Latent feature modeling allows capturing the latent structure responsible for generating the observed properties of a set of objects. It is often used to make predictions either for new values of interest or missing information in the original data, as well as to perform data exploratory analysis. However, although there is an extensive literature on latent feature models for homogeneous datasets, where all the attributes that describe each object are of the same (continuous or discrete) nature, there is a lack of work on latent feature modeling for heterogeneous databases. In this paper, we introduce a general Bayesian nonparametric latent feature model suitable for heterogeneous datasets, where the attributes describing each object can be either discrete, continuous or mixed variables. The proposed model presents several important properties. First, it accounts for heterogeneous data while keeping the properties of conjugate models, which allow us to infer the model in linear time with respect to the number of objects and attributes. Second, its Bayesian nonparametric nature allows us to automatically infer the model complexity from the data, i.e., the number of features necessary to capture the latent structure in the data. Third, the latent features in the model are binary-valued variables, easing the interpretability of the obtained latent features in data exploratory analysis. We show the flexibility of the proposed model by solving both prediction and data analysis tasks on several real-world datasets. Moreover, a software package of the GLFM is publicly available for other researcher to use and improve it.

Bayes Nets (or Bayesian Networks) give remarkable results in determining the effects of many variables on an outcome. They typically perform strongly even in cases when other methods falter or fail. These networks have had relatively little use with business-related problems, although they have worked successfully for years in fields such as scientific research, public safety, aircraft guidance systems and national defense. Importantly, they often outperform regression, particularly in determining variables' effects. Regression is one of the most august multivariate methods, and among the most studied and applied.

One more opportunity to implement data mining techniques in the health care industry will be helping the healthcare insurers to detect fraud transactions so that the other patients can receive better and more affordable healthcare services. This occurs when individuals deceive an insurance company to try to obtain money to which they are not entitled. It happens when someone puts false information on an insurance application and when false or misleading information is given or important information is omitted in an insurance transaction or claim. Apart from the data we have collected from the patients, we will be gathering one more dataset where we will be having the details of all hospitals in the locality, diagnosis, quality. Evaluation: Here we cannot accurately classify whether a transaction is default or not, because of the challenges faced while collecting the data related to the hospitals.

For massive data, the family of subsampling algorithms is popular to downsize the data volume and reduce computational burden. Existing studies focus on approximating the ordinary least squares estimate in linear regression, where statistical leverage scores are often used to define subsampling probabilities. In this paper, we propose fast subsampling algorithms to efficiently approximate the maximum likelihood estimate in logistic regression. We first establish consistency and asymptotic normality of the estimator from a general subsampling algorithm, and then derive optimal subsampling probabilities that minimize the asymptotic mean squared error of the resultant estimator. An alternative minimization criterion is also proposed to further reduce the computational cost. The optimal subsampling probabilities depend on the full data estimate, so we develop a two-step algorithm to approximate the optimal subsampling procedure. This algorithm is computationally efficient and has a significant reduction in computing time compared to the full data approach. Consistency and asymptotic normality of the estimator from a two-step algorithm are also established. Synthetic and real data sets are used to evaluate the practical performance of the proposed method.

We propose a novel online learning algorithm for Restricted Boltzmann Machines (RBM), namely, the Online Generative Discriminative Restricted Boltzmann Machine (OGD-RBM), that provides the ability to build and adapt the network architecture of RBM according to the statistics of streaming data. The OGD-RBM is trained in two phases: (1) an online generative phase for unsupervised feature representation at the hidden layer and (2) a discriminative phase for classification. The online generative training begins with zero neurons in the hidden layer, adds and updates the neurons to adapt to statistics of streaming data in a single pass unsupervised manner, resulting in a feature representation best suited to the data. The discriminative phase is based on stochastic gradient descent and associates the represented features to the class labels. We demonstrate the OGD-RBM on a set of multi-category and binary classification problems for data sets having varying degrees of class-imbalance. We first apply the OGD-RBM algorithm on the multi-class MNIST dataset to characterize the network evolution. We demonstrate that the online generative phase converges to a stable, concise network architecture, wherein individual neurons are inherently discriminative to the class labels despite unsupervised training. We then benchmark OGD-RBM performance to other machine learning, neural network and ClassRBM techniques for credit scoring applications using 3 public non-stationary two-class credit datasets with varying degrees of class-imbalance. We report that OGD-RBM improves accuracy by 2.5-3% over batch learning techniques while requiring at least 24%-70% fewer neurons and fewer training samples. This online generative training approach can be extended greedily to multiple layers for training Deep Belief Networks in non-stationary data mining applications without the need for a priori fixed architectures.

Many applications of Bayesian data analysis involve sensitive information, motivating methods which ensure that privacy is protected. We introduce a general privacy-preserving framework for Variational Bayes (VB), a widely used optimization-based Bayesian inference method. Our framework respects differential privacy, the gold-standard privacy criterion, and encompasses a large class of probabilistic models, called the Conjugate Exponential (CE) family. We observe that we can straightforwardly privatise VB's approximate posterior distributions for models in the CE family, by perturbing the expected sufficient statistics of the complete-data likelihood. For a broadly-used class of non-CE models, those with binomial likelihoods, we show how to bring such models into the CE family, such that inferences in the modified model resemble the private variational Bayes algorithm as closely as possible, using the Polya-Gamma data augmentation scheme. The iterative nature of variational Bayes presents a further challenge since iterations increase the amount of noise needed. We overcome this by combining: (1) an improved composition method for differential privacy, called the moments accountant, which provides a tight bound on the privacy cost of multiple VB iterations and thus significantly decreases the amount of additive noise; and (2) the privacy amplification effect of subsampling mini-batches from large-scale data in stochastic learning. We empirically demonstrate the effectiveness of our method in CE and non-CE models including latent Dirichlet allocation, Bayesian logistic regression, and sigmoid belief networks, evaluated on real-world datasets.

A classic approach for learning Bayesian networks from data is to identify a maximum a posteriori (MAP) network structure. In the case of discrete Bayesian networks, MAP networks are selected by maximising one of several possible Bayesian Dirichlet (BD) scores; the most famous is the Bayesian Dirichlet equivalent uniform (BDeu) score from Heckerman et al (1995). The key properties of BDeu arise from its uniform prior over the parameters of each local distribution in the network, which makes structure learning computationally efficient; it does not require the elicitation of prior knowledge from experts; and it satisfies score equivalence. In this paper we will review the derivation and the properties of BD scores, and of BDeu in particular, and we will link them to the corresponding entropy estimates to study them from an information theoretic perspective. To this end, we will work in the context of the foundational work of Giffin and Caticha (2007), who showed that Bayesian inference can be framed as a particular case of the maximum relative entropy principle. We will use this connection to show that BDeu should not be used for structure learning from sparse data, since it violates the maximum relative entropy principle; and that it is also problematic from a more classic Bayesian model selection perspective, because it produces Bayes factors that are sensitive to the value of its only hyperparameter. Using a large simulation study, we found in our previous work (Scutari, 2016) that the Bayesian Dirichlet sparse (BDs) score seems to provide better accuracy in structure learning; in this paper we further show that BDs does not suffer from the issues above, and we recommend to use it for sparse data instead of BDeu. Finally, will show that these issues are in fact different aspects of the same problem and a consequence of the distributional assumptions of the prior.

Predictive models learned from historical data are widely used to help companies and organizations make decisions. However, they may digitally unfairly treat unwanted groups, raising concerns about fairness and discrimination. In this paper, we study the fairness-aware ranking problem which aims to discover discrimination in ranked datasets and reconstruct the fair ranking. Existing methods in fairness-aware ranking are mainly based on statistical parity that cannot measure the true discriminatory effect since discrimination is causal. On the other hand, existing methods in causal-based anti-discrimination learning focus on classification problems and cannot be directly applied to handle the ranked data. To address these limitations, we propose to map the rank position to a continuous score variable that represents the qualification of the candidates. Then, we build a causal graph that consists of both the discrete profile attributes and the continuous score. The path-specific effect technique is extended to the mixed-variable causal graph to identify both direct and indirect discrimination. The relationship between the path-specific effects for the ranked data and those for the binary decision is theoretically analyzed. Finally, algorithms for discovering and removing discrimination from a ranked dataset are developed. Experiments using the real dataset show the effectiveness of our approaches.

An important metric of users' satisfaction and engagement within on-line streaming services is the user session length, i.e. the amount of time they spend on a service continuously without interruption. Being able to predict this value directly benefits the recommendation and ad pacing contexts in music and video streaming services. Recent research has shown that predicting the exact amount of time spent is highly nontrivial due to many external factors for which a user can end a session, and the lack of predictive covariates. Most of the other related literature on duration based user engagement has focused on dwell time for websites, for search and display ads, mainly for post-click satisfaction prediction or ad ranking. In this work we present a novel framework inspired by hierarchical Bayesian modeling to predict, at the moment of login, the amount of time a user will spend in the streaming service. The time spent by a user on a platform depends upon user-specific latent variables which are learned via hierarchical shrinkage. Our framework enjoys theoretical guarantees, naturally incorporates flexible parametric/nonparametric models on the covariates and is found to outperform state-of- the-art estimators in terms of efficiency and predictive performance on real world datasets.

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint and are not well-suited to general purpose optimization packages for their solution. In this paper, we introduce a fundamentally different strategy: We formulate the structure learning problem as a smooth, constrained optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting nonconvex, constrained program involves smooth functions whose gradients are easy to compute and only involve elementary matrix operations. By using existing black-box optimization routines, our method uses global search to find an optimal DAG and can be implemented in about 50 lines of Python and outperforms existing methods without imposing any structural constraints.