Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature

By Tom Gunter, Michael A. Osborne, Roman Garnett, Philipp Hennig, Stephen J. Roberts. 2014

121 citations recorded by the source corpus.

We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo (MCMC) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function). MCMC has provided approaches to numerical integration that deliver state-of-the-art inference, but can suffer from sample inefficiency and poor convergence diagnostics. Bayesian quadrature techniques offer a model-based solution to such problems, but their uptake has been hindered by prohibitive computation costs. We introduce a warped model for probabilistic integrands (likelihoods) that are known to be non-negative, permitting a cheap active learning scheme to optimally select sample locations. Our algorithm is demonstrated to offer faster convergence (in seconds) relative to simple Monte Carlo and annealed importance sampling on both synthetic and real-world examples.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, background): “ection. 5.2 Optimal design points Quadrature rules for optimal selection of design points D 0 have been proposed in the Bayesian quadrature (BQ) literature (O’Hagan, 1991; Huszar and Duvenaud, 2012; Gunter et al., 2014; 20 -2 -1 0 1 2 -1 0 1 2 x 1 x 2 Monte Carlo -2 -1 0 1 2 -1 0 1 2 x 1 x 2 Optimal”
Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees (2015, methodology): “This is not always the case: For example [12] approximately encodes non-negativity of f into BQ which leads to a dependence on f in the posterior variance.”
Variational Inference for Gaussian Process Modulated Poisson Processes (2014, methodology): “Since only f is conditioned on u and since q ( u ) is conjugate to p ( f | u ), we can write down in closed-form the resulting integral: We also obtain a KL-divergence term arising from our variational approximation, which is simply the KL-divergence between two Gaussians i Substituting q ( f )…”
Probabilistic numerics and uncertainty in computations (2015, methodology): “kelihood p(DS;M)) and a 19-dimensional rendering the quadrature problem challenging. As the probabilistic method, we use a recent algorithm: Warped, Sequential, Active Bayesian Integration (WSABI7) [9]. WSABI is a Bayesian quadrature algorithm that uses an internal 6The authors are grateful to H. Pa”