An implementation of a nonparametric Bayesian approach to solving binary classification problems on graphs is described. A hierarchical Bayesian approach with a randomly scaled Gaussian prior is considered. The prior uses the graph Laplacian to take into account the underlying geometry of the graph. A method based on a theoretically optimal prior and a more flexible variant using partial conjugacy are proposed. Two simulated data examples and two examples using real data are used in order to illustrate the proposed methods.

Consider the problem of estimating click probabilities for links between pages of a website, given a hyperlink graph and aggregate statistics on the number of times each page has been visited. Naively, one might expect that the probability of clicking on a particular link should be roughly proportional to the traffic of the link's target. However, this neglects important structural effects: a page's traffic is influenced by a) the number of incoming links, b) the traffic at the pages that link to it, and c) the traffic absorbed by competing links. In order to successfully infer click probabilities, it is therefore necessary to disentangle the preference for a page (i.e., the intrinsic propensity of a user to click on a link pointing to it) from the page's visibility (the exposure it gets from pages linking to it). Building upon recent work by Kumar et al. [2015], we present a statistical framework that tackles a general formulation of the problem: given a network (representing possible transitions between nodes) and the marginal traffic at each node, recover the transition probabilities.

We investigate the problem of learning discrete, undirected graphical models in a differentially private way. We show that the approach of releasing noisy sufficient statistics using the Laplace mechanism achieves a good trade-off between privacy, utility, and practicality. A naive learning algorithm that uses the noisy sufficient statistics "as is" outperforms general-purpose differentially private learning algorithms. However, it has three limitations: it ignores knowledge about the data generating process, rests on uncertain theoretical foundations, and exhibits certain pathologies. We develop a more principled approach that applies the formalism of collective graphical models to perform inference over the true sufficient statistics within an expectation-maximization framework. We show that this learns better models than competing approaches on both synthetic data and on real human mobility data used as a case study.

There is general consensus that learning representations is useful for a variety of reasons, e.g. efficient use of labeled data (semi-supervised learning), transfer learning and understanding hidden structure of data. Popular techniques for representation learning include clustering, manifold learning, kernel-learning, autoencoders, Boltzmann machines, etc. To study the relative merits of these techniques, it's essential to formalize the definition and goals of representation learning, so that they are all become instances of the same definition. This paper introduces such a formal framework that also formalizes the utility of learning the representation. It is related to previous Bayesian notions, but with some new twists. We show the usefulness of our framework by exhibiting simple and natural settings -- linear mixture models and loglinear models, where the power of representation learning can be formally shown. In these examples, representation learning can be performed provably and efficiently under plausible assumptions (despite being NP-hard), and furthermore: (i) it greatly reduces the need for labeled data (semi-supervised learning) and (ii) it allows solving classification tasks when simpler approaches like nearest neighbors require too much data (iii) it is more powerful than manifold learning methods.

Undirected graphical models are applied in genomics, protein structure prediction, and neuroscience to identify sparse interactions that underlie discrete data. Although Bayesian methods for inference would be favorable in these contexts, they are rarely used because they require doubly intractable Monte Carlo sampling. Here, we develop a framework for scalable Bayesian inference of discrete undirected models based on two new methods. The first is Persistent VI, an algorithm for variational inference of discrete undirected models that avoids doubly intractable MCMC and approximations of the partition function. The second is Fadeout, a reparameterization approach for variational inference under sparsity-inducing priors that captures a posteriori correlations between parameters and hyperparameters with noncentered parameterizations. We find that, together, these methods for variational inference substantially improve learning of sparse undirected graphical models in simulated and real problems from physics and biology.

We consider the problem of approximate Bayesian parameter inference in non-linear state-space models with intractable likelihoods. Sequential Monte Carlo with approximate Bayesian computations (SMC-ABC) is one approach to approximate the likelihood in this type of models. However, such approximations can be noisy and computationally costly which hinders efficient implementations using standard methods based on optimisation and Monte Carlo methods. We propose a computationally efficient novel method based on the combination of Gaussian process optimisation and SMC-ABC to create a Laplace approximation of the intractable posterior. We exemplify the proposed algorithm for inference in stochastic volatility models with both synthetic and real-world data as well as for estimating the Value-at-Risk for two portfolios using a copula model. We document speed-ups of between one and two orders of magnitude compared to state-of-the-art algorithms for posterior inference.

Relational data are usually highly incomplete in practice, which inspires us to leverage side information to improve the performance of community detection and link prediction. This paper presents a Bayesian probabilistic approach that incorporates various kinds of node attributes encoded in binary form in relational models with Poisson likelihood. Our method works flexibly with both directed and undirected relational networks. The inference can be done by efficient Gibbs sampling which leverages sparsity of both networks and node attributes. Extensive experiments show that our models achieve the state-of-the-art link prediction results, especially with highly incomplete relational data.

We introduce a novel stochastic version of the non-reversible, rejection-free Bouncy Particle Sampler (BPS), a Markov process whose sample trajectories are piecewise linear. The algorithm is based on simulating first arrival times in a doubly stochastic Poisson process using the thinning method, and allows efficient sampling of Bayesian posteriors in big datasets. We prove that in the BPS no bias is introduced by noisy evaluations of the log-likelihood gradient. On the other hand, we argue that efficiency considerations favor a small, controllable bias in the construction of the thinning proposals, in exchange for faster mixing. We introduce a simple regression-based proposal intensity for the thinning method that controls this trade-off. We illustrate the algorithm in several examples in which it outperforms both unbiased, but slowly mixing stochastic versions of BPS, as well as biased stochastic gradient-based samplers.

Computational results demonstrate that posterior sampling for reinforcement learning (PSRL) dramatically outperforms algorithms driven by optimism, such as UCRL2. We provide insight into the extent of this performance boost and the phenomenon that drives it. We leverage this insight to establish an $\tilde{O}(H\sqrt{SAT})$ Bayesian expected regret bound for PSRL in finite-horizon episodic Markov decision processes, where $H$ is the horizon, $S$ is the number of states, $A$ is the number of actions and $T$ is the time elapsed. This improves upon the best previous bound of $\tilde{O}(H S \sqrt{AT})$ for any reinforcement learning algorithm.

We reinterpret multiplicative noise in neural networks as auxiliary random variables that augment the approximate posterior in a variational setting for Bayesian neural networks. We show that through this interpretation it is both efficient and straightforward to improve the approximation by employing normalizing flows (Rezende & Mohamed, 2015) while still allowing for local reparametrizations (Kingma et al., 2015) and a tractable lower bound (Ranganath et al., 2015; Maaløe et al., 2016). In experiments we show that with this new approximation we can significantly improve upon classical mean field for Bayesian neural networks on both predictive accuracy as well as predictive uncertainty.