We propose a vector-valued regression problem whose solution is equivalent to the reproducing kernel Hilbert space (RKHS) embedding of the Bayesian posterior distribution. This equivalence provides a new understanding of kernel Bayesian inference. Moreover, the optimization problem induces a new regularization for the posterior embedding estimator, which is faster and has comparable performance to the squared regularization in kernel Bayes' rule. This regularization coincides with a former thresholding approach used in kernel POMDPs whose consistency remains to be established. Our theoretical work solves this open problem and provides consistency analysis in regression settings. Based on our optimizational formulation, we propose a flexible Bayesian posterior regularization framework which for the first time enables us to put regularization at the distribution level. We apply this method to nonparametric state-space filtering tasks with extremely nonlinear dynamics and show performance gains over all other baselines.

We present a fast variational Bayesian algorithm for performing non-negative matrix factorisation and tri-factorisation. We show that our approach achieves faster convergence per iteration and timestep (wall-clock) than Gibbs sampling and non-probabilistic approaches, and do not require additional samples to estimate the posterior. We show that in particular for matrix tri-factorisation convergence is difficult, but our variational Bayesian approach offers a fast solution, allowing the tri-factorisation approach to be used more effectively.

Cross-validation (CV) is a technique for evaluating the ability of statistical models/learning systems based on a given data set. Despite its wide applicability, the rather heavy computational cost can prevent its use as the system size grows. To resolve this difficulty in the case of Bayesian linear regression, we develop a formula for evaluating the leave-one-out CV error approximately without actually performing CV. The usefulness of the developed formula is tested by statistical mechanical analysis for a synthetic model. This is confirmed by application to a real-world supernova data set as well.

Many statistical models can be simulated forwards but have intractable likelihoods. Approximate Bayesian Computation (ABC) methods are used to infer properties of these models from data. Traditionally these methods approximate the posterior over parameters by conditioning on data being inside an $\epsilon$-ball around the observed data, which is only correct in the limit $\epsilon\!\rightarrow\!0$. Monte Carlo methods can then draw samples from the approximate posterior to approximate predictions or error bars on parameters. These algorithms critically slow down as $\epsilon\!\rightarrow\!0$, and in practice draw samples from a broader distribution than the posterior. We propose a new approach to likelihood-free inference based on Bayesian conditional density estimation. Preliminary inferences based on limited simulation data are used to guide later simulations. In some cases, learning an accurate parametric representation of the entire true posterior distribution requires fewer model simulations than Monte Carlo ABC methods need to produce a single sample from an approximate posterior.

Methods for unsupervised anomaly detection suffer from the fact that the data is unlabeled, making it difficult to assess the optimality of detection algorithms. Ensemble learning has shown exceptional results in classification and clustering problems, but has not seen as much research in the context of outlier detection. Existing methods focus on combining output scores of individual detectors, but this leads to outputs that are not easily interpretable. In this paper, we introduce a theoretical foundation for combining individual detectors with Bayesian classifier combination. Not only are posterior distributions easily interpreted as the probability distribution of anomalies, but bias, variance, and individual error rates of detectors are all easily obtained. Performance on real-world datasets shows high accuracy across varied types of time series data.

We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, using the L\'evy-Khintchine formula, that any shift-invariant kernel given by a bounded, continuous and symmetric probability density function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is characteristic. We also present some closure property of such characteristic kernels under addition, pointwise product, and convolution. Second, in developing various kernel mean algorithms, it is fundamental to compute the following values: (i) kernel mean values $m_P(x)$, $x \in \mathcal{X}$, and (ii) kernel mean RKHS inner products ${\left\langle m_P, m_Q \right\rangle_{\mathcal{H}}}$, for probability measures $P, Q$. If $P, Q$, and kernel $k$ are Gaussians, then computation (i) and (ii) results in Gaussian pdfs that is tractable. We generalize this Gaussian combination to more general cases in the class of infinitely divisible distributions. We then introduce a {\it conjugate} kernel and {\it convolution trick}, so that the above (i) and (ii) have the same pdf form, expecting tractable computation at least in some cases. As specific instances, we explore $\alpha$-stable distributions and a rich class of generalized hyperbolic distributions, where the Laplace, Cauchy and Student-t distributions are included.

Learning theory can be traced back to the late 60s and has attracted a great attention since. We refer to the monographs Devroye et al. (1996) and Vapnik (2000) for a survey. Most of the literature addresses the simplified case of i.i.d observations coupled with bounded loss functions. Many bounds on the excess risk holding with large probability were provided - these bounds are refered to as PAC learning bounds since Valiant (1984). In the late 90s, the PAC-Bayesian approach has been pioneered by Shawe-Taylor and Williamson (1997) and McAllester (1998, 1999). It consists in producing PAC bounds for a specific class of Bayesian-flavored estimators. Similarly to classical PAC results, most PAC-Bayesian bounds have been obtained with bounded loss functions (see Catoni, 2007, for some of the most accurate results). Note that Catoni (2004) provides bounds for unbouded loss, but still under very strong exponential moments assumptions. These assumptions were essentially not improved in the most recent works Guedj and Alquier (2013) and Bégin et al. (2016).

The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to derive the formulas for counting the sizes of Markov equivalence classes. We first introduce a new concept of core graph. The size of a Markov equivalence class of interest is a polynomial of the number of vertices given its core graph. Then, we discuss the recursive and explicit formula of the polynomial, and provide an algorithm to derive the size formula via symbolic computation for any given core graph. The proposed size formula derivation sheds light on the relationships between the size of a Markov equivalence class and its representation graph, and makes size counting efficient, even when the essential graphs contain non-sparse undirected subgraphs.

The highly variable dynamics of neocortical circuits observed in vivo have been hypothesized to represent a signature of ongoing stochastic inference but stand in apparent contrast to the deterministic response of neurons measured in vitro. Based on a propagation of the membrane autocorrelation across spike bursts, we provide an analytical derivation of the neural activation function that holds for a large parameter space, including the high-conductance state. On this basis, we show how an ensemble of leaky integrate-and-fire neurons with conductance-based synapses embedded in a spiking environment can attain the correct firing statistics for sampling from a well-defined target distribution. For recurrent networks, we examine convergence toward stationarity in computer simulations and demonstrate sample-based Bayesian inference in a mixed graphical model. This points to a new computational role of high-conductance states and establishes a rigorous link between deterministic neuron models and functional stochastic dynamics on the network level.

Independent component analysis (ICA) is the most popular method for blind source separation (BSS) with a diverse set of applications, such as biomedical signal processing, video and image analysis, and communications. Maximum likelihood (ML), an optimal theoretical framework for ICA, requires knowledge of the true underlying probability density function (PDF) of the latent sources, which, in many applications, is unknown. ICA algorithms cast in the ML framework often deviate from its theoretical optimality properties due to poor estimation of the source PDF. Therefore, accurate estimation of source PDFs is critical in order to avoid model mismatch and poor ICA performance. In this paper, we propose a new and efficient ICA algorithm based on entropy maximization with kernels, (ICA-EMK), which uses both global and local measuring functions as constraints to dynamically estimate the PDF of the sources with reasonable complexity. In addition, the new algorithm performs optimization with respect to each of the cost function gradient directions separately, enabling parallel implementations on multi-core computers. We demonstrate the superior performance of ICA-EMK over competing ICA algorithms using simulated as well as real-world data.