How to cluster event sequences generated via different point processes is an interesting and important problem in statistical machine learning. To solve this problem, we propose and discuss an effective model-based clustering method based on a novel Dirichlet mixture model of a special but significant type of point processes --- Hawkes process. The proposed model generates the event sequences with different clusters from the Hawkes processes with different parameters, and uses a Dirichlet process as the prior distribution of the clusters. We prove the identifiability of our mixture model and propose an effective variational Bayesian inference algorithm to learn our model. An adaptive inner iteration allocation strategy is designed to accelerate the convergence of our algorithm. Moreover, we investigate the sample complexity and the computational complexity of our learning algorithm in depth. Experiments on both synthetic and real-world data show that the clustering method based on our model can learn structural triggering patterns hidden in asynchronous event sequences robustly and achieve superior performance on clustering purity and consistency compared to existing methods.

Most generative models for clustering implicitly assume that the number of data points in each cluster grows linearly with the total number of data points. Finite mixture models, Dirichlet process mixture models, and Pitman--Yor process mixture models make this assumption, as do all other infinitely exchangeable clustering models. However, for some applications, this assumption is inappropriate. For example, when performing entity resolution, the size of each cluster should be unrelated to the size of the data set, and each cluster should contain a negligible fraction of the total number of data points. These applications require models that yield clusters whose sizes grow sublinearly with the size of the data set. We address this requirement by defining the microclustering property and introducing a new class of models that can exhibit this property. We compare models within this class to two commonly used clustering models using four entity-resolution data sets.

Recent years have witnessed substantial progress in understanding the behavior of EM for mixture models that are correctly specified. Given that model misspecification is common in practice, it is important to understand EM in this more general setting. We provide non-asymptotic guarantees for population and sample-based EM for parameter estimation under a few specific univariate settings of misspecified Gaussian mixture models. Due to misspecification, the EM iterates no longer converge to the true model and instead converge to the projection of the true model over the set of models being searched over. We provide two classes of theoretical guarantees: first, we characterize the bias introduced due to the misspecification; and second, we prove that population EM converges at a geometric rate to the model projection under a suitable initialization condition. This geometric convergence rate for population EM imply a statistical complexity of order $1/\sqrt{n}$ when running EM with $n$ samples.

We study the sample complexity of semi-supervised learning (SSL) and introduce new assumptions based on the mismatch between a mixture model learned from unlabeled data and the true mixture model induced by the (unknown) class conditional distributions. Under these assumptions, we establish an $\Omega(K\log K)$ labeled sample complexity bound without imposing parametric assumptions, where $K$ is the number of classes. Our results suggest that even in nonparametric settings it is possible to learn a near-optimal classifier using only a few labeled samples. Unlike previous theoretical work which focuses on binary classification, we consider general multiclass classification ($K> 2$), which requires solving a difficult permutation learning problem. This permutation defines a classifier whose classification error is controlled by the Wasserstein distance between mixing measures, and we provide finite-sample results characterizing the behaviour of the excess risk of this classifier. Finally, we describe three algorithms for computing these estimators based on a connection to bipartite graph matching, and perform experiments to illustrate the superiority of the MLE over the majority vote estimator.

Understanding and interpreting how machine learning (ML) models make decisions have been a big challenge. While recent research has proposed various technical approaches to provide some clues as to how an ML model makes individual predictions, they cannot provide users with an ability to inspect a model as a complete entity. In this work, we propose a novel technical approach that augments a Bayesian non-parametric regression mixture model with multiple elastic nets. Using the enhanced mixture model, we can extract generalizable insights for a target model through a global approximation. To demonstrate the utility of our approach, we evaluate it on different ML models in the context of image recognition. The empirical results indicate that our proposed approach not only outperforms the state-of-the-art techniques in explaining individual decisions but also provides users with an ability to discover the vulnerabilities of the target ML models.

The inference of the causal relationship between a pair of observed variables is a fundamental problem in science, and most existing approaches are based on one single causal model. In practice, however, observations are often collected from multiple sources with heterogeneous causal models due to certain uncontrollable factors, which renders causal analysis results obtained by a single model skeptical. In this paper, we generalize the Additive Noise Model (ANM) to a mixture model, which consists of a finite number of ANMs, and provide the condition of its causal identifiability. To conduct model estimation, we propose Gaussian Process Partially Observable Model (GPPOM), and incorporate independence enforcement into it to learn latent parameter associated with each observation. Causal inference and clustering according to the underlying generating mechanisms of the mixture model are addressed in this work. Experiments on synthetic and real data demonstrate the effectiveness of our proposed approach.

The underlyingmetricis,however,non-parametric.Wedevelopamaximumlikelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, andprovidethecorresponding EMalgorithm.

The application of reinforcement learning (RL) involving interactions with individuals has grown significantly in recent years.

Probabilistic modelling lies at the heart of machine learning.