Noisy-OR Models with Latent Confounding

Given a set of experiments in which varying subsets of observed variables are subject to intervention, we consider the problem of identifiability of causal models exhibiting latent confounding. While identifiability is trivial when each experiment intervenes on a large number of variables, the situation is more complicated when only one or a few variables are subject to intervention per experiment. For linear causal models with latent variables Hyttinen et al. (2010) gave precise conditions for when such data are sufficient to identify the full model. While their result cannot be extended to discrete-valued variables with arbitrary cause-effect relationships, we show that a similar result can be obtained for the class of causal models whose conditional probability distributions are restricted to a `noisy-OR' parameterization. We further show that identification is preserved under an extension of the model that allows for negative influences, and present learning algorithms that we test for accuracy, scalability and robustness

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

Experiment Selection for Causal Discovery

Randomized controlled experiments are often described as the most reliable tool available to scientists for discovering causal relationships among quantities of interest. However, it is often unclear how many and which different experiments are needed to identify the full (possibly cyclic) causal structure among some given (possibly causally insufficient) set of variables. Recent results in the causal discovery literature have explored various identifiability criteria that depend on the assumptions one is able to make about the underlying causal process, but these criteria are not directly constructive for selecting the optimal set of experiments. Fortunately, many of the needed constructions already exist in the combinatorics literature, albeit under terminology which is unfamiliar to most of the causal discovery community. In this paper we translate the theoretical results and apply them to the concrete problem of experiment selection. For a variety of settings we give explicit constructions of the optimal set of experiments and adapt some of the general combinatorics results to answer questions relating to the problem of experiment selection