Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.

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