Bayesian networks and causal models

In the previous sections we gave an introduction to the notion of causal depen-dencies and causal effects. There exist several possibilities to represent causal dependencies. We mentioned before the structural equation models that were among the first causal models. Instead, in our work, we use Bayesian networks.

Graphical models are often more intuitive when many parameters and inter de-pendencies need to be considered. In this section we will present Bayesian networks and expose the theory that motivated the use of this model to repre-sent causal dependencies. Different theorems based on graphical criteria are presented, and show the simplification of the problem of causal inference when Bayesian networks are used to model causal dependencies.

Bayesian networks

Bayesian networks [Dar09] are graphs consisting of a set of vertices and di-rected edges that represent conditional dependencies between a set of param-eters. Let us suppose that we want to represent our system,s, corresponding to a set of observed parameters,{Xi}i∈[1,N], with a Bayesian network. We cre-ate one vertex, Vi, for each parameter Xi and a directed edge, Ei,j, between Vi and Vj (denoted Vi → Vj) if the parameter Xj depends on the parameter Xi. The graph we obtain is a Bayesian network representing the dependencies between the parameters of our system s. Figure 1.5 presents the Bayesian network that represents the causal model of the variables X, Y andZ repre-sented by the structural equation model of Equation (1.3). One possible way to represent an atomic intervention on Xis given on Figure 1.6, whereXis not influenced anymore by its direct (or remote) causes and its value is defined by the intervention, that can be seen as the new and unique parent ofX.

In our work we are interested in Directed Acyclic Graph (DAG). DAGs are graphs where all the edges are oriented and that contain no cycle. However, as we will see in Section 1.3.4 where we present the algorithms we use for causal model inference, we will sometimes use a Partial Ancestral Graph (PAG). PAGs are partially oriented graphs, graphs where some of the edges are left unori-ented (Vi—Vj).

1.3. CAUSALITY 21

Figure 1.5:Example of a Bayesian network

Figure 1.6:Bayesian network representing an intervention onX

Causal model

Properties There exist several possibilities to model causal dependencies (Bayesian networks, Structural Equation Model are two common models). In our work, we are interested in the use of a Bayesian network to model the causal dependencies between the parameters of the system that is studied.

In this case, an edge between two nodes represents a causal dependency between the two corresponding parameters.

Bayesian networks provide a very concise and intuitive way to view the differ-ent relationships between the differdiffer-ent parameters of a system. One of the most interesting properties of causal models comes from their stability under intervention. As mentioned previously, the causal dependencies between the parameters of a system are still valid if we modify the behavior of this system.

Important works have been done [Pea09, SGS01] to simplify the causal study of a system by using its representation as a Bayesian network. In Section 1.3.4, we discuss the Bayesian network representation of a causal model. We also present some graphical criteria used in our work to predict the effect of an in-tervention on a system using data that was collected prior to this inin-tervention, without the need to perform additional experiments or observations.

The inference of the Bayesian network representing the causal dependencies between the parameters of the system that we want to model has been one of the biggest challenges of our work. Many algorithms rely on assumptions that are not verified in the domain of telecommunication networks. We believe that A causal approach to the study of Telecommunication networks

22 1. INTRODUCTION AND BACKGROUND the absence of the assumptions commonly made by these algorithms is not specific to the domain of telecommunication networks. Hence, the difficulties first faced and then solved should be of interest to a broad range of domains.

In the following paragraphs we present the different steps of the algorithms used to infer the causal model of a given system when we limit ourselves to passive observations exclusively. As shown in our work, it is very important to understand each of the methods being used and the assumption on which they rely to be able, if necessary, to validate them. In the event of a method relying on assumptions that are not verified for the domain on which we intend to apply it, it is necessary to whether mitigate these assumption or revise such method.

Inference Inference algorithms for graphical causal models can be divided roughly into two categories: The score-based algorithms and the constraint-based algorithms. The score-constraint-based algorithms solve an optimization problem:

The graph is built step by step and the progression is dictated by the opti-mization of an objective function matching the actual graph with the input data that represent our observations of the system. There has been an important progress in recent years in “exact Bayesian network inference”. Using the score-based approach, the exact posterior probability of subnetworks (edges, Markov blanket, . . . ) is computed to search for the best Bayesian network fitting the in-put data. [Koi06, KS04] propose an algorithm with lower than super-exponential complexity in the number of parameters of the system. These algorithms al-ways converge and allow its process to stop at any moment and obtain a graph with a lower bound on its accuracy. However, in these algorithms,completeness is assumed (all the variables are observed) and the parameters must follow a normal distribution. These hypotheses do not hold in our work.

Constraint-based algorithms build the graph in two steps. First, an undirected graph (skeleton) is built and in a second step the edges are oriented. The infer-ence of the skeleton starts with a fully connected graph (i.e., a graph in which every vertex is connected to any other vertex). Then we test all the possible conditional and unconditional independences between the different parame-ters. We remove an edge connecting two vertices when their corresponding parameters are found to be independent. This first phase consists of a series of independence tests. In the PC algorithm [SG91], the sequence of tests can be optimized to decrease the number of independences to test. In the PC al-gorithm, we start with a fully connected graph and test all the unconditional independences, removing an edge between two nodes whose corresponding parameters are found independent. For the parameters whose nodes are still adjacent, we test for each pair of adjacent nodes if there exists a conditioning set of size 1 that makes them independent. If such set exists we remove the edge connecting the corresponding two nodes, otherwise the edge is not re-moved. This step is repeated, increasing the conditioning set size by one at

1.3. CAUSALITY 23 each step, until the size of the conditioning set reaches the maximum degree of the current skeleton (the maximum number of adjacent vertices for a given vertex), which means that no more independence can be found. The second phase, the orientation of the edges, can be performed using two different meth-ods. One method orients the V-structures (subgraphsX−Z−Y where Xand Y are not adjacent) in a first step and then orients as many edges as possible without creating new colliders (in X → Z ← Y, Z is called a collider) or cy-cles [Pea09] in a second step. The second method, which is implemented in the kPC algorithm, is based on Weakly Additive Noise (WAN) models [TGS09]

and tests the independence of residuals by conditioning on their regressor. To determine the orientation of a given undirected edge, the regression from a child on one of his parent leads to a residual independent of its regressor, while this is not the case if the regression is made from a parent on one of its children (assuming the WAN property holds).

When we started our work, we used the Tetrad software [SGS01] to com-pare the graphs that could be obtained using score based methods with the graphs that could be obtained using constraint based methods. Tetrad imple-ments several algorithms that we describe in Section 1.3.5. However both, the constraint based algorithms and scored based algorithms are implemented in Tetrad based on the assumptions of linearity and normality. The parameters de-scribing the performance of the applications relying on telecommunication net-works do not verify these assumptions. Despite these limitations, it appeared that constraint based methods tend to infer more informative models whose dependencies are closer to our domain knowledge. While the experiments we made with the Tetrad software did not prove that score-based methods cannot give correct results for our data, we decided to use constraint-based methods in our work.

The performance of the kPC algorithm, which uses nonlinear regression, highly depends on the goodness of fit of the regression method used. As the depen-dencies between the parameters defining TCP performance do not follow any given functional model, the graphical causal models we obtain with the kPC al-gorithm suggest that the existing nonlinear regression functions could not cor-rectly capture the dependencies, as no orientations could be made in most of the cases. This is the case in the study of the Dropbox application performance that is presented in Chapter 4. The study of the Dropbox performance was one of the first study that was made in our work and the models we obtained with the kPC algorithm suggests us that the use of non linear regression to orient the edges of the causal dependencies in the second phase of the causal model inference was not accurate. Therefore, in our work we use a method relying on colliders to orient the edges of the Bayesian network representing the causal model of our system. Because our choice does not rely on theoretical results but on our observations, we also present the causal models obtained with the kPC algorithm in the study of the emulated network, Section 5.1.3, and in the A causal approach to the study of Telecommunication networks

24 1. INTRODUCTION AND BACKGROUND study of the FTP traffic, Section 5.2.2 that also show the inadequacy of this approach.

Remarks

It is worth noting that the PC algorithm is able to infer a causal model up to its independence equivalence class (called Markov Equivalence Class and rep-resented by a Partially Directed Acyclic Graph (PDAG) also called pattern). It often happens that several graphs exist that imply the detected independences (see Section 1.3.4 and the paragraph presenting the d-separation graphical cri-terion). As a consequence, the output of the PC algorithm is a graph where only the independences leading to a unique orientation of the edges are present.

The other edges are left unoriented. In our work we use the bnt toolbox imple-mentation of the PC algorithm [Mur01] and select the member of the equiva-lence class inferred by the PC algorithm based on our knowledge and under-standing of TCP networks.

It often happens that domain knowledge informs us on the presence of a causal dependence between two parameters of our model (X → Y) or, on the oppo-site, on the absence of a causal dependence between two parameters (X66→Y).

In [Mee95], the author presents an algorithm that, in the absence of latent vari-able, is able to infer a causal model that is consistent with both the detected independences and a prior background knowledge. The set of rules presented in [Mee95] is proven to be sound when no latent variable is present and the graphical causal model can be represented by a DAG, or its Markov Equiva-lence Class can be represented by a PDAG. In the event of latent variables present in the model, the inference of a causal model, where indirect depen-dencies can be present, is more complex [SMR95]. However, [BT12] also pro-poses a set of procedures to make use of prior domain knowledge to infer the Maximum Ancestral Graph that represents the (possibly indirect) dependencies between the parameters of the system that one is trying to model.²

An important notion in the problem of causal model inference is the one of faith-fulness. The faithfulness property is an assumption made by any causal model inference algorithm that stipulates that the properties exhibited by the underly-ing model, its independences in our case, are stable. This stability means that the properties we detect for the studied system are not entailed by the partic-ular circumstances under which the observations of such system happened, or, expressed differently, that such properties would still be valid if some of the system parameter values would be different. Summarized in a concise way, the faithfulness assumption means that the only independences entailed by

2Note that this prior domain knowledge can be used in existing implementations of graphical causal model inference algorithms present in tools such as Tetrad or the BayesNet Toolbox.

1.3. CAUSALITY 25 the system being studied are structural independences. The faithfulness as-sumption is related to the relation between causal dependencies and structural dependencies, by opposition to parametric dependencies. In the presence of deterministic dependencies, a given parameterization of the system can can-cel the influence of two parameters. If Y = 3X+ Z, and for a given value x of X we have Z = −1/3x, Y might seem independent of X. However such independence is not structural and, would the parameter values be different, such independence would not be true. Hence, such independence would not be stable and would violate the assumption of faithfulness.

When studying a system, one important step is the choice of the set of param-eters that will represent the system. The choice of these paramparam-eters can only be dictated by domain knowledge. However, some considerations need to be taken into account:

• When representing a system with a Bayesian network capturing the dif-ferent causal dependencies, a vertexVi, named according to a parameter Xi, also captures the action of all unobserved parameters that influence the parameter Xi (and no other parameters of the model). Therefore, it is not necessary to have all the parameters of all the mechanisms influ-encing the performance that one wants to model with causal model but, at least, one parameter per mechanism that rules the system behavior.

• In order to infer the causal model of our system we need to test the dif-ferent independences between the difdif-ferent parameters that represent our system. The presence of an unobserved variable influencing more than one observed variable is called a latent variable. The presence of latent variables makes the independence tests inaccurate. Even if there exist algorithms to detect the presence of a latent variables in a model [SMR95, SGS01], the model inferred by such an algorithm is not necessarily accurate and such a model cannot always be used to predict intervention. We will present some of the causal models obtained using the IC* algorithm [Pea09, Page 52]. These models often inform us on the possiblepresence of latent variables in our system. However, it is very dif-ficult to detect with certainty the presence or absence of latent variables in a set of parameters representing a system.

• Domain knowledge should dictate the choice of the parameters repre-senting the system being studied in order to obtain a system where all the mechanisms influencing its functioning are captured by parameters that we observe.

• In theory, one could start with an exhaustive list of parameters and re-move the parameters that are found unnecessary in the system model.

However, in practice, we do not always have access to all the parameters A causal approach to the study of Telecommunication networks

26 1. INTRODUCTION AND BACKGROUND that influence the behavior of the system. In addition, the algorithm exe-cution time is an important limitation that needs to be taken into account.

Section 3.2 discusses the independence test used in our work. The num-ber of independence tests grows very fast with the numnum-ber of parameters as we are dealing with a combinatorial problem. Increasing the number of parameters that represent our system also increases the size of the maximum conditioning set on which independences need to be tested in the constraint based algorithm used for the causal model inference. As presented in Figure 3.1 in Section 3.3.1, the completion time of the inde-pendence test used in our work increases very fast with the conditioning set size.

Finally, a last remark concerns the system observation and data acquisition.

An important fact when studying the parameter causal dependencies through statistical test is the presence ofoutliers. As we will see from the formulas in Section 1.3.4, outliers are important as they often represent points of interest for our predictions. We are often interested in predicting scenarios that differ from the one that are commonly observed. To do so, we need to have observed parameters in situations where they take values in ranges outside the area of the principal mass of their distribution. On the other hand, the presence of out-liers can make more complex the detection of structural, causal, dependencies if they are based on statistical tests. In our work, we often restrict our input data to the samples falling in the [P5,P95] during the stage of the causal model in-ference, whereP^X represents theX^thpercentile of a given parameter observed values.

Feature selection

A last point that we would like to insist on concerns the choice of the parameters that will form the nodes of the Bayesian network representing the causal model of your system. As mentioned previously, it is mainly domain knowledge that will suggest the parameters that impact the performance of the system being studied. In the event where our model does not include a parameter impacting more than one of the parameters being observed, such a parameter is called a latent variable. The presence of a latent variable makes the detection of the independence between two parameters more complex. In X← L →Y, where Lis a latent variable,XandYmight look dependent while they are not. Such a scenario makes a causal study more complex when there are back paths pass-ing through L. The causal theory in [Pea09, SGS01] handles the presence of latent variables. However, as illustrated by the models presented in Figure 5.3 and Figure 5.11 in Section 5.1.3 and Section 5.2.2 respectively, the detection of latent variables is often complex. In our work, we use domain knowledge to define a set of parameters that forms a system where all the dependences

1.3. CAUSALITY 27 are captured by the observed parameters, leaving ourselves the possibility to extend the list of parameters in the event of a possible latent variable prevent-ing the inference of our system causal model. An example of such scenario is given in Section 5.2.2.

Properties and theorems

In this section, we suppose that we have already inferred the Bayesian network representing the causal model of our system and we present the different prop-erties of such model. We also present the different graphical criteria that we use in our work to predict an intervention from passive observations made prior to the intervention and from the graphical causal model of our system.

D-separation The D-separation criterion is a graphical criterion to judge, from

D-separation The D-separation criterion is a graphical criterion to judge, from