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### Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression

We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation method in practice.

### Variational Inference for Nonparametric Bayesian Quantile Regression

Quantile regression deals with the problem of computing robust estimators when the conditional mean and standard deviation of the predicted function are inadequate to capture its variability. The technique has an extensive list of applications, including health sciences, ecology and finance. In this work we present a non-parametric method of inferring quantiles and derive a novel Variational Bayesian (VB) approximation to the marginal likelihood, leading to an elegant Expectation Maximisation algorithm for learning the model. Our method is nonparametric, has strong convergence guarantees, and can deal with nonsymmetric quantiles seamlessly. We compare the method to other parametric and non-parametric Bayesian techniques, and alternative approximations based on expectation propagation demonstrating the benefits of our framework in toy problems and real datasets.

### Variational Inference via $\chi$ Upper Bound Minimization

Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_{\chi}(p || q)$, the $\chi$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evidence, which we term the $\chi$ upper bound (CUBO). Minimizing the CUBO leads to improved posterior uncertainty, and it can also be used with the classical VI lower bound (ELBO) to provide a sandwich estimate of the model evidence. We study CHIVI on three models: probit regression, Gaussian process classification, and a Cox process model of basketball plays. When compared to expectation propagation and classical VI, CHIVI produces better error rates and more accurate estimates of posterior variance.

### Approximate Inference Turns Deep Networks into Gaussian Processes

Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain Gaussian posterior approximations for Bayesian DNNs are equivalent to GP posteriors. As a result, we can obtain a GP kernel and a nonlinear feature map simply by training the DNN. Surprisingly, the resulting kernel is the neural tangent kernel which has desirable theoretical properties for infinitely-wide DNNs. We show feature maps obtained on real datasets and demonstrate the use of the GP marginal likelihood to tune hyperparameters of DNNs. Our work aims to facilitate further research on combining DNNs and GPs in practical settings.

### PG-TS: Improved Thompson Sampling for Logistic Contextual Bandits

We address the problem of regret minimization in logistic contextual bandits, where a learner decides among sequential actions or arms given their respective contexts to maximize binary rewards. Using a fast inference procedure with Polya-Gamma distributed augmentation variables, we propose an improved version of Thompson Sampling, a Bayesian formulation of contextual bandits with near-optimal performance. Our approach, Polya-Gamma augmented Thompson Sampling (PG-TS), achieves state-of-the-art performance on simulated and real data. PG-TS explores the action space efficiently and exploits high-reward arms, quickly converging to solutions of low regret. Its explicit estimation of the posterior distribution of the context feature covariance leads to substantial empirical gains over approximate approaches. PG-TS is the first approach to demonstrate the benefits of Polya-Gamma augmentation in bandits and to propose an efficient Gibbs sampler for approximating the analytically unsolvable integral of logistic contextual bandits.