Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

We present a novel approach for explaining Gaussian processes (GPs) that can utilize the full analytical covariance structure present in GPs.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.

Bayesian Optimization (BO) is an effective framework for black-box optimization.

Neural Information Processing Systems http://nips.cc/

Efficient coding theory hasbeen used topredict howthe brain should allocate its limited resources in these scenarios by removing redundant information [1-4].