A Bootstrap Likelihood Approach to Bayesian Computation

Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler-Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

Bayesian analysis of immigration in Europe with generalized logistic regression

The number of immigrants moving to and settling in Europe has increased over the past decade, making migration one of the most topical and pressing issues in European politics. It is without a doubt that immigration has multiple impacts, in terms of economy, society and culture, on the European Union. It is fundamental to policy-makers to correctly evaluate people's attitudes towards immigration when designing integration policies. Of critical interest is to properly discriminate between subjects who are favourable towards immigration from those who are against it. Public opinions on migration are typically coded as binary responses in surveys. However, traditional methods, such as the standard logistic regression, may suffer from computational issues and are often not able to accurately model survey information. In this paper we propose an efficient Bayesian approach for modelling binary response data based on the generalized logistic regression. We show how the proposed approach provides an increased flexibility compared to traditional methods, due to its ability to capture heavy and light tails. The power of our methodology is tested through simulation studies and is illustrated using European Social Survey data on immigration collected in different European countries in 2016-2017.

On Bayesian Testing of Additive Conjoint Measurement Axioms Using Synthetic Likelihood

This article introduces a Bayesian method for testing the axioms of additive conjoint measurement. The method is based on an importance sampling algorithm that performs likelihood-free, approximate Bayesian inference using a synthetic likelihood to overcome the analytical intractability of this testing problem. This new method improves upon previous methods because it provides an omnibus test of the entire hierarchy of cancellation axioms, beyond double cancellation. It does so while accounting for the posterior uncertainty that is inherent in the empirical orderings that are implied by these axioms, together. The new method is illustrated through a test of the cancellation axioms on a classic survey data set, and through the analysis of simulated data.