Abstract--In this paper we propose a novel approach for learning from data using rule based fuzzy inference systems where the model parameters are estimated using Bayesian inference and Markov Chain Monte Carlo (MCMC) techniques. We show the applicability of the method for regression and classification tasks using synthetic data-sets and also a real world example in the financial services industry. Then we demonstrate how the method can be extended for knowledge extraction to select the individual rules in a Bayesian way which best explains the given data. Finally we discuss the advantages and pitfalls of using this method over state-of-the-art techniques and highlight the specific class of problems where this would be useful. ROBABILITY theory and fuzzy logic have been shown to be complementary [1] and various works have looked at the symbiotic integration of these two paradigms [2], [3] including the recently introduced concept of Z-numbers [4]. Historically fuzzy logic has been applied to problems involving imprecision in linguistic variables, while probability theory has been used for quantifying uncertainty in a wide range of disciplines. V arious generalisations and extensions of fuzzy sets have been proposed to incorporate uncertainty and vagueness which arise from multiple sources. For example, the type-2 fuzzy [5], [6] sets and type-n fuzzy sets [5] can include uncertainty while defining the membership functions themselves. Intuitionistic fuzzy sets [7] additionally introduce the degree of non-membership of an element to take into account that there might be some hesitation degree and the degree of membership and non-membership of an element might not always add to one. Non-stationary fuzzy sets [8] can model variation of opinion over time by defining a collection of type 1 fuzzy sets and an explicit relationship between them. Fuzzy multi-sets [9] generalise crisp sets where multiple occurrences of an element are permitted. Hesitant fuzzy sets [10] have been proposed from the motivation that the problem of assigning a degree of membership to an element is not because of a margin of error (like Atanassov's intuitionistic fuzzy sets) or a possibility distribution on possibility values (e.g. Formally these can be viewed as fuzzy multi-sets but with a different interpretation.

Factorial Hidden Markov Models (FHMMs) are powerful models for sequential data but they do not scale well with long sequences. We propose a scalable inference and learning algorithm for FHMMs that draws on ideas from the stochastic variational inference, neural network and copula literatures. Unlike existing approaches, the proposed algorithm requires no message passing procedure among latent variables and can be distributed to a network of computers to speed up learning. Our experiments corroborate that the proposed algorithm does not introduce further approximation bias compared to the proven structured mean-field algorithm, and achieves better performance with long sequences and large FHMMs.

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high dimensional. These rules are based on the assumption that the integrand has a certain degree of smoothness, which is expressed as that the integrand belongs to a certain reproducing kernel Hilbert space (RKHS). However, this assumption can be violated in practice (e.g., when the integrand is a black box function), and no general theory has been established for the convergence of kernel quadratures in such misspecified settings. Our contribution is in proving that kernel quadratures can be consistent even when the integrand does not belong to the assumed RKHS, i.e., when the integrand is less smooth than assumed. Specifically, we derive convergence rates that depend on the (unknown) lesser smoothness of the integrand, where the degree of smoothness is expressed via powers of RKHSs or via Sobolev spaces.

Two semimetrics on probability distributions are proposed, given as the sum of differences of expectations of analytic functions evaluated at spatial or frequency locations (i.e, features). The features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound on test power for a statistical test using these features. The result is a parsimonious and interpretable indication of how and where two distributions differ locally. An empirical estimate of the test power criterion converges with increasing sample size, ensuring the quality of the returned features. In real-world benchmarks on high-dimensional text and image data, linear-time tests using the proposed semimetrics achieve comparable performance to the state-of-the-art quadratic-time maximum mean discrepancy test, while returning human-interpretable features that explain the test results.

In PU learning, a binary classifier is trained from positive (P) and unlabeled (U) data without negative (N) data. Although N data is missing, it sometimes outperforms PN learning (i.e., ordinary supervised learning). Hitherto, neither theoretical nor experimental analysis has been given to explain this phenomenon. In this paper, we theoretically compare PU (and NU) learning against PN learning based on the upper bounds on estimation errors. We find simple conditions when PU and NU learning are likely to outperform PN learning, and we prove that, in terms of the upper bounds, either PU or NU learning (depending on the class-prior probability and the sizes of P and N data) given infinite U data will improve on PN learning. Our theoretical findings well agree with the experimental results on artificial and benchmark data even when the experimental setup does not match the theoretical assumptions exactly.

We propose and study a new model for reinforcement learning with rich observations, generalizing contextual bandits to sequential decision making. These models require an agent to take actions based on observations (features) with the goal of achieving long-term performance competitive with a large set of policies. To avoid barriers to sample-efficient learning associated with large observation spaces and general POMDPs, we focus on problems that can be summarized by a small number of hidden states and have long-term rewards that are predictable by a reactive function class. In this setting, we design and analyze a new reinforcement learning algorithm, Least Squares Value Elimination by Exploration. We prove that the algorithm learns near optimal behavior after a number of episodes that is polynomial in all relevant parameters, logarithmic in the number of policies, and independent of the size of the observation space. Our result provides theoretical justification for reinforcement learning with function approximation.

This paper introduces the variational R\'enyi bound (VR) that extends traditional variational inference to R\'enyi's alpha-divergences. This new family of variational methods unifies a number of existing approaches, and enables a smooth interpolation from the evidence lower-bound to the log (marginal) likelihood that is controlled by the value of alpha that parametrises the divergence. The reparameterization trick, Monte Carlo approximation and stochastic optimisation methods are deployed to obtain a tractable and unified framework for optimisation. We further consider negative alpha values and propose a novel variational inference method as a new special case in the proposed framework. Experiments on Bayesian neural networks and variational auto-encoders demonstrate the wide applicability of the VR bound.

We consider the problem of decentralized clustering and estimation over multi-task networks, where agents infer and track different models of interest. The agents do not know beforehand which model is generating their own data. They also do not know which agents in their neighborhood belong to the same cluster. We propose a decentralized clustering algorithm aimed at identifying and forming clusters of agents of similar objectives, and at guiding cooperation to enhance the inference performance. One key feature of the proposed technique is the integration of the learning and clustering tasks into a single strategy. We analyze the performance of the procedure and show that the error probabilities of types I and II decay exponentially to zero with the step-size parameter. While links between agents following different objectives are ignored in the clustering process, we nevertheless show how to exploit these links to relay critical information across the network for enhanced performance. Simulation results illustrate the performance of the proposed method in comparison to other useful techniques.

The U.S. government wants to find out if artificial intelligence can help doctors diagnose and treat breast cancer more effectively. In an effort to find targeted treatments for particularly invasive types of breast cancer that don't respond well to existing drugs, the Department of Defense announced this week that it is enlisting the biopharma company Berg Health to use AI for drug discovery. The partnership supports the White House's Cancer Moonshot initiative to screen up to 250,000 patient samples in search of new biological indicators, or biomarkers, of the earliest signs of cancer. While the death rate from breast cancer has dropped steadily over the past two decades, it remains the second-biggest killer among cancers in U.S. women, according to the National Cancer Institute. Under the partnership, Berg will have access to the DoD's Clinical Breast Care Project, a bank of 13,600 samples of both healthy and diseased tissue from nearly 8,000 patients.

Apple made several new product announcements on Thursday, most notably a new MacBook Pro laptop with a touchscreen strip above the keyboard. Apple introduced a pair of new MacBook Pro laptops with what's called the "Touch Strip." It's a touchscreen bar that sits above the keyboard and takes the place of the function keys. The strip offers different functionality depending on the software being used -- when you're using iTunes, for instance, you would see playback controls. The new laptops also have Apple's Touch ID fingerprint scanner, enabling Apple Pay as well as quick user-swapping on shared computers.