Markus Bläser

Benito van der Zander, Marcel Wienöbst, Markus Bläser et al.

Linear structural equation models represent direct causal effects as directed edges and confounding factors as bidirected edges. An open problem is to identify the causal parameters from correlations between the nodes. We investigate models, whose directed component forms a tree, and show that there, besides classical instrumental variables, missing cycles of bidirected edges can be used to identify the model. They can yield systems of quadratic equations that we explicitly solve to obtain one or two solutions for the causal parameters of adjacent directed edges. We show how multiple missing cycles can be combined to obtain a unique solution. This results in an algorithm that can identify instances that previously required approaches based on Gröbner bases, which have doubly-exponential time complexity in the number of structural parameters.

Julian Dörfler, Benito van der Zander, Markus Bläser et al.

Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from a combination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix. In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By standard simulation results, this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gröbner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the task asking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for $\forall R$, the co-class of the existential theory of the reals. In particular, this problem is $coNP$-hard. To our best knowledge, this is the first hardness result for some notion of identifiability.

Aaryan Gupta, Markus Bläser

Linear structural causal models (SCMs) are used to express and analyse the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. (AISTATS'22, PLMR 151, pp. 6770--6792, 2022) give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gröbner basis approach, which has doubly-exponential time complexity in the number of structural parameters. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively.

Sanyam Agarwal, Markus Bläser, Mridul Gupta

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.

Julian Dörfler, Benito van der Zander, Markus Bläser et al.

The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: probabilistic (i.e. purely observational), interventional, and counterfactual, that reflect the progressive sophistication of human thought regarding causation. We investigate the computational complexity aspects of reasoning in this framework focusing mainly on satisfiability problems expressed in probabilistic and causal languages across the PCH. That is, given a system of formulas in the standard probabilistic and causal languages, does there exist a model satisfying the formulas? Our main contribution is to prove the exact computational complexities showing that languages allowing addition and marginalization (via the summation operator) yield NP^PP, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. These are the first results to demonstrate a strictly increasing complexity across the PCH: from probabilistic to causal and counterfactual reasoning. On the other hand, in the case of full languages, i.e. allowing addition, marginalization, and multiplication, we show that the satisfiability for the counterfactual level remains the same as for the probabilistic and causal levels, solving an open problem in the field.

Sanyam Agarwal, Markus Bläser

Zhang et al. (ICML 2021, PLMR 139, pp. 12447-1245) introduced probabilistic generating circuits (PGCs) as a probabilistic model to unify probabilistic circuits (PCs) and determinantal point processes (DPPs). At a first glance, PGCs store a distribution in a very different way, they compute the probability generating polynomial instead of the probability mass function and it seems that this is the main reason why PGCs are more powerful than PCs or DPPs. However, PGCs also allow for negative weights, whereas classical PCs assume that all weights are nonnegative. One of the main insights of our paper is that the negative weights are responsible for the power of PGCs and not the different representation. PGCs are PCs in disguise, in particular, we show how to transform any PGC into a PC with negative weights with only polynomial blowup. PGCs were defined by Zhang et al. only for binary random variables. As our second main result, we show that there is a good reason for this: we prove that PGCs for categorial variables with larger image size do not support tractable marginalization unless NP = P. On the other hand, we show that we can model categorial variables with larger image size as PC with negative weights computing set-multilinear polynomials. These allow for tractable marginalization. In this sense, PCs with negative weights strictly subsume PGCs.

Markus Bläser, Julian Dörfler, Maciej Liśkiewicz et al.

We study the complexity of satisfiability problems in probabilistic and causal reasoning. Given random variables $X_1, X_2,\ldots$ over finite domains, the basic terms are probabilities of propositional formulas over atomic events $X_i = x_i$, such as $P(X_1 = x_1)$ or $P(X_1 = x_1 \vee X_2 = x_2)$. The basic terms can be combined using addition (yielding linear terms) or multiplication (polynomial terms). The probabilistic satisfiability problem asks whether a joint probability distribution satisfies a Boolean combination of (in)equalities over such terms. Fagin et al. (1990) showed that for basic and linear terms, this problem is NP-complete, making it no harder than Boolean satisfiability, while Mossé et al. (2022) proved that for polynomial terms, it is complete for the existential theory of the reals. Pearl's Causal Hierarchy (PCH) extends the probabilistic setting with interventional and counterfactual reasoning, enriching the expressiveness of languages. However, Mossé et al. (2022) found that satisfiability complexity remains unchanged. Van der Zander et al. (2023) showed that introducing a marginalization operator to languages induces a significant increase in complexity. We extend this line of work by adding two new dimensions to the problem by constraining the models. First, we fix the graph structure of the underlying structural causal model, motivated by settings like Pearl's do-calculus, and give a nearly complete landscape across different arithmetics and PCH levels. Second, we study small models. While earlier work showed that satisfiable instances admit polynomial-size models, this is no longer guaranteed with compact marginalization. We characterize the complexities of satisfiability under small-model constraints across different settings.

Oliver Broadrick, Sanyam Agarwal, Guy Van den Broeck et al.

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can be commonly expressed by polynomial size arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\textsf{FP}\neq\#\textsf{P}$ (an assumption implied by $\textsf{P} \neq \textsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small circuits for that function's multilinear representation.

Benito van der Zander, Markus Bläser, Maciej Liśkiewicz

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ$\exists$R, which can be viewed as a succinct variant of the well-studied class $\exists$R, and show that the problems we consider are complete for succ$\exists$R. Our results imply even stronger algorithmic limitations than were proven by Fagin, Halpern, and Megiddo (1990) and Mossé, Ibeling, and Icard (2022) for some variants of the standard languages used commonly in probabilistic and causal inference.