Causal Graphs Underlying Generative Models: Path to Learning with Limited Data

Training generative models that capture rich semantics of the data and interpreting the latent representations encoded by such models are very important problems in unsupervised learning. In this work, we provide a simple algorithm that relies on perturbation experiments on latent codes of a pre-trained generative autoencoder to uncover a causal graph that is implied by the generative model. We leverage pre-trained attribute classifiers and perform perturbation experiments to check for influence of a given latent variable on a subset of attributes. Given this, we show that one can fit an effective causal graph that models a structural equation model between latent codes taken as exogenous variables and attributes taken as observed variables. One interesting aspect is that a single latent variable controls multiple overlapping subsets of attributes unlike conventional approach that tries to impose full independence. Using a pre-trained RNN-based generative autoencoder trained on a dataset of peptide sequences, we demonstrate that the learnt causal graph from our algorithm between various attributes and latent codes can be used to predict a specific property for sequences which are unseen. We compare prediction models trained on either all available attributes or only the ones in the Markov blanket and empirically show that in both the unsupervised and supervised regimes, typically, using the predictor that relies on Markov blanket attributes generalizes better for out-of-distribution sequences

Bayesian Network Approximation from Local Structures

This work is focused on the problem of Bayesian network structure learning. There are two main areas in this field which are here discussed.The first area is a theoretical one. We consider some aspects of the Bayesian network structure learning hardness. In particular we prove that the problem of finding a Bayesian network structure with a minimal number of edges encoding the joint probability distribution of a given dataset is NP-hard. This result can be considered as a significantly different than the standard one view on the NP-hardness of the Bayesian network structure learning. The most notable so far results in this area are focused mainly on the specific characterization of the problem, where the aim is to find a Bayesian network structure maximizing some given probabilistic criterion. These criteria arise from quite advanced considerations in the area of statistics, and in particular their interpretation might be not intuitive---especially for the people not familiar with the Bayesian networks domain. In contrary the proposed here criterion, for which the NP-hardness is proved, does not require any advanced knowledge and it can be easily understandable.The second area is related to concrete algorithms. We focus on one of the most interesting branch in history of Bayesian network structure learning methods, leading to a very significant solutions. Namely we consider the branch of local Bayesian network structure learning methods, where the main aim is to gather first of all some information describing local properties of constructed networks, and then use this information appropriately in order to construct the whole network structure. The algorithm which is the root of this branch is focused on the important local characterization of Bayesian networks---so called Markov blankets. The Markov blanket of a given attribute consists of such other attributes which in the probabilistic sense correspond to the maximal in strength and minimal in size set of its causes. The aforementioned first algorithm in the considered here branch is based on one important observation. Subject to appropriate assumptions it is possible to determine the optimal Bayesian network structure by examining relations between attributes only within the Markov blankets. In the case of datasets derived from appropriately sparse distributions, where Markov blanket of each attribute has a limited by some common constant size, such procedure leads to a well time scalable Bayesian network structure learning approach.The Bayesian network local learning branch has mainly evolved in direction of reducing the gathered local information into even smaller and more reliably learned patterns. This reduction has raised from the parallel progress in the Markov blankets approximation field.The main result of this dissertation is the proposal of Bayesian network structure learning procedure which can be placed into the branch of local learning methods and which leads to the fork in its root in fact. The fundamental idea is to appropriately aggregate learned over the Markov blankets local knowledge not in the form of derived dependencies within these blankets---as it happens in the root method, but in the form of local Bayesian networks. The user can thanks to this have much influence on the character of this local knowledge---by choosing appropriate to his needs Bayesian network structure learning method used in order to learn the local structures. The merging approach of local structures into a global one is justified theoretically and evaluated empirically, showing its ability to enhance even very advanced Bayesian network structure learning algorithms, when applying them locally in the proposed scheme.Praca ta skupia się na problemie uczenia struktury sieci bayesowskiej. Są dwa główne pola w tym temacie, które są tutaj omówione.Pierwsze pole ma charakter teoretyczny. Rozpatrujemy pewne aspekty trudności uczenia struktury sieci bayesowskiej. W szczególności pokozujemy, że problem wyznaczenia struktury sieci bayesowskiej o minimalnej liczbie krawędzi kodującej w sobie łączny rozkład prawdopodobieństwa atrybutów danej tabeli danych jest NP-trudny. Rezultat ten może być postrzegany jako istotnie inne od standardowego spojrzenie na NP-trudność uczenia struktury sieci bayesowskiej. Najbardziej znaczące jak dotąd rezultaty w tym zakresie skupiają się głównie na specyficznej charakterystyce problemu, gdzie celem jest wyznaczenie struktury sieci bayesowskiej maksymalizującej pewne zadane probabilistyczne kryterium. Te kryteria wywodzą się z dość zaawansowanych rozważań w zakresie statystyki i w szczególności mogą nie być intuicyjne---szczególnie dla ludzi niezaznajomionych z dziedziną sieci bayesowskich. W przeciwieństwie do tego zaproponowane tutaj kryterium, dla którego została wykazana NP-trudność, nie wymaga żadnej zaawansowanej wiedzy i może być łatwo zrozumiane.Drugie pole wiąże się z konkretnymi algorytmami. Skupiamy się na jednej z najbardziej interesujących gałęzi w historii metod uczenia struktur sieci bayesowskich, prowadzącej do bardzo znaczących rozwiązań. Konkretnie rozpatrujemy gałąź metod lokalnego uczenia struktur sieci bayesowskich, gdzie głównym celem jest zebranie w pierwszej kolejności pewnych informacji opisujących lokalne własności konstruowanych sieci, a następnie użycie tych informacji w odpowiedni sposób celem konstrukcji pełnej struktury sieci. Algorytm będący korzeniem tej gałęzi skupia się na ważnej lokalnej charakteryzacji sieci bayesowskich---tak zwanych kocach Markowa. Koc Markowa dla zadanego atrybutu składa się z tych pozostałych atrybutów, które w sensie probabilistycznym odpowiadają maksymalnymu w sile i minimalnemu w rozmiarze zbiorowi jego przyczyn. Wspomniany pierwszy algorytm w rozpatrywanej tu gałęzi opiera się na jednej istotnej obserwacji. Przy odpowiednich założeniach możliwe jest wyznaczenie optymalnej struktury sieci bayesowskiej poprzez badanie relacji między atrybutami jedynie w obrębie koców Markowa. W przypadku zbiorów danych wywodzących się z odpowiednio rzadkiego rozkładu, gdzie koc Markowa każdego atrybutu ma ograniczony przez pewną wspólną stałą rozmiar, taka procedura prowadzi do dobrze skalowalnego czasowo podejścia uczenia struktury sieci bayesowskiej.Gałąź lokalnego uczenia sieci bayesowskich rozwinęła się głównie w kierunku redukcji zbieranych lokalnych informacji do jeszcze mniejszych i bardziej niezawodnie wyuczanych wzorców. Redukcja ta wyrosła na bazie równoległego rozwoju w dziedzinie aproksymacji koców Markowa.Głównym rezultatem tej rozprawy jest zaproponowanie procedury uczenia struktury sieci bayesowskiej, która może być umiejscowiona w gałęzi metod lokalnego uczenia i która faktycznie wyznacza rozgałęzienie w jego korzeniu. Fundamentalny pomysł polega tu na tym, żeby odpowiednio agregować wyuczoną w obrębie koców Markowa lokalną wiedzę nie w formie wyprowadzonych zależności w obrębie tych koców---tak jak to się dzieje w przypadku metody - korzenia, ale w formie lokalnych sieci bayesowskich. Użytkownik może mieć dzięki temu duży wpływ na charakter tej lokalnej wiedzy---poprzez wybór odpowiedniej dla jego potrzeb metody uczenia struktury sieci bayesowskiej użytej w celu wyznaczenia lokalnych struktur. Procedura scalenia lokalnych modeli celem utworzenia globalnego jest uzasadniona teoretycznie oraz zbadana eksperymentalnie, pokazując jej zdolność do poprawienia nawet bardzo zaawansowanych algorytmów uczenia struktury sieci bayesowskiej, gdy zastosuje się je lokalnie w ramach zaproponowanego schematu

Multiple Quantitative Trait Analysis Using Bayesian Networks

Models for genome-wide prediction and association studies usually target a single phenotypic trait. However, in animal and plant genetics it is common to record information on multiple phenotypes for each individual that will be genotyped. Modeling traits individually disregards the fact that they are most likely associated due to pleiotropy and shared biological basis, thus providing only a partial, confounded view of genetic effects and phenotypic interactions. In this paper we use data from a Multiparent Advanced Generation Inter-Cross (MAGIC) winter wheat population to explore Bayesian networks as a convenient and interpretable framework for the simultaneous modeling of multiple quantitative traits. We show that they are equivalent to multivariate genetic best linear unbiased prediction (GBLUP), and that they are competitive with single-trait elastic net and single-trait GBLUP in predictive performance. Finally, we discuss their relationship with other additive-effects models and their advantages in inference and interpretation. MAGIC populations provide an ideal setting for this kind of investigation because the very low population structure and large sample size result in predictive models with good power and limited confounding due to relatedness.Comment: 28 pages, 1 figure, code at http://www.bnlearn.com/research/genetics1

Efficient computational strategies to learn the structure of probabilistic graphical models of cumulative phenomena

Structural learning of Bayesian Networks (BNs) is a NP-hard problem, which is further complicated by many theoretical issues, such as the I-equivalence among different structures. In this work, we focus on a specific subclass of BNs, named Suppes-Bayes Causal Networks (SBCNs), which include specific structural constraints based on Suppes' probabilistic causation to efficiently model cumulative phenomena. Here we compare the performance, via extensive simulations, of various state-of-the-art search strategies, such as local search techniques and Genetic Algorithms, as well as of distinct regularization methods. The assessment is performed on a large number of simulated datasets from topologies with distinct levels of complexity, various sample size and different rates of errors in the data. Among the main results, we show that the introduction of Suppes' constraints dramatically improve the inference accuracy, by reducing the solution space and providing a temporal ordering on the variables. We also report on trade-offs among different search techniques that can be efficiently employed in distinct experimental settings. This manuscript is an extended version of the paper "Structural Learning of Probabilistic Graphical Models of Cumulative Phenomena" presented at the 2018 International Conference on Computational Science

Uncovering Meanings of Embeddings via Partial Orthogonality

Machine learning tools often rely on embedding text as vectors of real numbers. In this paper, we study how the semantic structure of language is encoded in the algebraic structure of such embeddings. Specifically, we look at a notion of ``semantic independence'' capturing the idea that, e.g., ``eggplant'' and ``tomato'' are independent given ``vegetable''. Although such examples are intuitive, it is difficult to formalize such a notion of semantic independence. The key observation here is that any sensible formalization should obey a set of so-called independence axioms, and thus any algebraic encoding of this structure should also obey these axioms. This leads us naturally to use partial orthogonality as the relevant algebraic structure. We develop theory and methods that allow us to demonstrate that partial orthogonality does indeed capture semantic independence. Complementary to this, we also introduce the concept of independence preserving embeddings where embeddings preserve the conditional independence structures of a distribution, and we prove the existence of such embeddings and approximations to them