arXiv subjects

Tuomas Sivula

Publications and source records attributed to Tuomas Sivula.

5 recordsLinked to original sources

arXiv2022

Unbiased estimator for the variance of the leave-one-out cross-validation estimator for a Bayesian normal model with fixed variance

When evaluating and comparing models using leave-one-out cross-validation (LOO-CV), the uncertainty of the estimate is typically assessed using the variance of the sampling distribution. Considering the uncertainty is important, as the variability of the estimate can be high in some cases. An important result by Bengio and Grandvalet (2004) states that no general unbiased variance estimator can be constructed, that would apply for any utility or loss measure and any model. We show that it is possible to construct an unbiased estimator considering a specific predictive performance measure and model. We demonstrate an unbiased sampling distribution variance estimator for the Bayesian normal model with fixed model variance using the expected log pointwise predictive density (elpd) utility score. This example demonstrates that it is possible to obtain improved, problem-specific, unbiased estimators for assessing the uncertainty in LOO-CV estimation.

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arXiv2020

Uncertainty in Bayesian Leave-One-Out Cross-Validation Based Model Comparison

It is useful to estimate the expected predictive performance of models planned to be used for prediction. We focus on leave-one-out cross-validation (LOO-CV), which has become a popular method for estimating predictive performance of Bayesian models. Given two models, we are interested in comparing the predictive performances and associated uncertainty, which can also be used to compute the probability of one model having better predictive performance than the other model. We study the properties of the Bayesian LOO-CV estimator and the related uncertainty quantification for the predictive performance difference, and analyse when a normal approximation of this uncertainty is well calibrated and whether taking into account higher moments could improve the approximation. We provide new results of the properties both theoretically in the linear regression case and empirically for hierarchical linear, latent linear, and spline models and discuss the challenges. We show that problematic cases include: comparing models with similar predictions, misspecified models, and small data. In these cases, there is a weak connection between the distributions of the LOO-CV estimator and its error. We show that that the problematic skewness of the error distribution for the difference, which occurs when the models make similar predictions, does not fade away when the data size grows to infinity in certain situations. Based on the results, we also provide some practical recommendations for the users of Bayesian LOO-CV for comparing predictive performance of models.

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arXiv2019

Expectation propagation as a way of life: A framework for Bayesian inference on partitioned data

A common divide-and-conquer approach for Bayesian computation with big data is to partition the data, perform local inference for each piece separately, and combine the results to obtain a global posterior approximation. While being conceptually and computationally appealing, this method involves the problematic need to also split the prior for the local inferences; these weakened priors may not provide enough regularization for each separate computation, thus eliminating one of the key advantages of Bayesian methods. To resolve this dilemma while still retaining the generalizability of the underlying local inference method, we apply the idea of expectation propagation (EP) as a framework for distributed Bayesian inference. The central idea is to iteratively update approximations to the local likelihoods given the state of the other approximations and the prior. The present paper has two roles: we review the steps that are needed to keep EP algorithms numerically stable, and we suggest a general approach, inspired by EP, for approaching data partitioning problems in a way that achieves the computational benefits of parallelism while allowing each local update to make use of relevant information from the other sites. In addition, we demonstrate how the method can be applied in a hierarchical context to make use of partitioning of both data and parameters. The paper describes a general algorithmic framework, rather than a specific algorithm, and presents an example implementation for it.

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arXiv2019

Meta-analysis of Bayesian analyses

Meta-analysis aims to generalize results from multiple related statistical analyses through a combined analysis. While the natural outcome of a Bayesian study is a posterior distribution, traditional Bayesian meta-analyses proceed by combining summary statistics (i.e., point-valued estimates) computed from data. In this paper, we develop a framework for combining posterior distributions from multiple related Bayesian studies into a meta-analysis. Importantly, the method is capable of reusing pre-computed posteriors from computationally costly analyses, without needing the implementation details from each study. Besides providing a consensus across studies, the method enables updating the local posteriors post-hoc and therefore refining them by sharing statistical strength between the studies, without rerunning the original analyses. We illustrate the wide applicability of the framework by combining results from likelihood-free Bayesian analyses, which would be difficult to carry out using standard methodology.

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arXiv2016

Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models

The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several Bayesian leave-one-out (LOO) cross-validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data. Our main objective is to assess the accuracy of the approximative LOO cross-validation estimators. That is, for each method (Laplace and EP) we compare the approximate fast computation with the exact brute force LOO computation. Secondarily, we evaluate the accuracy of the Laplace and EP approximations themselves against a ground truth established through extensive Markov chain Monte Carlo simulation. Our empirical results show that the approach based upon a Gaussian approximation to the LOO marginal distribution (the so-called cavity distribution) gives the most accurate and reliable results among the fast methods.

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