Distributional Equivalence and Structure Learning for Bow-free Acyclic Path Diagrams

We consider the problem of structure learning for bow-free acyclic path diagrams (BAPs). BAPs can be viewed as a generalization of linear Gaussian DAG models that allow for certain hidden variables. We present a first method for this problem using a greedy score-based search algorithm. We also prove some necessary and some sufficient conditions for distributional equivalence of BAPs which are used in an algorithmic ap- proach to compute (nearly) equivalent model structures. This allows us to infer lower bounds of causal effects. We also present applications to real and simulated datasets using our publicly available R-package

D'ya like DAGs? A Survey on Structure Learning and Causal Discovery

Causal reasoning is a crucial part of science and human intelligence. In order to discover causal relationships from data, we need structure discovery methods. We provide a review of background theory and a survey of methods for structure discovery. We primarily focus on modern, continuous optimization methods, and provide reference to further resources such as benchmark datasets and software packages. Finally, we discuss the assumptive leap required to take us from structure to causality.Comment: 35 page

Model selection and local geometry

We consider problems in model selection caused by the geometry of models close to their points of intersection. In some cases---including common classes of causal or graphical models, as well as time series models---distinct models may nevertheless have identical tangent spaces. This has two immediate consequences: first, in order to obtain constant power to reject one model in favour of another we need local alternative hypotheses that decrease to the null at a slower rate than the usual parametric $n^{-1/2}$ (typically we will require $n^{-1/4}$ or slower); in other words, to distinguish between the models we need large effect sizes or very large sample sizes. Second, we show that under even weaker conditions on their tangent cones, models in these classes cannot be made simultaneously convex by a reparameterization. This shows that Bayesian network models, amongst others, cannot be learned directly with a convex method similar to the graphical lasso. However, we are able to use our results to suggest methods for model selection that learn the tangent space directly, rather than the model itself. In particular, we give a generic algorithm for learning Bayesian network models

Syy-seuraussuhteiden oppiminen piilomuuttujien vaikutuksessa

The causal relationships determining the behaviour of a system under study are inherently directional: by manipulating a cause we can control its effect, but an effect cannot be used to control its cause. Understanding the network of causal relationships is necessary, for example, if we want to predict the behaviour in settings where the system is subject to different manipulations. However, we are rarely able to directly observe the causal processes in action; we only see the statistical associations they induce in the collected data. This thesis considers the discovery of the fundamental causal relationships from data in several different learning settings and under various modeling assumptions. Although the research is mostly theoretical, possible application areas include biology, medicine, economics and the social sciences. Latent confounders, unobserved common causes of two or more observed parts of a system, are especially troublesome when discovering causal relations. The statistical dependence relations induced by such latent confounders often cannot be distinguished from directed causal relationships. Possible presence of feedback, that induces a cyclic causal structure, provides another complicating factor. To achieve informative learning results in this challenging setting, some restricting assumptions need to be made. One option is to constrain the functional forms of the causal relationships to be smooth and simple. In particular, we explore how linearity of the causal relations can be effectively exploited. Another common assumption under study is causal faithfulness, with which we can deduce the lack of causal relations from the lack of statistical associations. Along with these assumptions, we use data from randomized experiments, in which the system under study is observed under different interventions and manipulations. In particular, we present a full theoretical foundation of learning linear cyclic models with latent variables using second order statistics in several experimental data sets. This includes sufficient and necessary conditions on the different experimental settings needed for full model identification, a provably complete learning algorithm and characterization of the underdetermination when the data do not allow for full model identification. We also consider several ways of exploiting the faithfulness assumption for this model class. We are able to learn from overlapping data sets, in which different (but overlapping) subsets of variables are observed. In addition, we formulate a model class called Noisy-OR models with latent confounding. We prove sufficient and worst case necessary conditions for the identifiability of the full model and derive several learning algorithms. The thesis also suggests the optimal sets of experiments for the identification of the above models and others. For settings without latent confounders, we develop a Bayesian learning algorithm that is able to exploit non-Gaussianity in passively observed data.Syy-seuraussuhteet, jotka viime kädessä määrittävät tutkittavan järjestelmän toiminnan, ovat suunnattuja: syyhyn puuttumalla voimme vaikuttaa seuraukseen, mutta seuraukseen puuttumalla ei voida vaikuttaa syyhyn. Syy-seuraussuhteiden verkon tunteminen on ensiarvoisen tärkeää, erityisesti jos haluamme todella ymmärtää miten järjestelmä toimii, esimerkiksi kun sitä manipuloidaan tai muutetaan. Useimmiten syy-seuraus mekanismien toimintaa ei voida suoraan nähdä, ainostaan mekanismien aikaansaamat tilastolliset riippuvuudet havaitaan. Tässä väitöskirjassa esitellään menetelmiä syy-seuraussuhteiden oppimiseen havaituista riippuvuuksista tilastollisessa datassa, erilaisissa ympäristöissä ja tilanteissa. Tutkimuksen lähtökohta on teoreettinen, mahdollisia sovelluskohteita voi löytyä mm. biologiasta, lääketieteestä, taloustieteestä ja yhteiskuntatieteestä. Erityinen hankaluus syy-seuraussuhteiden oppimisen kannalta ovat piilomuuttujat, jotka vastaavat tutkittavan järjestelmän mittaamattomia osia. Piilomuuttujat voivat saada aikaan tilastollisia riippuvuuksia, joita on vaikea erottaa syy-seuraussuhteiden aiheuttamista riippuvuuksista. Syy-seuraussuhdeverkot voivat myös pitää sisällään syklejä. Jotta seuraussuhteita voidaan oppia näissä tilanteissa, tarvitaan muita yksinkertaistavia oletuksia. Yksittäisten seuraussuhteiden kompleksisuutta voidaan rajoittaa esimerkiksi lineaariseksi. Myös niin kutsuttu uskollisuusoletus, jonka mukaan eri seuraussuhteet eivät täysin kumoa toistensa vaikutusta, on hyödyllinen. Jossain tapauksissa tutkittavasta järjestelmästä saadaan havaintoja siihen itse vaikuttaen, esimerkiksi satunnaistetuissa kokeissa. Väitöskirjassa esitellään useita oppimismenetelmiä, useissa eri oppimistilainteissa, eri oletusten vallitessa. Syy-seuraussuhteita opitaan käyttäen erilaisissa koetilanteissa havaittua dataa. Erityisesti tarkastellaan teoreettisesti mitä seuraussuhteita voidaan oppia missäkin tilanteessa ja mitä ei. Väitöskirjassa esitellään myös optimaalisia koejärjestelyitä