Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Probabilistic integral circuits (PICs) have been recently introduced as probabilistic models enjoying the key ingredient behind expressive generative models: continuous latent variables (LVs). PICs are symbolic computational graphs defining continuous LV models as hierarchies of functions that are summed and multiplied together, or integrated over some LVs. They are tractable if LVs can be analytically integrated out, otherwise they can be approximated by tractable probabilistic circuits (PC) encoding a hierarchical numerical quadrature process, called QPCs. So far, only tree-shaped PICs have been explored, and training them via numerical quadrature requires memory-intensive processing at scale. In this paper, we address these issues, and present: (i) a pipeline for building DAG-shaped PICs out of arbitrary variable decompositions, (ii) a procedure for training PICs using tensorized circuit architectures, and (iii) neural functional sharing techniques to allow scalable training. In extensive experiments, we showcase the effectiveness of functional sharing and the superiority of QPCs over traditional PCs.

Probabilistic models based on continuous latent spaces, such as variational autoencoders, can be understood as uncountable mixture models where components depend continuously on the latent code. They have proven to be expressive tools for generative and probabilistic modelling, but are at odds with tractable probabilistic inference, that is, computing marginals and conditionals of the represented probability distribution. Meanwhile, tractable probabilistic models such as probabilistic circuits (PCs) can be understood as hierarchical discrete mixture models, and thus are capable of performing exact inference efficiently but often show subpar performance in comparison to continuous latent-space models. In this paper, we investigate a hybrid approach, namely continuous mixtures of tractable models with a small latent dimension. While these models are analytically intractable, they are well amenable to numerical integration schemes based on a finite set of integration points. With a large enough number of integration points the approximation becomes de-facto exact. Moreover, for a finite set of integration points, the integration method effectively compiles the continuous mixture into a standard PC. In experiments, we show that this simple scheme proves remarkably effective, as PCs learnt this way set new state of the art for tractable models on many standard density estimation benchmarks.

Probabilistic circuits (PCs) are a prominent representation of probability distributions with tractable inference. While parameter learning in PCs is rigorously studied, structure learning is often more based on heuristics than on principled objectives. In this paper, we develop Bayesian structure scores for deterministic PCs, i.e., the structure likelihood with parameters marginalized out, which are well known as rigorous objectives for structure learning in probabilistic graphical models. When used within a greedy cutset algorithm, our scores effectively protect against overfitting and yield a fast and almost hyper-parameter-free structure learner, distinguishing it from previous approaches. In experiments, we achieve good trade-offs between training time and model fit in terms of log-likelihood. Moreover, the principled nature of Bayesian scores unlocks PCs for accommodating frameworks such as structural expectation-maximization.

Density estimation could be viewed as a core component in machine learning, since a good estimator could be used to solve many tasks such as classification, regression, and imputing missing values. The main challenge of density estimation is balancing the model expressiveness and its learning and inference complexity. Probabilistic circuits (PCs) model a probability distribution as a computational graph. By imposing specific structural properties on such models many inference tasks become tractable. However, learning PCs usually relies on greedy and time consuming procedures. In this paper we propose a new unified approach to efficiently learn PCs having several structural properties. We introduce extremely randomized PCs (XPCs), PCs with a random structure. We show their advantage on standard density estimation benchmarks when compared to other density estimators.