The difficulty of multi-class classification generally increases with the number of classes. Using data from a subset of the classes, can we predict how well a classifier will scale with an increased number of classes? Under the assumption that the classes are sampled exchangeably, and under the assumption that the classifier is generative (e.g. QDA or Naive Bayes), we show that the expected accuracy when the classifier is trained on $k$ classes is the $k-1$st moment of a \emph{conditional accuracy distribution}, which can be estimated from data. This provides the theoretical foundation for performance extrapolation based on pseudolikelihood, unbiased estimation, and high-dimensional asymptotics. We investigate the robustness of our methods to non-generative classifiers in simulations and one optical character recognition example.

L. Schwaller · S. Robin Abstract We consider the problem of change-point detection in multivariate time-series. The multivariate distribution of the observations is supposed to follow a graphical model, whose graph and parameters are affected by abrupt changes throughout time. We demonstrate that it is possible to perform exact Bayesian inference whenever one considers a simple class of undirected graphs called spanning trees as possible structures. We are then able to integrate on the graph and segmentation spaces at the same time by combining classical dynamic programming with algebraic results pertaining to spanning trees. In particular, we show that quantities such as posterior distributions for change-points or posterior edge probabilities over time can efficiently be obtained. We illustrate our results on both synthetic and experimental data arising from biology and neuroscience. Keywords change-point detection, exact Bayesian inference, graphical model, multivariate time-series, spanning tree 1 Introduction We are interested in time-series data where several variables are observed throughout time. An assumption often made in multivariate settings is that there exists an underlying network describing the dependences between the different variables. When modelling time-series data, one is faced with a choice: shall this network be considered stationary or not? Taking the example of genomic data, it might for instance be un-L. This network might slowly evolve, or undergo abrupt changes leading to the initialisation of new morphological development stages in the organism of interest. Here, we focus our interest on the second scenario. The inference of the dependence structure ruling a multivariate time-series was first performed under the assumption that this structure was stationary ( e.g.

This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is associated with a linear subspace on which observations vary maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds for various dimensionality reduction problems, explore the connection between the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the Grassmannian for the first time. We delineate in which situations either manifold should be considered. Further, matrix manifold models are used to yield scientific insight in the context of cognitive neuroscience, and we conclude that our methods are suitable for basic inference as well as accurate prediction. All datasets and computer programs are publicly available at http://www.ics.uci.edu/

The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizing flow, whereby a simple initial density is transformed into a more complex one by applying a sequence of invertible transformations until a desired level of complexity is attained. We use this view of normalizing flows to develop categories of finite and infinitesimal flows and provide a unified view of approaches for constructing rich posterior approximations. We demonstrate that the theoretical advantages of having posteriors that better match the true posterior, combined with the scalability of amortized variational approaches, provides a clear improvement in performance and applicability of variational inference.

Probabilistic inference over large data sets is a challenging data management problem since exact inference is generally #P-hard and is most often solved approximately with sampling-based methods today. This paper proposes an alternative approach for approximate evaluation of conjunctive queries with standard relational databases: In our approach, every query is evaluated entirely in the database engine by evaluating a fixed number of query plans, each providing an upper bound on the true probability, then taking their minimum. We provide an algorithm that takes into account important schema information to enumerate only the minimal necessary plans among all possible plans. Importantly, this algorithm is a strict generalization of all known PTIME self-join-free conjunctive queries: A query is in PTIME if and only if our algorithm returns one single plan. Furthermore, our approach is a generalization of a family of efficient ranking methods from graphs to hypergraphs. We also adapt three relational query optimization techniques to evaluate all necessary plans very fast. We give a detailed experimental evaluation of our approach and, in the process, provide a new way of thinking about the value of probabilistic methods over non-probabilistic methods for ranking query answers. We also note that the techniques developed in this paper apply immediately to lifted inference from statistical relational models since lifted inference corresponds to PTIME plans in probabilistic databases.

Nowadays, there is broad consensus that the ability to think probabilistically is a fundamental component of scientific literacy. The aim of this class is to introduce the relevant models, skills, and tools, by combining mathematics with conceptual understanding and intuition.

The Conference on Uncertainty in Artificial Intelligence (UAI) is one of the premier international conferences on research related to knowledge representation, learning, and reasoning in the presence of uncertainty. UAI is supported by the Association for Uncertainty in Artificial Intelligence (AUAI). Invited speakers will discuss Statistics, Machine Learning & the Detection of Gravitational Waves; Online optimization of power networks; Structured Prediction and Deep Learning: Past, Present, and Future; and Total positivity and Markov structures. Workshops will include Causation: Foundation to Application, Bayesian Applications Workshop, and Machine Learning for Health. Tutorials will include Discrete Sampling and Integration in High Dimensional Spaces, Parallel and High-performance Computing for Speeding up Machine Learning Algorithms, Integrative Logic-Based Causal Discovery, and Reasoning Under Uncertainty with Subjective Logic.

The Conference on Uncertainty in Artificial Intelligence (UAI) is one of the premier international conferences on research related to knowledge representation, learning, and reasoning in the presence of uncertainty. UAI is supported by the Association for Uncertainty in Artificial Intelligence (AUAI). Invited speakers will discuss Statistics, Machine Learning & the Detection of Gravitational Waves; Online optimization of power networks; Structured Prediction and Deep Learning: Past, Present, and Future; and Total positivity and Markov structures. Workshops will include Causation: Foundation to Application, Bayesian Applications Workshop, and Machine Learning for Health. Tutorials will include Discrete Sampling and Integration in High Dimensional Spaces, Parallel and High-performance Computing for Speeding up Machine Learning Algorithms, Integrative Logic-Based Causal Discovery, and Reasoning Under Uncertainty with Subjective Logic.

Learning Granger causality for general point processes is a very challenging task. In this paper, we propose an effective method, learning Granger causality, for a special but significant type of point processes --- Hawkes process. We reveal the relationship between Hawkes process's impact function and its Granger causality graph. Specifically, our model represents impact functions using a series of basis functions and recovers the Granger causality graph via group sparsity of the impact functions' coefficients. We propose an effective learning algorithm combining a maximum likelihood estimator (MLE) with a sparse-group-lasso (SGL) regularizer. Additionally, the flexibility of our model allows to incorporate the clustering structure event types into learning framework. We analyze our learning algorithm and propose an adaptive procedure to select basis functions. Experiments on both synthetic and real-world data show that our method can learn the Granger causality graph and the triggering patterns of the Hawkes processes simultaneously.

Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most statistical guarantees are only for random scan. While it has been conjectured that the mixing times of random scan and systematic scan do not differ by more than a logarithmic factor, we show by counterexample that this is not the case, and we prove that that the mixing times do not differ by more than a polynomial factor under mild conditions. To prove these relative bounds, we introduce a method of augmenting the state space to study systematic scan using conductance.