Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. 10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

Intersectoral and spatial spill-overs of firms’ bankruptcy in Spain

This paper employs provincial data to study the spatial and intersectoral spill-overs in aggregate failure rates in Spain, by using an Integrated Nested Laplace Approximation. The analysis is based on NUTS3 data over the time span 2005Q1-2013Q4. By speculating on the effects of the Spanish financial crisis, we document empirical evidence of the presence of spatial spill-overs among neighboring counties. Furthermore, some intersectoral spill-overs are also detected: we observe that Industry and Agriculture exhibit a positive impact on the Service sector. These results can be useful to design proper policy rules to better manage the spread of bankruptcies over time and space.

Effectiveness of the protected areas on the Mornington Peninsula for the common resident shorebird species using citizen science data

In the absence of comprehensive survey data this study used citizen science bird counts, extracted from the Atlas of Living Australia, to assess which species benefit most from protected areas. This was done by fitting temporal models using the Integrated Laplace Approximation (INLA) method.The trends for five resident shorebird species were compared to the Australian Pied Oystercatcher, with significantly steeper upward trends identified for the Black-fronted Dotterel, Red-capped Dotterel and Red-kneed Dotterel. Steeper upward trends were observed in protected than unprotected areas for the Black-fronted Dotterel, Masked Lapwing and Red-kneed Dotterel.This work suggests that, with some limitations, statistical models can be used with citizen science data for monitoring the persistence of resident shorebirds and for investigating factors that are impacting these data. The results for the Dotterel species in protected areas are particularly encouraging.

A Comparison of Nonergodic Ground-Motion Models based on Geographically Weighted Regression and the Integrated Nested Laplace Approximation

Different nonergodic Ground-Motion Models based on spatially varying coefficient models are compared for ground-motion data in Italy. The models are based different methodologies: Multi-source geographically weighted regression (Caramenti et al., 2020), and Bayesian hierarchical models estimated with the integrated nested Laplace approximation (Rue et al., 2009). The different models are compared in terms of their predictive performance, their spatial coefficients, and their predictions. Models that include spatial terms perform slightly better than a simple base model that includes only event and station terms, in terms of out-of sample error based on cross-validation. The Bayesian spatial models have slightly lower generalization error, which can be attributed to the fact that they can include random effects for events and stations. The different methodologies give rise to different dependencies of the spatially varying terms on event and station locations, leading to between-model uncertainty in their predictions, which should be accommodated in a nonergodic seismic hazard assessment.

New error bounds for Laplace approximation via Stein's method

We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren \cite {pike} for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

Efficient and flexible Integration of variant characteristics in rare variant association studies using integrated nested Laplace approximation

Rare variants are thought to play an important role in the etiology of complex diseases and may explain a significant fraction of the missing heritability in genetic disease studies. Next-generation sequencing facilitates the association of rare variants in coding or regulatory regions with complex diseases in large cohorts at genome-wide scale. However, rare variant association studies (RVAS) still lack power when cohorts are small to medium-sized and if genetic variation explains a small fraction of phenotypic variance. Here we present a novel Bayesian rare variant Association Test using Integrated Nested Laplace Approximation (BATI). Unlike existing RVAS tests, BATI allows integration of individual or variant-specific features as covariates, while efficiently performing inference based on full model estimation. We demonstrate that BATI outperforms established RVAS methods on realistic, semi-synthetic whole-exome sequencing cohorts, especially when using meaningful biological context, such as functional annotation. We show that BATI achieves power above 70% in scenarios in which competing tests fail to identify risk genes, e.g. when risk variants in sum explain less than 0.5% of phenotypic variance. We have integrated BATI, together with five existing RVAS tests in the ‘Rare Variant Genome Wide Association Study’ (rvGWAS) framework for data analyzed by whole-exome or whole genome sequencing. rvGWAS supports rare variant association for genes or any other biological unit such as promoters, while allowing the analysis of essential functionalities like quality control or filtering. Applying rvGWAS to a Chronic Lymphocytic Leukemia study we identified eight candidate predisposition genes, including EHMT2 and COPS7A.