the Gaussian alternative. The new density estimator is implemented by the esaddle R package, which is freely available on the Comprehensive R Archive Network (CRAN).

What the paper shows and why it matters

(AI-generated)

Many simulation models have no computable likelihood, which rules out standard fitting. Synthetic likelihood (Wood 2010) works around this by reducing observed and simulated data to a few summary statistics and comparing them through a likelihood built from the simulations. It needs much less tuning than approximate Bayesian computation (ABC) but it assumes that the summary statistics are approximately multivariate normal. This assumption can be problematic, and transformations that make statistics normal are hard to find for all parameter values. The paper sets out to drop the normality requirement while keeping the method’s scalability.

The paper shows that synthetic likelihood can do without its normality assumption by using a more flexible density estimator for the summary statistics. The new estimator, the extended empirical saddlepoint (EES) density, is built directly from the simulated statistics and blended with a Gaussian one (mixing the two at the level of the cumulant generating function), so that it is defined even for values outside the range spanned by the simulations. One parameter sets the blend and is chosen by cross-validation, that is, by testing on held-out simulations. The paper proves that maximising the synthetic likelihood yields consistent parameter estimates under either the new or the Gaussian density, under stated regularity conditions. In a boom-and-bust population model and in the forest model Formind (with deliberately strongly skewed statistics), EES-based synthetic likelihood gave lower estimation errors than the Gaussian version; in the population model it also gave lower errors than ABC for three of the four parameters. In a toy example, the paper argues from an analytical calculation that ABC would need rapidly growing effort to match EES as the number of statistics rises.

The results mean that where summary statistics are clearly non-normal, synthetic likelihood can give more accurate estimates, which, the paper argues, frees users to concentrate on choosing informative statistics. The paper names two open points: whether extra assumptions would allow conclusions about the efficiency of the estimates, and whether one Gaussian-process speed-up, which models only the first two moments, can take higher moments into account. The method is available in the R package esaddle. An et al. (2020) built a Bayesian version and, in their test examples, found a semi-parametric alternative of their own to be mostly more accurate.

References

  • Wood (2010). Statistical inference for noisy nonlinear ecological dynamic systems. Nature 466, 1102–1104. doi:10.1038/nature09319
  • An et al. (2020). Robust Bayesian synthetic likelihood via a semi-parametric approach. Statistics and Computing 30, 543–557. doi:10.1007/s11222-019-09904-x