In this paper, we take a statistical decision-theoretic viewpoint on social choice, putting a focus on the decision to be made on behalf of a system of agents. In our framework, we are given a statistical ranking model, a decision space, and a loss function defined on (parameter, decision) pairs, and formulate social choice mechanisms as decision rules that minimize expected loss. This suggests a general framework for the design and analysis of new social choice mechanisms. We compare Bayesian estimators, which minimize Bayesian expected loss, for the Mallows model and the Condorcet model respectively, and the Kemeny rule. We consider various normative properties, in addition to computational complexity and asymptotic behavior. In particular, we show that the Bayesian estimator for the Condorcet model satisfies some desired properties such as anonymity, neutrality, and monotonicity, can be computed in polynomial time, and is asymptotically different from the other two rules when the data are generated from the Condorcet model for some ground truth parameter.

One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more than two decades ago), provide numerical solutions to the nonlinear state estimation problems arising in SSMs. When combined with additional identification techniques, these algorithms provide solid solutions to the nonlinear system identification problem. We describe two general strategies for creating such combinations and discuss why SMC is a natural tool for implementing these strategies.

In machine learning, there is an ongoing revival of the use of variational Bayes (VB) to deal with complex probabilistic models with hidden variables. The revival is driven by the use of stochastic methods to approximate the VB lower bound and associated gradients. See for example [1, 2, 5]. The advantages include automated inference [3] and also that they are applicable to a much wider class of probabilistic models. While the basic recipes are limited in the flexibility of the approximate hidden-variable posteriors, there are ongoing efforts to make them more flexible.

A number of people are found in a room and involved in loud conversations in groups, just as it would happen in a cocktail party. There might also be some background noise, which could be music, car noise from outside, etc. Each person in this room is therefore forced to listen to a mixture of speech sounds coming from various directions, along with some noise. These sounds may come directly to one's ear or have first suffered a sequence of reverberations because of their reflections on the room's walls. The problem of focusing one's listening attention on a particular speaker among this cacophony of conversations and noise has been known as the cocktail party problem [6]. It consists of separating a mixture of speech signals of different characteristics with noise added to it. The signals are a-priori unknown (one listens only to a combination of them) as is also the way they have been mixed. The above scenario is a good analog for many other examples of situations that demand for a separation of mixed signals with no presupposed knowledge on the signals and the system mixing them.

KECA roughly reduces to a sorting of the importance of kernel eigenvectors by entropy instead of by variance as in Kernel Principal Components Analysis. In this work, we propose an extension of the KECA method, named Optimized KECA (OKECA), that directly extracts the optimal features retaining most of the data entropy by means of compacting the information in very few features (often in just one or two). The proposed method produces features which have higher expressive power . In particular, it is based on the Independent Component Analysis (ICA) framework, and introduces an extra rotation to the eigen-decomposition, which is optimized via gradient ascent search. This maximum entropy preservation suggests that OKECA features are more efficient than KECA features for density estimation. In addition, a critical issue in both methods is the selection of the kernel parameter since it critically affects the resulting performance. Here we analyze the most common kernel length-scale selection criteria. Results of both methods are illustrated in different synthetic and real problems. Results show that 1) OKECA returns projections with more expressive power than KECA, 2) the most successful rule for estimating the kernel parameter is based on maximum likelihood, and 3) OKECA is more robust to the selection of the length-scale parameter in kernel density estimation.

A variety of real-world tasks involve the classification of images into pre-determined categories. Designing image classification algorithms that exhibit robustness to acquisition noise and image distortions, particularly when the available training data are insufficient to learn accurate models, is a significant challenge. This dissertation explores the development of discriminative models for robust image classification that exploit underlying signal structure, via probabilistic graphical models and sparse signal representations. Probabilistic graphical models are widely used in many applications to approximate high-dimensional data in a reduced complexity set-up. Learning graphical structures to approximate probability distributions is an area of active research. Recent work has focused on learning graphs in a discriminative manner with the goal of minimizing classification error. In the first part of the dissertation, we develop a discriminative learning framework that exploits the complementary yet correlated information offered by multiple representations (or projections) of a given signal/image. Specifically, we propose a discriminative tree-based scheme for feature fusion by explicitly learning the conditional correlations among such multiple projections in an iterative manner. Experiments reveal the robustness of the resulting graphical model classifier to training insufficiency.

The Efficient Global Optimization (EGO) algorithm uses a conditional Gaus-sian Process (GP) to approximate an objective function known at a finite number of observation points and sequentially adds new points which maximize the Expected Improvement criterion according to the GP. The important factor that controls the efficiency of EGO is the GP covariance function (or kernel) which should be chosen according to the objective function. Traditionally, a pa-rameterized family of covariance functions is considered whose parameters are learned through statistical procedures such as maximum likelihood or cross-validation. However, it may be questioned whether statistical procedures for learning covariance functions are the most efficient for optimization as they target a global agreement between the GP and the observations which is not the ultimate goal of optimization. Furthermore, statistical learning procedures are computationally expensive. The main alternative to the statistical learning of the GP is self-adaptation, where the algorithm tunes the kernel parameters based on their contribution to objective function improvement. After questioning the possibility of self-adaptation for kriging based optimizers, this paper proposes a novel approach for tuning the length-scale of the GP in EGO: At each iteration, a small ensemble of kriging models structured by their length-scales is created. All of the models contribute to an iterate in an EGO-like fashion. Then, the set of models is densified around the model whose length-scale yielded the best iterate and further points are produced. Numerical experiments are provided which motivate the use of many length-scales. The tested implementation does not perform better than the classical EGO algorithm in a sequential context but show the potential of the approach for parallel implementations.

Review fraud is a pervasive problem in online commerce, in which fraudulent sellers write or purchase fake reviews to manipulate perception of their products and services. Fake reviews are often detected based on several signs, including 1) they occur in short bursts of time; 2) fraudulent user accounts have skewed rating distributions. However, these may both be true in any given dataset. Hence, in this paper, we propose an approach for detecting fraudulent reviews which combines these 2 approaches in a principled manner, allowing successful detection even when one of these signs is not present. To combine these 2 approaches, we formulate our Bayesian Inference for Rating Data (BIRD) model, a flexible Bayesian model of user rating behavior. Based on our model we formulate a likelihood-based suspiciousness metric, Normalized Expected Surprise Total (NEST). We propose a linear-time algorithm for performing Bayesian inference using our model and computing the metric. Experiments on real data show that BIRDNEST successfully spots review fraud in large, real-world graphs: the 50 most suspicious users of the Flipkart platform flagged by our algorithm were investigated and all identified as fraudulent by domain experts at Flipkart.

We introduce overdispersed black-box variational inference, a method to reduce the variance of the Monte Carlo estimator of the gradient in black-box variational inference. Instead of taking samples from the variational distribution, we use importance sampling to take samples from an overdispersed distribution in the same exponential family as the variational approximation. Our approach is general since it can be readily applied to any exponential family distribution, which is the typical choice for the variational approximation. We run experiments on two non-conjugate probabilistic models to show that our method effectively reduces the variance, and the overhead introduced by the computation of the proposal parameters and the importance weights is negligible. We find that our overdispersed importance sampling scheme provides lower variance than black-box variational inference, even when the latter uses twice the number of samples.

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use.