### Flexible Models for Microclustering with Application to Entity Resolution

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.

### Microclustering: When the Cluster Sizes Grow Sublinearly with the Size of the Data Set

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 tasks, this assumption is undesirable. For example, when performing entity resolution, the size of each cluster is often unrelated to the size of the data set. Consequently, each cluster contains a negligible fraction of the total number of data points. Such tasks therefore require models that yield clusters whose sizes grow sublinearly with the size of the data set. We address this requirement by defining the \emph{microclustering property} and introducing a new model that exhibits this property. We compare this model to several commonly used clustering models by checking model fit using real and simulated data sets.

### Performance Bounds for Graphical Record Linkage

Record linkage involves merging records in large, noisy databases to remove duplicate entities. It has become an important area because of its widespread occurrence in bibliometrics, public health, official statistics production, political science, and beyond. Traditional linkage methods directly linking records to one another are computationally infeasible as the number of records grows. As a result, it is increasingly common for researchers to treat record linkage as a clustering task, in which each latent entity is associated with one or more noisy database records. We critically assess performance bounds using the Kullback-Leibler (KL) divergence under a Bayesian record linkage framework, making connections to Kolchin partition models. We provide an upper bound using the KL divergence and a lower bound on the minimum probability of misclassifying a latent entity. We give insights for when our bounds hold using simulated data and provide practical user guidance.

### Non-exchangeable random partition models for microclustering

Many popular random partition models, such as the Chinese restaurant process and its two-parameter extension, fall in the class of exchangeable random partitions, and have found wide applicability in model-based clustering, population genetics, ecology or network analysis. While the exchangeability assumption is sensible in many cases, it has some strong implications. In particular, Kingman's representation theorem implies that the size of the clusters necessarily grows linearly with the sample size; this feature may be undesirable for some applications, as recently pointed out by Miller et al. (2015). We present here a flexible class of non-exchangeable random partition models which are able to generate partitions whose cluster sizes grow sublinearly with the sample size, and where the growth rate is controlled by one parameter. Along with this result, we provide the asymptotic behaviour of the number of clusters of a given size, and show that the model can exhibit a power-law behavior, controlled by another parameter. The construction is based on completely random measures and a Poisson embedding of the random partition, and inference is performed using a Sequential Monte Carlo algorithm. Additionally, we show how the model can also be directly used to generate sparse multigraphs with power-law degree distributions and degree sequences with sublinear growth. Finally, experiments on real datasets emphasize the usefulness of the approach compared to a two-parameter Chinese restaurant process.

### d-blink: Distributed End-to-End Bayesian Entity Resolution

Entity resolution (ER) (record linkage or de-duplication) is the process of merging together noisy databases, often in the absence of a unique identifier. A major advancement in ER methodology has been the application of Bayesian generative models. Such models provide a natural framework for clustering records to unobserved (latent) entities, while providing exact uncertainty quantification and tight performance bounds. Despite these advancements, existing models do not scale to realistically-sized databases (larger than 1000 records) and they do not incorporate probabilistic blocking. In this paper, we propose "distributed Bayesian linkage" or d-blink -- the first scalable and distributed end-to-end Bayesian model for ER, which propagates uncertainty in blocking, matching and merging. We make several novel contributions, including: (i) incorporating probabilistic blocking directly into the model through auxiliary partitions; (ii) support for missing values; (iii) a partially-collapsed Gibbs sampler; and (iv) a novel perturbation sampling algorithm (leveraging the Vose-Alias method) that enables fast updates of the entity attributes. Finally, we conduct experiments on five data sets which show that d-blink can achieve significant efficiency gains -- in excess of 300$\times$ -- when compared to existing non-distributed methods.