• Clarified lower and upper reachability definitions for IMCs and their relations • Lower hitting probabilities as unique solution of nonlinear fixed-point equation • Efficient iterative algorithms to compute lower and upper hitting probabilities We study the computation of lower and upper probabilities of hitting a target set of states for imprecise Markov chains, where transition uncertainty is modelled by a convex set of transition matrices. In the precise case, hitting probabilities are the minimal nonnegative solution of a linear system and admit a closed-form expression. We investigate the notion of reachability in the imprecise setting. The literature review highlights several different definitions of lower reachability; thus, we explore the relations among them and present examples to clarify their logical implications. Using this revised definition of reachability for imprecise Markov chains, we partition the state space into classes of states whose hitting probabilities are trivially zero or one, and those which require further computation. For these nontrivial states, we show that the lower hitting probability is the unique solution of a nonlinear fixed-point equation, while the same does not hold for upper hitting probabilities. For the practical computation of lower and upper hitting probabilities, we propose iterative algorithms that alternate between solving a linear system and choosing an extreme point from the set of transition matrices. Numerical experiments demonstrate that, in practice, these algorithms converge in substantially fewer iterations than the theoretically established worst-case bound.

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 circuits are a class of probabilistic generative models that allow us to compute different types of probabilistic queries in polynomial time. Unlike many of the mainstream approaches for generative modeling, they can compute exact likelihoods, marginals, and expectations. Yet, assessing the reliability of their inferences is not straightforward. Credal probabilistic circuits are the imprecise counterpart of probabilistic circuits allowing, among other queries, computations of cautious inferences and sensitivity analyses. In this work, we propose an efficient algorithm to compute the lower and upper expectations for factorizing functions using a credal probabilistic circuit. We discuss under what structural assumptions and types of factorizing functions the algorithm works. We prove that such algorithm has polynomial time complexity in the input size. In the general case, we prove that computing cautious inferences using credal probabilistic circuits is an NP-hard problem, yet the proposed algorithm can be used as an approximation. Some experiments show how the approximation degrades with the complexity of the model structure.

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.