Uplift modeling estimates the causal effect of an intervention as the difference between potential outcomes under treatment and control, whereas counterfactual identification aims to recover the joint distribution of these potential outcomes (e.g., "Would this customer still have churned had we given them a marketing offer?"). This joint counterfactual distribution provides richer information than the uplift but is harder to estimate. However, the two approaches are synergistic: uplift models can be leveraged for counterfactual estimation. We propose a counterfactual estimator that fits a bivariate beta distribution to predicted uplift scores, yielding posterior distributions over counterfactual outcomes. Our approach requires no causal assumptions beyond those of uplift modeling. Simulations show the efficacy of the approach, which can be applied, for example, to the problem of customer churn in telecom, where it reveals insights unavailable to standard ML or uplift models alone.

This unsupervised learning method is widely used in various applications, including image analysis, information retrieval, text analysis, bioinformatics, and many more [1, 2, 3, 4]. Clustering helps uncover the underlying structure of the data, facilitates data summarization, and sometimes serves as a preprocessing step for other algorithms [2]. Despite its widespread use, one of the primary challenges many traditional clustering algorithms face is that they often assume that the data points form clusters with convex shapes. For example, centroid-based algorithms like k -means and distribution-based models like Gaussian Mixture Models (GMM) typically produce clusters that are hyperspherical or ellipsoidal [5]. Although this assumption simplifies the clustering process, it restricts the flexibility of these models to handle complex data distributions that do not conform to convex shapes.

Data clustering groups data points into components so that similar points are within the same component. Data clustering is commonly used for data exploration and is sometimes used as a preprocessing step for later analysis [1]. In this paper, the multivariate beta mixture model (MBMM), a new probabilistic model for soft clustering, is proposed. As the MBMM is a mixture model, it shares many properties with the Gaussian mixture model (GMM), including its soft cluster assignment and parametric modeling. In addition, the MBMM allows the generation of new (synthetic) instances based on a generative process. Because the beta distribution is highly flexible (e.g., unimodal, bimodal, straight line, or exponentially increasing or decreasing), MBMM can fit data with versatile shapes.

Long Short-Term Memory (LSTM) infers the long term dependency through a cell state maintained by the input and the forget gate structures, which models a gate output as a value in [0,1] through a sigmoid function. However, due to the graduality of the sigmoid function, the sigmoid gate is not flexible in representing multi-modality or skewness. Besides, the previous models lack correlation modeling between the gates, which would be a new method to adopt domain knowledge. This paper proposes a new gate structure with the bivariate Beta distribution. The proposed gate structure enables hierarchical probabilistic modeling on the gates within the LSTM cell, so the modelers can customize the cell state flow. Also, we observed that our structured flexible gate modeling is enabled by the probability density estimation. Moreover, we theoretically show and empirically experiment that the bivariate Beta distribution gate structure alleviates the gradient vanishing problem. We demonstrate the effectiveness of bivariate Beta gate structure on the sentence classification, image classification, polyphonic music modeling, and image caption generation.

Nonnegative Matrix Factorization (NMF) aims to factorize a matrix into two optimized nonnegative matrices appropriate for the intended applications. The method has been widely used for unsupervised learning tasks, including recommender systems (rating matrix of users by items) and document clustering (weighting matrix of papers by keywords). However, traditional NMF methods typically assume the number of latent factors (i.e., dimensionality of the loading matrices) to be fixed. This assumption makes them inflexible for many applications. In this paper, we propose a nonparametric NMF framework to mitigate this issue by using dependent Indian Buffet Processes (dIBP). In a nutshell, we apply a correlation function for the generation of two stick weights associated with each pair of columns of loading matrices, while still maintaining their respective marginal distribution specified by IBP. As a consequence, the generation of two loading matrices will be column-wise (indirectly) correlated. Under this same framework, two classes of correlation function are proposed (1) using Bivariate beta distribution and (2) using Copula function. Both methods allow us to adopt our work for various applications by flexibly choosing an appropriate parameter settings. Compared with the other state-of-the art approaches in this area, such as using Gaussian Process (GP)-based dIBP, our work is seen to be much more flexible in terms of allowing the two corresponding binary matrix columns to have greater variations in their non-zero entries. Our experiments on the real-world and synthetic datasets show that three proposed models perform well on the document clustering task comparing standard NMF without predefining the dimension for the factor matrices, and the Bivariate beta distribution-based and Copula-based models have better flexibility than the GP-based model.