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1. Compact Representation of Uncertainty in Clustering

   Greenberg, Craig; Monath, Nicholas; Kobren, Ari; Flaherty, Patrick; McGregor, Andrew; McCallum, Andrew (January 2018, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018)

   For many classic structured prediction problems, probability distributions over the dependent variables can be efficiently computed using widely-known algorithms and data structures (such as forward-backward, and its corresponding trellis for exact probability distributions in Markov models). However, we know of no previ- ous work studying efficient representations of exact distributions over clusterings. This paper presents definitions and proofs for a dynamic-programming inference procedure that computes the partition function, the marginal probability of a cluster, and the MAP clustering—all exactly. Rather than the N th Bell number, these exact solutions take time and space proportional to the substantially smaller powerset of N . Indeed, we improve upon the time complexity of the algorithm introduced by Kohonen and Corander [11] for this problem by a factor of N. While still large, this previously unknown result is intellectually interesting in its own right, makes feasible exact inference for important real-world small data applications (such as medicine), and provides a natural stepping stone towards sparse-trellis approximations that enable further scalability (which we also explore). In experi- ments, we demonstrate the superiority of our approach over approximate methods in analyzing real-world gene expression data used in cancer treatment. 

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2. A conditional density estimation partition model using logistic Gaussian processes

   https://doi.org/10.1093/biomet/asz064  

   Payne, R D; Guha, N; Ding, Y; Mallick, B K (December 2019, Biometrika)

   Summary Conditional density estimation seeks to model the distribution of a response variable conditional on covariates. We propose a Bayesian partition model using logistic Gaussian processes to perform conditional density estimation. The partition takes the form of a Voronoi tessellation and is learned from the data using a reversible jump Markov chain Monte Carlo algorithm. The methodology models data in which the density changes sharply throughout the covariate space, and can be used to determine where important changes in the density occur. The Markov chain Monte Carlo algorithm involves a Laplace approximation on the latent variables of the logistic Gaussian process model which marginalizes the parameters in each partition element, allowing an efficient search of the approximate posterior distribution of the tessellation. The method is consistent when the density is piecewise constant in the covariate space or when the density is Lipschitz continuous with respect to the covariates. In simulation and application to wind turbine data, the model successfully estimates the partition structure and conditional distribution. 

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3. Analytical Probability Distributions and EM-Learning for Deep Generative Networks

   Balestriero, Randall; Paris, Sebastien; Baraniuk, Richard G. (January 2020, Advances in neural information processing systems)

   Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation. 

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4. Analytical Probability Distributions and EM-Learning for Deep Generative Networks

   Balestriero, Randall; Paris, Sebastien; Baraniuk, Richard G. (December 2020, Advances in neural information processing systems)

   null (Ed.)

   Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation. 

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5. Practical Consequences of the Bias in the Laplace Approximation to Marginal Likelihood for Hierarchical Models

   https://doi.org/10.3390/e27030289  

   Lele, Subhash R; Glen, C George; Ponciano, José Miguel (March 2025, Entropy)

   Due to the high dimensional integration over latent variables, computing marginal likelihood and posterior distributions for the parameters of a general hierarchical model is a difficult task. The Markov Chain Monte Carlo (MCMC) algorithms are commonly used to approximate the posterior distributions. These algorithms, though effective, are computationally intensive and can be slow for large, complex models. As an alternative to the MCMC approach, the Laplace approximation (LA) has been successfully used to obtain fast and accurate approximations to the posterior mean and other derived quantities related to the posterior distribution. In the last couple of decades, LA has also been used to approximate the marginal likelihood function and the posterior distribution. In this paper, we show that the bias in the Laplace approximation to the marginal likelihood has substantial practical consequences. 

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