Add to Chrome
Add to Firefox
Add to EdgeErik Quaeghebeur
Probabilistic Integral Circuits
Oct 25, 2023
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.
E(n)-equivariant Graph Neural Cellular Automata
Jan 25, 2023
Cellular automata (CAs) are computational models exhibiting rich dynamics emerging from the local interaction of cells arranged in a regular lattice. Graph CAs (GCAs) generalise standard CAs by allowing for arbitrary graphs rather than regular lattices, similar to how Graph Neural Networks (GNNs) generalise Convolutional NNs. Recently, Graph Neural CAs (GNCAs) have been proposed as models built on top of standard GNNs that can be trained to approximate the transition rule of any arbitrary GCA. Existing GNCAs are anisotropic in the sense that their transition rules are not equivariant to translation, rotation, and reflection of the nodes' spatial locations. However, it is desirable for instances related by such transformations to be treated identically by the model. By replacing standard graph convolutions with E(n)-equivariant ones, we avoid anisotropy by design and propose a class of isotropic automata that we call E(n)-GNCAs. These models are lightweight, but can nevertheless handle large graphs, capture complex dynamics and exhibit emergent self-organising behaviours. We showcase the broad and successful applicability of E(n)-GNCAs on three different tasks: (i) pattern formation, (ii) graph auto-encoding, and (iii) simulation of E(n)-equivariant dynamical systems.
Continuous Mixtures of Tractable Probabilistic Models
Sep 21, 2022
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 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, which allows them to perform exact inference, but often they 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, using a finite set of integration points, the approximation method can be compiled into a PC performing `exact inference in an approximate model'. In experiments, we show that this simple scheme proves remarkably effective, as PCs learned this way set new state-of-the-art for tractable models on many standard density estimation benchmarks.
Sensitivity analysis for finite Markov chains in discrete time
Aug 09, 2014
When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This is done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations. We also study how the inferred credal set about the state at time n evolves as n->infinity: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.
Characterizing the Set of Coherent Lower Previsions with a Finite Number of Constraints or Vertices
Mar 15, 2012The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a reformulation of this criterion that only uses a finite number of constraints. Any such lower prevision is coherent if it lies within the convex polytope defined by these constraints. The vertices of this polytope are the extreme coherent lower previsions for the given set of gambles. Our reformulation makes it possible to compute them. We show how this is done and illustrate the procedure and its results.
Exchangeability and sets of desirable gambles
Dec 09, 2010
Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti's finite and infinite representation theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.