Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions

LazyIter: A Fast Algorithm for Counting Markov Equivalent DAGs and Designing Experiments

The causal relationships among a set of random variables are commonly represented by a Directed Acyclic Graph (DAG), where there is a directed edge from variable $X$ to variable $Y$ if $X$ is a direct cause of $Y$. From the purely observational data, the true causal graph can be identified up to a Markov Equivalence Class (MEC), which is a set of DAGs with the same conditional independencies between the variables. The size of an MEC is a measure of complexity for recovering the true causal graph by performing interventions. We propose a method for efficient iteration over possible MECs given intervention results. We utilize the proposed method for computing MEC sizes and experiment design in active and passive learning settings. Compared to previous work for computing the size of MEC, our proposed algorithm reduces the time complexity by a factor of $O(n)$ for sparse graphs where $n$ is the number of variables in the system. Additionally, integrating our approach with dynamic programming, we design an optimal algorithm for passive experiment design. Experimental results show that our proposed algorithms for both computing the size of MEC and experiment design outperform the state of the art.Comment: 11 pages, 2 figures, ICM

Efficient Sampling and Counting of Graph Structures related to Chordal Graphs

Counting problems aim to count the number of solutions for a given input, for example, counting the number of variable assignments that satisfy a Boolean formula. Sampling problems aim to produce a random object from a desired distribution, for example, producing a variable assignment drawn uniformly at random from all assignments that satisfy a Boolean formula. The problems of counting and sampling of graph structures on different types of graphs have been studied for decades for their great importance in areas like complexity theory and statistical physics. For many graph structures such as independent sets and acyclic orientations, it is widely believed that no exact or approximate (with an arbitrarily small error) polynomial-time algorithms on general graphs exist. Therefore, the research community studies various types of graphs, aiming either to design a polynomial-time counting or sampling algorithm for such inputs, or to prove a corresponding inapproximability result. Chordal graphs have been studied widely in both AI and theoretical computer science, but their study from the counting perspective has been relatively limited. Previous works showed that some graph structures can be counted in polynomial time on chordal graphs, when their counting on general graphs is provably computationally hard. The main objective of this thesis is to design and analyze counting and sampling algorithms for several well-known graph structures, including independent sets and different types of graph orientations, on chordal graphs. Our contributions can be described from two perspectives: evaluating the performances of some well-known sampling techniques, such as Markov chain Monte Carlo, on chordal graphs; and showing that the chordality does make those counting problems polynomial-time solvable

Counting and Sampling Markov Equivalent Directed Acyclic Graphs

Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2(n)n(4)) arithmetic operations, improving a previous super-exponential upper bound. The second requires O (k! 2(k) k(2)n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.Peer reviewe

Counting and Sampling Directed Acyclic Graphs for Learning Bayesian Networks

Bayesian networks are probabilistic models that represent dependencies between random variables via directed acyclic graphs (DAGs). They provide a succinct representation for the joint distribution in cases where the dependency structure is sparse. Specifying the network by hand is often unfeasible, and thus it would be desirable to learn the model from observed data over the variables. In this thesis, we study computational problems encountered in different approaches to learning Bayesian networks. All of the problems involve counting or sampling DAGs under various constraints. One important computational problem in the fully Bayesian approach to structure learning is the problem of sampling DAGs from the posterior distribution over all the possible structures for the Bayesian network. From the typical modeling assumptions it follows that the distribution is modular, which means that the probability of each DAG factorizes into per-node weights, each of which depends only on the parent set of the node. For this problem, we give the first exact algorithm with a time complexity bound exponential in the number of nodes, and thus polynomial in the size of the input, which consists of all the possible per-node weights. We also adapt the algorithm such that it outperforms the previous methods in the special case of sampling DAGs from the uniform distribution. We also study the problem of counting the linear extensions of a given partial order, which appears as a subroutine in some importance sampling methods for modular distributions. This problem is a classic example of a #P-complete problem that can be approximately solved in polynomial time by reduction to sampling linear extensions uniformly at random. We present two new randomized approximation algorithms for the problem. The first algorithm extends the applicable range of an exact dynamic programming algorithm by using sampling to reduce the given instance into an easier instance. The second algorithm is obtained by combining a novel, Markov chain-based exact sampler with the Tootsie Pop algorithm, a recent generic scheme for reducing counting into sampling. Together, these two algorithms speed up approximate linear extension counting by multiple orders of magnitude in practice. Finally, we investigate the problem of counting and sampling DAGs that are Markov equivalent to a given DAG. This problem is important in learning causal Bayesian networks, because distinct Markov equivalent DAGs cannot be distinguished only based on observational data, yet they are different from the causal viewpoint. We speed up the state-of-the-art recursive algorithm for the problem by using dynamic programming. We also present a new, tree decomposition-based algorithm, which runs in linear time if the size of the maximum clique is bounded.Bayes-verkot mallintavat satunnaismuuttujien välisiä tilastollisia suhteita suunnattuina syklittöminä verkkoina, joissa solmut vastaavat satunnaismuuttujia ja kaaret niiden välisiä riippuvuuksia. Verkkorakenne havainnollistaa muuttujien kuvaaman ilmiön rakennetta ja mahdollistaa muuttujien yhteisjakauman esittämisen tiiviissä muodossa. Vaikka Bayes-verkko voidaan periaatteessa rakentaa käsin, se on epäkäytännöllistä, mikäli muuttujia on paljon tai mallinnettavaa ilmiötä ei ymmärretä täydellisesti. Tämän takia on hyödyllistä oppia verkon rakenne ilmiöstä kerätyn datan perusteella. Väitöskirjassa tutkitaan laskennallisia ongelmia, jotka liittyvät Bayes-verkon rakenteen oppimiseen. Kaikki nämä ongelmat koskevat suunnattujen syklittömien verkkojen laskemista tai satunnaisotantaa erilaisilla rajoitteilla. Yksi keskeinen ongelma Bayes-verkon rakenteen oppimisessa on rakenteen poiminta posteriorisatunnaisjakaumasta, joka painottaa parhaiten dataa vastaavia rakenteita. Väitöskirjassa esitellään tähän ongelmaan ensimmäinen eksakti algoritmi, joka hyödyntämällä posteriorijakauman erityisominaisuuksia saavuttaa polynomisen aikavaativuuden suhteessa jakauman määrittelevän tietorakenteen kokoon. Algoritmi tarjoaa myös aiempia algoritmeja tehokkaamman tavan suunnattujen syklittömien verkkojen poimintaan tasajakaumasta. Toinen väitöskirjassa tutkittu ongelma on osittaisjärjestyksen lineaariekstensioiden laskenta. Tämä ongelma tiedetään kuuluvaksi vaikeiden laskentaongelmien #P-luokkaan, mutta se voidaan silti ratkaista likimäärisesti polynomisessa ajassa palauttamalla se vastaavaan satunnaisotantaongelmaan. Väitöskirja esittelee kaksi uutta likimääräistä satunnaisalgoritmia lineaariekstensioiden laskentaan. Ensimmäinen algoritmi muuttaa tunnetun eksaktin laskenta-algoritmin likimääräiseksi yhdistämällä siihen satunnaisotokseen perustuvaa arviointia. Toinen algoritmi palauttaa laskentaongelman uuteen Markovin ketjuihin perustuvan satunnaisotantamenetelmään. Yhdessä nämä kaksi algoritmia nopeuttavat käytännön tapauksissa likimääräistä lineaariekstensioiden laskentaa usealla kertaluokalla. Työn loppuosassa tutkitaan tietyssä Markov-ekvivalenssiluokassa olevien suunnattujen syklittömien verkkojen laskenta- ja satunnaisotantaongelmia. Ongelma on tärkeä Bayes-verkkojen käytössä kausaalisten riippuvuuksien mallintamiseen, koska Markov-ekvivalentteja rakenteita ei voi erottaa pelkästään havaintodatan perusteella, vaikka ne ovat kausaalisesta näkökulmasta erilaisia. Työssä esitellään tapa nopeuttaa parasta tunnettua algoritmia dynaamisen ohjelmoinnin avulla. Tämän lisäksi väitöskirja esittelee uuden verkon puuhajotelmaan perustuvan menetelmän, jonka aikavaativuus on lineaarinen, mikäli verkon suurimman klikin koko on rajoitettu