The article is about algorithms for learning Bayesian hierarchical models, the computational complexity of which scales linearly with the number of observations and the number of parameters in the model. It focuses on crossed random effect and nested multilevel models, which are used ubiquitously in applied sciences, and illustrates the methodology on two challenging real data analyses on predicting electoral results and real estate prices respectively. The posterior dependence in both classes is sparse: in crossed random effects models it resembles a random graph, whereas in nested multilevel models it is tree-structured. For each class we develop a framework for scalable computation based on collapsed Gibbs sampling and belief propagation respectively. We provide a number of negative (for crossed) and positive (for nested) results for the scalability (or lack thereof) of methods based on sparse linear algebra, which are relevant also to Laplace approximation methods for such models. Our numerical experiments compare with off-the-shelf variational approximations and Hamiltonian Monte Carlo. Our theoretical results, although partial, are useful in suggesting interesting methodologies and lead to conclusions that our numerics suggest to hold well beyond the scope of the underlying assumptions.
We propose a new methodology for predicting electoral results that combines a fundamental model and national polls within an evidence synthesis framework. Although novel, the methodology builds upon basic statistical structures, largely modern analysis of variance type models, and it is carried out in open-source software. The methodology is motivated by the specific challenges of forecasting elections with the participation of new political parties, which is becoming increasingly common in the post-2008 European panorama. Our methodology is also particularly useful for the allocation of parliamentary seats, since the vast majority of available opinion polls predict at national level whereas seats are allocated at local level. We illustrate the advantages of our approach relative to recent competing approaches using the 2015 Spanish Congressional Election. In general, the predictions of our model outperform the alternative specifications, including hybrid models that combine fundamental and polls' models. Our forecasts are, in relative terms, particularly accurate in predicting the seats obtained by each political party.
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