arXiv subjects

Patrick Forré

Publications and source records attributed to Patrick Forré.

At least 19 recordsLinked to original sources

arXiv2026

Transitional Conditional Independence

Statistical models contain variables that are not random: parameters, treatments, environments, design points. Ordinary conditional independence cannot express relations involving such variables. To apply it one must first put a distribution on them, and that changes the meaning of the statement. This paper introduces transitional conditional independence. It relates three variables on a Markov kernel $K(W|T)$ with non-stochastic input $T$, and is defined by a single factorization: \[ X\perp\!\!\perp_{K(W|T)} Y |Z \quad :\iff \quad \exists\, Q(X|Z):\; K(X,Y,Z|T) = Q(X|Z)\otimes K(Y,Z|T).\] The relation asserts a Markov kernel $Q(X|Z)$ that is the same for every input $t$. It therefore yields a factorization rather than an almost-sure identity between conditional expectations, and it needs no distribution on the input space. The relation is asymmetric. We show that the asymmetry is essential: symmetrizing it destroys the statements it was built to make. We prove left and right versions of all separoid rules except Symmetry. Ten of them hold on arbitrary measurable spaces, the remaining ones under one condition on the spaces involved, and we give criteria for when Symmetry itself holds. We axiomatize the resulting structure and show that it arises from any symmetric separoid by a shift. We give several applications. Ancillarity, sufficiency and adequacy become factorizations that hold pointwise in the parameter, without a prior and without null sets; the theorems of Fisher--Neyman and of Basu take this form. The invariance hypothesis of invariant prediction, $Y \perp\!\!\perp E | X_S$, receives its intended meaning: one kernel predicts $Y$ from $X_S$ in every environment $E$. And Bayesian networks with non-stochastic input nodes satisfy a directed global Markov property whose graphical id-separation criterion returns a factorization of Markov kernels, on arbitrary input spaces.

math.ST↗

arXiv2026

Are Bayesian networks typically faithful?

Faithfulness is a common assumption in causal inference, often motivated by the fact that the faithful parameters of linear Gaussian and discrete Bayesian networks are typical, and the folklore belief that this should also hold for other classes of Bayesian networks. We address this open question by showing that among all Bayesian networks over a given DAG, the faithful Bayesian networks are indeed `typical': they constitute a dense, open set with respect to the total variation metric. This does not directly imply that faithfulness is typical in restricted classes of Bayesian networks that are often considered in statistical applications. To this end we consider the class of Bayesian networks parametrised by conditional exponential families, for which we show that under regularity conditions, the faithful parameters constitute a dense and open set, the unfaithful parameters have Lebesgue measure zero, and the induced faithful distributions are open and dense in the weak topology. This extends the existing results for linear Gaussian and discrete Bayesian networks. We also show for nonparametric classes of Bayesian networks with uniformly equicontinuous and uniformly bounded conditional densities that the faithful Bayesian networks are open and dense in the weak topology. All these results also hold for Bayesian networks with latent variables, if faithfulness is only required to hold with respect to the latent projection. Finally, for the considered conditional exponential family parametrisations and nonparametric conditional density models, the topological properties of conditional independence imply the existence of a consistent conditional independence test. Together with the topological properties of faithfulness, this implies that sound constraint-based causal discovery algorithms like PC and FCI are consistent on an open and dense -- and hence `typical' -- set of Bayesian networks.

math.ST↗

arXiv2026

The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/α$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $α$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $Λ(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E Λ(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^β/\mathbb{E}_{P_0}[R^β]$: the ceiling as $β\downarrow0$, $R$ at $β=1$. Which $β$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/α)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(β>1)$ below it and by flattening above.

math.ST↗

arXiv2026

Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses

In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}α$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $α$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}α$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.

math.ST↗

arXiv2026

A Direct Route to Markov Chain Convergence via Asymptotic Equivalence with the Target

For a Markov kernel $T$ with an invariant probability measure $π$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $π$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid π)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(π\mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $π$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $π$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $σ$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.

math.PR↗

arXiv2026

Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming

We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $Ω_n := \mathbb{R}^n \times Ω$ of a Cartesian space with the universal Hilbert cube $Ω$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $Ω_n \to \mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $Ω$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $ω\mapsto (W(ω),Φ(ω))$, which turn measurability of a smooth family from an axiom into a consequence. $\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $Π\dashv \flat \dashv \natural \dashv \sharp \dashv Λ$, making $\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\mathsf{P}(\mathcal{X})$ as push-forwards of $\mathcal{X}$-plots at every test object, $\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\mathbf{SSS}$ -- functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting -- and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.

math.CT↗

arXiv2026

Foundations of Independent Component Analysis

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

math.ST↗

arXiv2026

Quasi-Measurable Spaces: A Convenient Foundation of Probability Theory

We introduce the categories of quasi-measurable spaces, mild generalizations of quasi-Borel spaces that allow for general sample spaces and less restrictive random variables. Each such category is complete, cocomplete, cartesian closed and, as we show, even locally cartesian closed, a quasitopos and a Heyting category, and so models an extensional dependent type theory with a typed intuitionistic first-order logic. We put one probability monad in the foreground, the push-forward probability monad $\mathcal{P}$, and isolate the two properties of the sample space its theory needs: a Fubini property, making $\mathcal{P}$ commutative, and a product isomorphism $Ω\times Ω\cong Ω$, making it strong. It is affine always. Our main interest is the category of quasi-universal spaces, where the sample space is the universal Hilbert cube $[0,1]^\mathbb{N}$ with the $σ$-algebra of all universally measurable subsets. That completion is what buys explicit descriptions of the induced $σ$-algebras, as intersections of Lebesgue-complete $σ$-algebras, and for countably separated spaces as the completion of any countable separating family, which quasi-Borel spaces do not admit. All push-forward probability monads collapse into one here, and on standard quasi-universal spaces the Giry monad joins them. We prove a Fubini theorem, a disintegration theorem for Markov kernels, Kolmogorov extension theorems and a conditional de Finetti theorem, and translate them into properties of the associated Markov category. Finally we formalize causal Bayesian networks over quasi-universal spaces and prove a global Markov property, translating transitional conditional independence into this setting and establishing its asymmetric separoid rules. Exponential objects then put variables and causal mechanisms on the same graphical footing, in partially generic causal Bayesian networks.

math.PR↗

arXiv2026

AVQ-Attention: Adaptive Vector-Quantized Attention

The $\mathcal{O}(N^2)$ complexity of attention over $N$ tokens remains a computational bottleneck in transformer models. Vector-Quantized (VQ) attention reduces this to $\mathcal{O}(MN)$ by representing keys with $M$ codewords, but applies uniform codebook capacity regardless of where attention mass concentrates: high-attention regions of key space may be coarsely approximated while low-attention regions waste representational capacity. We propose Adaptive Vector-Quantized (AVQ) Attention, which adaptively allocates codebook capacity based on attention importance. Starting from a small set of codewords, our method identifies the most important codes during the forward pass and refines them with pre-learned child codewords, achieving fine-grained quantization where it matters most while maintaining coarse quantization elsewhere. We develop an implementation using custom Triton kernels that enables the full adaptive refinement process, including importance scoring, child codeword insertion, and parent contribution replacement, to be carried out within the tiled computation paradigm of Flash Attention with minimal overhead. Our approach maintains $\mathcal{O}(MN)$ complexity while achieving improved accuracy-efficiency trade-offs compared to fixed-codebook VQ-attention.

cs.LG↗

arXiv2026

Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs

Möbius inversion and Shapley values are two mathematical tools for characterizing and decomposing higher-order structure in complex systems. The former defines higher-order interactions as discrete derivatives over a partial order; the latter provides a principled way to attribute those interactions back to the `atomic' elements of the system. Both have found wide application, from combinatorics and cooperative game theory to machine learning and explainable AI. We generalize both tools simultaneously in two orthogonal directions: 1) from real-valued functions to functions valued in any abelian group (in particular, vector-valued functions), and 2) from partial orders and lattices to directed acyclic multigraphs (DAMGs) and weighted versions thereof. The classical axioms, linearity, efficiency, null player, and symmetry, which uniquely characterize Shapley values on lattices, are insufficient in this more general setting. We resolve this by introducing projection operators that recursively re-attribute higher-order synergies down to the roots of the graph, and by proposing two natural axioms: weak elements (coalitions with zero synergy can be removed without affecting any attribution) and flat hierarchy (on graphs with no intermediate hierarchy, attributions are distributed proportionally to edge counts). Together with linearity, these three axioms uniquely determine the Shapley values via a simple explicit formula, while automatically implying efficiency, null player, symmetry, and a novel projection property. The resulting framework recovers all existing lattice-based definitions as special cases, and naturally handles settings, such as games on non-lattice partial orders, which were previously out of reach. The extension to vector-valued functions and general DAMG-structured hierarchies opens new application areas in machine learning, natural language processing, and explainable AI.

cs.GT↗

arXiv2026

Abstract Markov Random Fields

Markov random fields are known to be fully characterized by properties of their information diagrams, or I-diagrams. In particular, for Markov random fields, regions in the I-diagram corresponding to disconnected vertex sets in the graph vanish. Recently, I-diagrams have been generalized to F-diagrams, for a larger class of functions F satisfying the chain rule beyond Shannon entropy, such as Kullback-Leibler divergence and cross-entropy. In this work, we generalize the notion and characterization of Markov random fields to this larger class of functions F and investigate preliminary applications. We define F-independences, F-mutual independences, and F-Markov random fields and characterize them by their F-diagram. In the process, we also define F-dual total correlation and prove that its vanishing is equivalent to F-mutual independence. We then apply our results to information functions F that are applied to probability mass functions. We show that if the probability distributions of a set of random variables are Markov random fields for the same graph, then we formally recover the notion of an F-Markov random field for that graph. We then study the Kullback-Leibler diagrams on specific Markov chains, leading to a visual representation of the second law of thermodynamics and a simple explicit derivation of the decomposition of the evidence lower bound for diffusion models.

cs.IT↗

arXiv2025

Modeling Human Beliefs about AI Behavior for Scalable Oversight

As AI systems advance beyond human capabilities, scalable oversight becomes critical: how can we supervise AI that exceeds our abilities? A key challenge is that human evaluators may form incorrect beliefs about AI behavior in complex tasks, leading to unreliable feedback and poor value inference. To address this, we propose modeling evaluators' beliefs to interpret their feedback more reliably. We formalize human belief models, analyze their theoretical role in value learning, and characterize when ambiguity remains. To reduce reliance on precise belief models, we introduce "belief model covering" as a relaxation. This motivates our preliminary proposal to use the internal representations of adapted foundation models to mimic human evaluators' beliefs. These representations could be used to learn correct values from human feedback even when evaluators misunderstand the AI's behavior. Our work suggests that modeling human beliefs can improve value learning and outlines practical research directions for implementing this approach to scalable oversight.

cs.AI↗

arXiv2025

Frame-based Equivariant Diffusion Models for 3D Molecular Generation

Recent methods for molecular generation face a trade-off: they either enforce strict equivariance with costly architectures or relax it to gain scalability and flexibility. We propose a frame-based diffusion paradigm that achieves deterministic E(3)-equivariance while decoupling symmetry handling from the backbone. Building on this paradigm, we investigate three variants: Global Frame Diffusion (GFD), which assigns a shared molecular frame; Local Frame Diffusion (LFD), which constructs node-specific frames and benefits from additional alignment constraints; and Invariant Frame Diffusion (IFD), which relies on pre-canonicalized invariant representations. To enhance expressivity, we further utilize EdgeDiT, a Diffusion Transformer with edge-aware attention. On the QM9 dataset, GFD with EdgeDiT achieves state-of-the-art performance, with a test NLL of -137.97 at standard scale and -141.85 at double scale, alongside atom stability of 98.98%, and molecular stability of 90.51%. These results surpass all equivariant baselines while maintaining high validity and uniqueness and nearly 2x faster sampling compared to EDM. Altogether, our study establishes frame-based diffusion as a scalable, flexible, and physically grounded paradigm for molecular generation, highlighting the critical role of global structure preservation.

cs.LG↗

arXiv2025

The Perils of Optimizing Learned Reward Functions: Low Training Error Does Not Guarantee Low Regret

In reinforcement learning, specifying reward functions that capture the intended task can be very challenging. Reward learning aims to address this issue by learning the reward function. However, a learned reward model may have a low error on the data distribution, and yet subsequently produce a policy with large regret. We say that such a reward model has an error-regret mismatch. The main source of an error-regret mismatch is the distributional shift that commonly occurs during policy optimization. In this paper, we mathematically show that a sufficiently low expected test error of the reward model guarantees low worst-case regret, but that for any fixed expected test error, there exist realistic data distributions that allow for error-regret mismatch to occur. We then show that similar problems persist even when using policy regularization techniques, commonly employed in methods such as RLHF. We hope our results stimulate the theoretical and empirical study of improved methods to learn reward models, and better ways to measure their quality reliably.

cs.LG↗

arXiv2025

AdS-GNN -- a Conformally Equivariant Graph Neural Network

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

cs.LG↗

arXiv2025

Clifford Group Equivariant Diffusion Models for 3D Molecular Generation

This paper explores leveraging the Clifford algebra's expressive power for $\E(n)$-equivariant diffusion models. We utilize the geometric products between Clifford multivectors and the rich geometric information encoded in Clifford subspaces in \emph{Clifford Diffusion Models} (CDMs). We extend the diffusion process beyond just Clifford one-vectors to incorporate all higher-grade multivector subspaces. The data is embedded in grade-$k$ subspaces, allowing us to apply latent diffusion across complete multivectors. This enables CDMs to capture the joint distribution across different subspaces of the algebra, incorporating richer geometric information through higher-order features. We provide empirical results for unconditional molecular generation on the QM9 dataset, showing that CDMs provide a promising avenue for generative modeling.

cs.LG↗

arXiv2025

Information Decomposition Diagrams Applied beyond Shannon Entropy: A Generalization of Hu's Theorem

In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information relate to each other. This is answered by Hu's theorem, which is widely known in the form of information diagrams: it relates shapes in a Venn diagram to information functions, thus establishing a bridge from set theory to information theory. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as a function satisfying the chain rule of information. This abstract viewpoint allows to prove a generalization of Hu's theorem. It applies to Shannon and Tsallis entropy, (Tsallis) Kullback-Leibler Divergence, cross-entropy, Kolmogorov complexity, submodular information functions, and the generalization error in machine learning. Our result implies for Chaitin's Kolmogorov complexity that the interaction complexities of all degrees are in expectation close to Shannon interaction information. For well-behaved probability distributions on increasing sequence lengths, this shows that the per-bit expected interaction complexity and information asymptotically coincide, thus showing a strong bridge between algorithmic and classical information theory.

cs.IT↗

arXiv2024

Robust Multi-view Co-expression Network Inference

Unraveling the co-expression of genes across studies enhances the understanding of cellular processes. Inferring gene co-expression networks from transcriptome data presents many challenges, including spurious gene correlations, sample correlations, and batch effects. To address these complexities, we introduce a robust method for high-dimensional graph inference from multiple independent studies. We base our approach on the premise that each dataset is essentially a noisy linear mixture of gene loadings that follow a multivariate $t$-distribution with a sparse precision matrix, which is shared across studies. This allows us to show that we can identify the co-expression matrix up to a scaling factor among other model parameters. Our method employs an Expectation-Maximization procedure for parameter estimation. Empirical evaluation on synthetic and gene expression data demonstrates our method's improved ability to learn the underlying graph structure compared to baseline methods.

stat.ML↗