A practical method for Bayesian training of feed-forward neural networks using sophisticated Monte Carlo methods is presented and evaluated. In reasonably small amounts of computer time this approach outperforms other state-of-the-art methods on 5 datalimited tasksfrom real world domains. 1 INTRODUCTION Bayesian learning uses a prior on model parameters, combines this with information from a training set, and then integrates over the resulting posterior to make predictions. Withthis approach, we can use large networks without fear of overfitting, allowing us to capture more structure in the data, thus improving prediction accuracy andeliminating the tedious search (often performed using cross validation) for the model complexity that optimises the bias/variance tradeoff. In this approach the size of the model is limited only by computational considerations. The application of Bayesian learning to neural networks has been pioneered by MacKay (1992), who uses a Gaussian approximation to the posterior weight distribution.

Dang, Khue-Dung, Quiroz, Matias, Kohn, Robert, Tran, Minh-Ngoc, Villani, Mattias

Hamiltonian Monte Carlo (HMC) has recently received considerable attention in the literature due to its ability to overcome the slow exploration of the parameter space inherent in random walk proposals. In tandem, data subsampling has been extensively used to overcome the computational bottlenecks in posterior sampling algorithms that require evaluating the likelihood over the whole data set, or its gradient. However, while data subsampling has been successful in traditional MCMC algorithms such as Metropolis-Hastings, it has been demonstrated to be unsuccessful in the context of HMC, both in terms of poor sampling efficiency and in producing highly biased inferences. We propose an efficient HMC-within-Gibbs algorithm that utilizes data subsampling to speed up computations and simulates from a slightly perturbed target, which is within $O(m^{-2})$ of the true target, where $m$ is the size of the subsample. We also show how to modify the method to obtain exact inference on any function of the parameters. Contrary to previous unsuccessful approaches, we perform subsampling in a way that conserves energy but for a modified Hamiltonian. We can therefore maintain high acceptance rates even for distant proposals. We apply the method for simulating from the posterior distribution of a high-dimensional spline model for bankruptcy data and document speed ups of several orders of magnitude compare to standard HMC and, moreover, demonstrate a negligible bias.

Ghahramani, Zoubin, Rasmussen, Carl E.

We investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation ofprior knowledge, such as smoothness of the integrand, into the estimation. In a simple problem we show that this outperforms any classical importance sampling method. We also attempt more challenging multidimensionalintegrals involved in computing marginal likelihoods ofstatistical models (a.k.a.

Luo, Rui, Zhang, Qiang, Liu, Yuanyuan

We propose a new sampler that integrates the protocol of parallel tempering with the Nos\'e-Hoover (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially facilitates deep Bayesian learning on large datasets where complex multimodal posteriors and mini-batch gradient are encountered.