We consider the problem of functional matrix factorization, finding low-dimensional structure in a matrix where every entry is a noisy function evaluated at a set of discrete points. Such problems arise frequently in drug discovery, where biological samples form the rows, candidate drugs form the columns, and entries contain the dose-response curve of a sample treated at different concentrations of a drug. We propose Bayesian Tensor Filtering (BTF), a hierarchical Bayesian model of matrices of functions. BTF captures the smoothness in each individual function while also being locally adaptive to sharp discontinuities. The BTF model is agnostic to the likelihood of the underlying observations, making it flexible enough to handle many different kinds of data. We derive efficient Gibbs samplers for three classes of likelihoods: (i) Gaussian, for which updates are fully conjugate; (ii) Binomial and related likelihoods, for which updates are conditionally conjugate through P{\'o}lya--Gamma augmentation; and (iii) Black-box likelihoods, for which updates are non-conjugate but admit an analytic truncated elliptical slice sampling routine. We compare BTF against a state-of-the-art method for dynamic Poisson matrix factorization, showing BTF better reconstructs held out data in synthetic experiments. Finally, we build a dose-response model around BTF and show on real data from a multi-sample, multi-drug cancer study that BTF outperforms the current standard approach in biology. Code for BTF is available at https://github.com/tansey/functionalmf.

A graphical model is a structured representation of the data generating process. The traditional method to reason over random variables is to perform inference in this graphical model. However, in many cases the generating process is only a poor approximation of the much more complex true data generating process, leading to suboptimal estimation. The subtleties of the generative process are however captured in the data itself and we can "learn to infer", that is, learn a direct mapping from observations to explanatory latent variables. In this work we propose a hybrid model that combines graphical inference with a learned inverse model, which we structure as in a graph neural network, while the iterative algorithm as a whole is formulated as a recurrent neural network. By using cross-validation we can automatically balance the amount of work performed by graphical inference versus learned inference. We apply our ideas to the Kalman filter, a Gaussian hidden Markov model for time sequences, and show, among other things, that our model can estimate the trajectory of a noisy chaotic Lorenz Attractor much more accurately than either the learned or graphical inference run in isolation.

Estimating the effect of a treatment on a given outcome, conditioned on a vector of covariates, is central in many applications. However, learning the impact of a treatment on a continuous temporal response, when the covariates suffer extensively from measurement error and even the timing of the treatments is uncertain, has not been addressed. We introduce a novel data-driven method that can estimate treatment-response trajectories in this challenging scenario. We model personalized treatment-response curves as a combination of parametric response functions, hierarchically sharing information across individuals, and a sparse Gaussian process for the baseline trend. Importantly, our model considers measurement error not only in treatment covariates, but also in treatment times, a problem which arises in practice for example when treatment information is based on self-reporting. In a challenging and timely problem of estimating the impact of diet on continuous blood glucose measurements, our model leads to significant improvements in estimation accuracy and prediction.

The bits-back argument suggests that latent variable models can be turned into lossless compression schemes. Translating the bits-back argument into efficient and practical lossless compression schemes for general latent variable models, however, is still an open problem. Bits-Back with Asymmetric Numeral Systems (BB-ANS), recently proposed by Townsend et al. (2019), makes bits-back coding practically feasible for latent variable models with one latent layer, but it is inefficient for hierarchical latent variable models. In this paper we propose Bit-Swap, a new compression scheme that generalizes BB-ANS and achieves strictly better compression rates for hierarchical latent variable models with Markov chain structure. Through experiments we verify that Bit-Swap results in lossless compression rates that are empirically superior to existing techniques. Our implementation is available at https://github.com/fhkingma/bitswap.

Recent advances in variational inference enable the modelling of highly structured joint distributions, but are limited in their capacity to scale to the high-dimensional setting of stochastic neural networks. This limitation motivates a need for scalable parameterizations of the noise generation process, in a manner that adequately captures the dependencies among the various parameters. In this work, we address this need and present the Kronecker Flow, a generalization of the Kronecker product to invertible mappings designed for stochastic neural networks. We apply our method to variational Bayesian neural networks on predictive tasks, PAC-Bayes generalization bound estimation, and approximate Thompson sampling in contextual bandits. In all setups, our methods prove to be competitive with existing methods and better than the baselines.

Unlike human speakers, typical text-to-speech (TTS) systems are unable to produce multiple distinct renditions of a given sentence. This has previously been addressed by adding explicit external control. In contrast, generative models are able to capture a distribution over multiple renditions and thus produce varied renditions using sampling. Typical neural TTS models learn the average of the data because they minimise mean squared error. In the context of prosody, taking the average produces flatter, more boring speech: an "average prosody". A generative model that can synthesise multiple prosodies will, by design, not model average prosody. We use variational autoencoders (VAE) which explicitly place the most "average" data close to the mean of the Gaussian prior. We propose that by moving towards the tails of the prior distribution, the model will transition towards generating more idiosyncratic, varied renditions. Focusing here on intonation, we investigate the trade-off between naturalness and intonation variation and find that typical acoustic models can either be natural, or varied, but not both. However, sampling from the tails of the VAE prior produces much more varied intonation than the traditional approaches, whilst maintaining the same level of naturalness.

In experimental design, we are given $n$ vectors in $d$ dimensions, and our goal is to select $k\ll n$ of them to perform expensive measurements, e.g., to obtain labels/responses, for a linear regression task. Many statistical criteria have been proposed for choosing the optimal design, with popular choices including A- and D-optimality. If prior knowledge is given, typically in the form of a $d\times d$ precision matrix $\mathbf A$, then all of the criteria can be extended to incorporate that information via a Bayesian framework. In this paper, we demonstrate a new fundamental connection between Bayesian experimental design and determinantal point processes, the latter being widely used for sampling diverse subsets of data. We use this connection to develop new efficient algorithms for finding $(1+\epsilon)$-approximations of optimal designs under four optimality criteria: A, C, D and V. Our algorithms can achieve this when the desired subset size $k$ is $\Omega(\frac{d_{\mathbf A}}{\epsilon} + \frac{\log 1/\epsilon}{\epsilon^2})$, where $d_{\mathbf A}\leq d$ is the $\mathbf A$-effective dimension, which can often be much smaller than $d$. Our results offer direct improvements over a number of prior works, for both Bayesian and classical experimental design, in terms of algorithm efficiency, approximation quality, and range of applicable criteria.

Latent Block Models are used for probabilistic biclustering, which is shown to be an effective method for analyzing various relational data sets. However, there has been no statistical test method for determining the row and column cluster numbers of Latent Block Models. Recent studies have constructed statistical-test-based methods for Stochastic Block Models, in which we assume that the observed matrix is a square symmetric matrix and that the cluster assignments are the same for rows and columns. In this paper, we develop a goodness-of-fit test for Latent Block Models, which tests whether an observed data matrix fits a given set of row and column cluster numbers, or it consists of more clusters in at least one direction of row and column. To construct the test method, we use a result from random matrix theory for a sample covariance matrix. We show experimentally the effectiveness of our proposed method, by showing the asymptotic behavior of the test statistic and the test accuracy.

This work considers the sample complexity of obtaining an $\epsilon$-optimal policy in a discounted Markov Decision Process (MDP), given only access to a generative model. In this model, the learner accesses the underlying transition model via a sampling oracle that provides a sample of the next state, when given any state-action pair as input. In this work, we study the effectiveness of the most natural plug-in approach to model-based planning: we build the maximum likelihood estimate of the transition model in the MDP from observations and then find an optimal policy in this empirical MDP. We ask arguably the most basic and unresolved question in model-based planning: is the na\"ive "plug-in" approach, non-asymptotically, minimax optimal in the quality of the policy it finds, given a fixed sample size? With access to a generative model, we resolve this question in the strongest possible sense: our main result shows that \emph{any} high accuracy solution in the plug-in model constructed with $N$ samples, provides an $\epsilon$-optimal policy in the true underlying MDP. In comparison, all prior (non-asymptotically) minimax optimal results use model-free approaches, such as the Variance Reduced Q-value iteration algorithm (Sidford et al 2018), while the best known model-based results (e.g. Azar et al 2013) require larger sample sample sizes in their dependence on the planning horizon or the state space. Notably, we show that the model-based approach allows the use of \emph{any} efficient planning algorithm in the empirical MDP, which simplifies the algorithm design as this approach does not tie the algorithm to the sampling procedure. The core of our analysis is a novel "absorbing MDP" construction to address the statistical dependency issues that arise in the analysis of model-based planning approaches, a construction which may be helpful more generally.

As machine learning is increasingly used to make real-world decisions, recent research efforts aim to define and ensure fairness in algorithmic decision making. Existing methods often assume a fixed set of observable features to define individuals, but lack a discussion of certain features not being observed at test time. In this paper, we study fairness of naive Bayes classifiers, which allow partial observations. In particular, we introduce the notion of a discrimination pattern, which refers to an individual receiving different classifications depending on whether some sensitive attributes were observed. Then a model is considered fair if it has no such pattern. We propose an algorithm to discover and mine for discrimination patterns in a naive Bayes classifier, and show how to learn maximum-likelihood parameters subject to these fairness constraints. Our approach iteratively discovers and eliminates discrimination patterns until a fair model is learned. An empirical evaluation on three real-world datasets demonstrates that we can remove exponentially many discrimination patterns by only adding a small fraction of them as constraints.