Nonparametric models are versatile, albeit computationally expensive, tool for modeling mixture models. In this paper, we introduce spectral methods for the two most popular nonparametric models: the Indian Buffet Process (IBP) and the Hierarchical Dirichlet Process (HDP). We show that using spectral methods for the inference of nonparametric models are computationally and statistically efficient. In particular, we derive the lower-order moments of the IBP and the HDP, propose spectral algorithms for both models, and provide reconstruction guarantees for the algorithms. For the HDP, we further show that applying hierarchical models on dataset with hierarchical structure, which can be solved with the generalized spectral HDP, produces better solutions to that of flat models regarding likelihood performance.

In this paper, we derive a Bayesian model order selection rule by using the exponentially embedded family method, termed Bayesian EEF. Unlike many other Bayesian model selection methods, the Bayesian EEF can use vague proper priors and improper noninformative priors to be objective in the elicitation of parameter priors. Moreover, the penalty term of the rule is shown to be the sum of half of the parameter dimension and the estimated mutual information between parameter and observed data. This helps to reveal the EEF mechanism in selecting model orders and may provide new insights into the open problems of choosing an optimal penalty term for model order selection and choosing a good prior from information theoretic viewpoints. The important example of linear model order selection is given to illustrate the algorithms and arguments. Lastly, the Bayesian EEF that uses Jeffreys prior coincides with the EEF rule derived by frequentist strategies. This shows another interesting relationship between the frequentist and Bayesian philosophies for model selection.

RADITIONAL rule based fuzzy systems encode expert opinion in the form of IF-THEN rules and optimise some performance metric (typically mean squared error in predicting a data-set) to obtain the hyper-parameters of the model (like the numeric values defining the shape of the membership functions etc.) [2]-[4]. The rule base is one of the core elements driving the model and slight change in the rule base would significantly affect the performance of the fuzzy inference system. Many automated methods have been proposed where the rule base structure and parameters can be automatically determined [5]-[7]. However interpretability of such models is an issue and various methods have been proposed to alleviate the issue [8]. In the present paper however, we are interested in the actual metric for comparison between different models having different rule bases derived from expert opinion. The comparison metric can nevertheless be embedded within an automated framework to evolve the best model if so required. The most straight forward way of comparing the fuzzy rule bases is to optimise the model parameters based on the prediction error (e.g.

In many scientific and engineering applications, we are tasked with the optimisation of an expensive to evaluate black box function $f$. Traditional settings for this problem assume just the availability of this single function. However, in many cases, cheap approximations to $f$ may be obtainable. For example, the expensive real world behaviour of a robot can be approximated by a cheap computer simulation. We can use these approximations to eliminate low function value regions cheaply and use the expensive evaluations of $f$ in a small but promising region and speedily identify the optimum. We formalise this task as a \emph{multi-fidelity} bandit problem where the target function and its approximations are sampled from a Gaussian process. We develop MF-GP-UCB, a novel method based on upper confidence bound techniques. In our theoretical analysis we demonstrate that it exhibits precisely the above behaviour, and achieves better regret than strategies which ignore multi-fidelity information. Empirically, MF-GP-UCB outperforms such naive strategies and other multi-fidelity methods on several synthetic and real experiments.

We consider a nonlinear state-space model with the state transition and observation functions expressed as basis function expansions. The coefficients in the basis function expansions are learned from data. Using a connection to Gaussian processes we also develop priors on the coefficients, for tuning the model flexibility and to prevent overfitting to data, akin to a Gaussian process state-space model. The priors can alternatively be seen as a regularization, and helps the model in generalizing the data without sacrificing the richness offered by the basis function expansion. To learn the coefficients and other unknown parameters efficiently, we tailor an algorithm using state-of-the-art sequential Monte Carlo methods, which comes with theoretical guarantees on the learning. Our approach indicates promising results when evaluated on a classical benchmark as well as real data.

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intractable because of two problems: (1) it includes the evaluation of the partition function and (2) it is formulated in the form of functional optimization. We avoid Problem (1) by using Bethe approximation. Bethe approximation is an approximation technique equivalent to the loopy belief propagation. Problem (2) can be solved by using orthonormal function expansion. Orthonormal function expansion can reduce a functional optimization problem to a function optimization problem. Our method can provide an analytic form of the solution of the inverse problem within the framework of Bethe approximation.

In real-time monitoring of hospital patients, high-quality inference of patients' health status using all information available from clinical covariates and lab tests are essential to enable successful medical interventions and improve patient outcomes. In this work, we develop and explore a Bayesian nonparametric model based on Gaussian process (GP) regression for hospital patient monitoring. Our method, MedGP, incorporates 24 clinical and lab covariates and supports a rich reference data set from which the relationships between these observed covariates may be inferred and exploited for high-quality inference of patient state over time. To do this, we develop a highly structured sparse GP kernel to enable tractable computation over tens of thousands of time points while estimating correlations among clinical covariates, patients, and periodicity in high-dimensional time series measurements of physiological signals. We apply MedGP to data from hundreds of thousands of patients treated at the Hospital of the University of Pennsylvania. MedGP has a number of benefits over current methods, including (i) not requiring an alignment of the time series data, (ii) quantifying confidence intervals in the predictions, (iii) exploiting a vast and rich database of patients, and (iv) providing interpretable relationships among clinical covariates. We evaluate and compare results from MedGP on the task of online state prediction for three different patient subgroups. Keywords: Gaussian processes, electronic health records, sparse time series analysis, spectral mixture kernel, kernel density estimation.

Managers of US National Forests must decide what policy to apply for dealing with lightning-caused wildfires. Conflicts among stakeholders (e.g., timber companies, home owners, and wildlife biologists) have often led to spirited political debates and even violent eco-terrorism. One way to transform these conflicts into multi-stakeholder negotiations is to provide a high-fidelity simulation environment in which stakeholders can explore the space of alternative policies and understand the tradeoffs therein. Such an environment needs to support fast optimization of MDP policies so that users can adjust reward functions and analyze the resulting optimal policies. This paper assesses the suitability of SMAC---a black-box empirical function optimization algorithm---for rapid optimization of MDP policies. The paper describes five reward function components and four stakeholder constituencies. It then introduces a parameterized class of policies that can be easily understood by the stakeholders. SMAC is applied to find the optimal policy in this class for the reward functions of each of the stakeholder constituencies. The results confirm that SMAC is able to rapidly find good policies that make sense from the domain perspective. Because the full-fidelity forest fire simulator is far too expensive to support interactive optimization, SMAC is applied to a surrogate model constructed from a modest number of runs of the full-fidelity simulator. To check the quality of the SMAC-optimized policies, the policies are evaluated on the full-fidelity simulator. The results confirm that the surrogate values estimates are valid. This is the first successful optimization of wildfire management policies using a full-fidelity simulation. The same methodology should be applicable to other contentious natural resource management problems where high-fidelity simulation is extremely expensive.

Datasets with hundreds of variables and many missing values are commonplace. In this setting, it is both statistically and computationally challenging to detect true predictive relationships between variables and also to suppress false positives. This paper proposes an approach that combines probabilistic programming, information theory, and non-parametric Bayes. It shows how to use Bayesian non-parametric modeling to (i) build an ensemble of joint probability models for all the variables; (ii) efficiently detect marginal independencies; and (iii) estimate the conditional mutual information between arbitrary subsets of variables, subject to a broad class of constraints. Users can access these capabilities using BayesDB, a probabilistic programming platform for probabilistic data analysis, by writing queries in a simple, SQL-like language. This paper demonstrates empirically that the method can (i) detect context-specific (in)dependencies on challenging synthetic problems and (ii) yield improved sensitivity and specificity over baselines from statistics and machine learning, on a real-world database of over 300 sparsely observed indicators of macroeconomic development and public health.

This paper introduces a method for efficiently inferring a high-dimensional distributed quantity from a few observations. The quantity of interest (QoI) is approximated in a basis (dictionary) learned from a training set. The coefficients associated with the approximation of the QoI in the basis are determined by minimizing the misfit with the observations. To obtain a probabilistic estimate of the quantity of interest, a Bayesian approach is employed. The QoI is treated as a random field endowed with a hierarchical prior distribution so that closed-form expressions can be obtained for the posterior distribution. The main contribution of the present work lies in the derivation of \emph{a representation basis consistent with the observation chain} used to infer the associated coefficients. The resulting dictionary is then tailored to be both observable by the sensors and accurate in approximating the posterior mean. An algorithm for deriving such an observable dictionary is presented. The method is illustrated with the estimation of the velocity field of an open cavity flow from a handful of wall-mounted point sensors. Comparison with standard estimation approaches relying on Principal Component Analysis and K-SVD dictionaries is provided and illustrates the superior performance of the present approach.