Modeling cumulative biological phenomena with Suppes-Bayes Causal Networks

Modeling cumulative biological phenomena with Suppes-Bayes Causal Networks

Daniele Ramazzotti Daniele Ramazzotti Department of Pathology, Stanford University, Stanford, CA 94305, USA 22email: daniele.ramazzotti@stanford.eduAlex Graudenzi Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy Giulio Caravagna School of Informatics, University of Edinburgh, Edinburgh, UK Marco Antoniotti Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy    Alex Graudenzi Daniele Ramazzotti Department of Pathology, Stanford University, Stanford, CA 94305, USA 22email: daniele.ramazzotti@stanford.eduAlex Graudenzi Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy Giulio Caravagna School of Informatics, University of Edinburgh, Edinburgh, UK Marco Antoniotti Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy    Giulio Caravagna Daniele Ramazzotti Department of Pathology, Stanford University, Stanford, CA 94305, USA 22email: daniele.ramazzotti@stanford.eduAlex Graudenzi Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy Giulio Caravagna School of Informatics, University of Edinburgh, Edinburgh, UK Marco Antoniotti Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy    Marco Antoniotti Daniele Ramazzotti Department of Pathology, Stanford University, Stanford, CA 94305, USA 22email: daniele.ramazzotti@stanford.eduAlex Graudenzi Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy Giulio Caravagna School of Informatics, University of Edinburgh, Edinburgh, UK Marco Antoniotti Department of Informatics, Systems and Communication, University of Milan-Bicocca, Milan, Italy
Received: date / Accepted: date
Abstract

Several diseases related to cell proliferation are characterized by the accumulation of somatic DNA changes, with respect to wildtype conditions. Cancer and HIV are two common examples of such diseases, where the mutational load in the cancerous/viral population increases over time. In these cases, selective pressures are often observed along with competition, co-operation and parasitism among distinct cellular clones. Recently, we presented a mathematical framework to model these phenomena, based on a combination of Bayesian inference and Suppes’ theory of probabilistic causation, depicted in graphical structures dubbed Suppes-Bayes Causal Networks (SBCNs). SBCNs are generative probabilistic graphical models that recapitulate the potential ordering of accumulation of such DNA changes during the progression of the disease. Such models can be inferred from data by exploiting likelihood-based model-selection strategies with regularization. In this paper we discuss the theoretical foundations of our approach and we investigate in depth the influence on the model-selection task of: the poset based on Suppes’ theory and different regularization strategies. Furthermore, we provide an example of application of our framework to HIV genetic data highlighting the valuable insights provided by the inferred SBCN.

Keywords:
Cumulative Phenomena Bayesian Graphical Models Probabilistic Causality

1 Introduction

A number of diseases are characterized by the accumulation of genomic lesions in the DNA of a population of cells. Such lesions are often classified as mutations, if they involve one or few nucleotides, or chromosomal alterations, if they involve wider regions of a chromosome. The effect of these lesions, occurring randomly and inherited through cell divisions (i.e., they are somatic), is that of inducing a phenotypic change in the cells. If the change is advantageous, then the clonal population might enjoy a fitness advantage over competing clones. In some cases, a natural selection process will tend to select the clones with more advantageous and inheritable traits. This particular picture can be framed in terms of Darwinian evolution as a scenario of survival of the fittest where, however, the prevalence of multiple heterogenous populations is often observed burrell2013causes.

Cancer and HIV are two diseases where the mutational111From now on, we will use the term mutation to refer to the types of genomic lesions mentioned above. load in the cancerours/viral population of cells increases over time, and drives phenotypic switches and disease progression. In this paper we specifically focus on these desease, but many biological systems present similar characteristics, see weinreich2006darwinian; poelwijk2007empirical; lozovsky2009stepwise.

The emergence and development of cancer can be characterized as an evolutionary process involving a large population of cells, heterogeneous both in their genomes and in their epigenomes. In fact, genetic and epigenetic random alterations commonly occurring in any cell, can occasionally be beneficial to the neoplastic cells and confer to these clones a functional selective advantage. During clonal evolution, clones are generally selected for increased proliferation and survival, which may eventually allow the cancer clones to outgrow the competing cells and, in turn, may lead to invasion and metastasis nowell1976clonal; merlo2006cancer. By means of such a multi-step stochastic process, cancer cells acquire over time a set of biological capabilities, i.e., hallmarks hanahan2000hallmarks; hanahan2011hallmarks. However, not all the alterations are involved in this acquisition; as a matter of fact, in solid tumors, we observe an average of to genes displaying a somatic mutations vogelstein2013cancer. But only some of them are involved in the hallmark acquisition, i.e., drivers, while the remaining ones are present in the cancer clones without increasing their fitness, i.e., passengers vogelstein2013cancer.

The onset of AIDS is characterized by the collapse of the immune system after a prolonged asymptomatic period, but its progression’s mechanistic basis is still unknown. It was recently hypothesized that the elevated turnover of lymphocytes throughout the asymptomatic period results in the accumulation of deleterious mutations, which impairs immunological function, replicative ability and viability of lymphocytes galvani2005role; seifert2015framework. The failure of the modern combination therapies (i.e., highly active antiretroviral therapy) of the disease is mostly due to the capability of the virus to escape from drug pressure by developing drug resistance. This mechanism is determined by HIV’s high rates of replication and mutation. In fact, under fixed drug pressure, these mutations are virtually non-reversible because they confer a strong selective advantage to viral populations perrin1998hiv; vandamme1999managing.

In the last decades huge technological advancements led to the development of Next Generation Sequencing (NGS) techniques. These allow, in different forms and with different technological characteristics, to read out genomes from single cells or bulk navin2014cancer; wang2014clonal; gerlinger2012intratumor; gerlinger2014genomic. Thus, we can use these technologies to quantify the presence of mutations in a sample. With this data at hand, we can therefore investigate the problem of inferring a Progression Model (PM) that recapitulates the ordering of accumulation of mutations during disease origination and development caravagna2016picnic. This problem allows different formulation according to the type of diseases that we are considering, the type of NGS data that we are processing and other factors. We point the reader to beerenwinkel2015cancer; caravagna2016picnic for a review on progression models.

This work is focused on a particular class of mathematical models that are becoming successful to represent such mutational ordering. These are called Suppes-Bayes Causal Networks222The first use of these networks appears in ramazzotti2015capri, and it’s earliest formal definition in bonchi2015exposing. (SBCN, bonchi2015exposing), and derive from a more general class of models, Bayesian Networks (BN, koller2009probabilistic), that has been successfully exploited to model cancer and HIV progressions desper1999inferring; beerenwinkel2007conjunctive; gerstung2009quantifying. SBCNs are probabilistic graphical models that are derived within a statistical framework based on Patrick Suppes’ theory of probabilistic causation suppes1970probabilistic. Thus, the main difference between standard Bayesian Networks and SBCNs is the encoding in the model of a set of causal axioms that describe the accumulation process. Both SBCNs and BNs are generative probabilistic models that induces a distribution of observing a particular mutational signature in a sample. But, the distribution induced by a SBCN is also consistent with the causal axioms and, in general, is different from the distribution induced by a standard BN ramazzotti2015capri.

The motivation for adopting a causal framework on top of standard BNs is that, in the particular case of cumulative biological phenomena, SBCNs allow better inferential algorithms and data-analysis pipelines to be developed loohuis2014inferring; ramazzotti2015capri; caravagna2016picnic. Extensive studies in the inference of cancer progression have indeed shown that model-selection strategies to extract SBCNs from NGS data achieve better performance than algorithms that infer BNs. In fact, SBCN’s inferential algorithms have higher rate of detection of true positive ordering relations, and higher rate of filtering out false positive ones. In general, these algorithms also show better scalability, resistance to noise in the data, and ability to work with datasets with few samples (see, e.g., loohuis2014inferring; ramazzotti2015capri).

In this paper we give a formal definition of SBCNs, we assess their relevance in modeling cumulative phenomena and investigate the influence of Suppes’ poset, and distinct maximum likelihood regularization strategies for model-selection. We do this by carrying out extensive synthetic tests in operational settings that are representative of different possible types of progressions, and data harbouring signals from heterogenous populations.

2 Suppes-Bayes Causal Networks

In suppes1970probabilistic, Suppes introduced the notion of prima facie causation. A prima facie relation between a cause and its effect is verified when the following two conditions are true: temporal priority (TP), i.e., any cause happens before its effect and probability raising (PR), i.e., the presence of the cause raises the probability of observing its effect.

Definition 1 (Probabilistic causation, suppes1970probabilistic)

For any two events and , occurring respectively at times and , under the mild assumptions that , the event is called a prima facie cause of if it occurs before and raises the probability of , i.e.,

TP: (1)
PR: (2)

While the notion of prima facie causation has known limitations in the context of the general theories of causality hitchcock2012probabilistic, this formulation seems to intuitively characterizes the dynamics of phenomena driven by the monotonic accumulation of events. In these cases, in fact, a temporal order among the events is implied and, furthermore, the occurrence of an early event positively correlates to the subsequent occurrence of a later one. Thus, this approach seems appropriate to capture the notion of selective advantage emerging from somatic mutations that accumulate during, e.g., cancer or HIV progression.

Let us now consider a graphical representation of the aforementioned dynamics in terms of a Bayesian graphical model.

Definition 2 (Bayesian network, koller2009probabilistic)

The pair is a Bayesian Network (BN), where is a directed acyclic graph (DAG) of nodes and arcs, and is a distribution induced over the nodes by the graph. Let be random variables and the edges/arcs encode the conditional dependencies among the variables. Define, for any , the parent set , then defines the joint probability distribution induced by the BN as follow:

(3)

All in all, a BN is a statistical model which succinctly represents the conditional dependencies among a set of random variables through a DAG. More precisely, a BN represents a factorization of the joint distribution in terms of marginal (when ) and conditional probabilities .

We now consider a common situation when we deal with data (i.e., observations) obtained at one (or a few) points in time, rather than through a timeline. In this case, we are resticted to work with cross-sectional data, where no explicit information of time is provided. Therefore, we can model the dynamics of cumulative phenomena by means of a specific set of the general BNs where the nodes represent the accumulating events as Bernoulli random variables taking values in {0, 1} based on their occurrence: the value of the variable is if the event is observed and otherwise. We then define a dataset of cross-sectional samples over Bernoulli random variables as follow.

(4)

In order to extend BNs to account for Suppes’ theory of probabilistic causation we need to estimate for any variable its timing . Because we are dealing with cumulative phenomena and, in the most general case, data that do not harbour any evidente temporal information333In many cases, the data that we can access are cross-sectional, meaning that the samples are collected at independent and unknown time-points. For this reason, we have to resort on the simplest possible approach to estimate timings. However, in the case we were provided with explicit observations of time, the temporal priority would be directly and, yet, more efficiently assessable., we can use the marginal probability as a proxy for (see also the commentary at the end of this section). In cancer and HIV, for instance, this makes sense since mutations are inherited through cells divisions, and thus will fixate in the clonal populations during disease progression, i.e., they are persistent.

Definition 3 (Suppes-Bayes Causal Network, bonchi2015exposing)

A BN is a SBCN if and only if, for any edge , Suppes’ conditions (Definition 1) hold, that is

(5)

It should be noted that SBCNs and BNs have the same likelihood function. Thus, SBCNs do not embed any constraint of the cumulative process in the likelihood computation, while approaches based on cumulative BNs do gerstung2009quantifying. Instead, the structure of the model, , is consistent with the causal model à-la-Suppes and, of course, this in turn reflects in the induced distribution. Even though this difference seem subtle, this is arguably the most interesting advantage of SBCNs over ad-hoc BNs for cumulative phenomena.

Model-selection to infer a network from data.

The structure of a BN (or of a SBCN) can be inferred from a data matrix , as well as the parameters of the conditional distributions that define . The model-selection task is that of inferring such information from data; in general, we expect different models (i.e., edges) if we infer a SBCN or a BN, as SBCNs encode Suppes’ additional constraints.

The general structural learning, i.e., the model-selection problem, for BNs is NP-HARD koller2009probabilistic, hence one needs to resort on approximate strategies. For each BN a log-likelihood function can be used to search in the space of structures (i.e., the set of edges ), together with a regularization function that penalizes overly complicated models. The network’s structure is then inferred by solving the following optimization problem

(6)

Moreover, the parameters of the conditional distributions can be computed by maximum-likelihood estimation for the set of edges ; the overall solution is locally optimal koller2009probabilistic.

Model-selection for SBCNs works in this very same way, but constrains the search for valid solutions (see, e.g., ramazzotti2015capri). In particular, it scans only the subset of edges that are consistent with Definition 1 – while a BN search will look for the full space. To filter pairs of edges, Suppes’ conditions can be estimated from the data with solutions based, for instance, on bootstrap estimates ramazzotti2015capri. The resulting model will satisfy, by construction, the conditions of probabilistic causation. It has been shown that if the underlying phenomenon that produced our data is characterised by an accumulation, then the inference of a SBCN, rather than a BN, leads to much better precision and recall loohuis2014inferring; ramazzotti2015capri.

We conclude this Section by discussing in details the characteristics of the SBCNs and, in particular, to which extent they are capable of modeling cumulative phenomena.

Temporal priority.

Suppes’ first constraint (“event is temporally preceding event ”, Definition 1) assumes an underlying temporal (partial) order among the events/ variables of the SBCN, that we need to compute.

Cross-sectional data, unfortunately, are not provided with an explicit measure of time and hence needs to be estimated from data 444We notice that in the case we were provided with explicit observations of time, would be directly and, yet, more efficiently assessable.. The cumulative nature of the phenomenon that we want to model leads to a simple estimation of : the temporal priority TP is assessed in terms of marginal frequencies ramazzotti2015capri,

(7)

Thus, more frequent events, i.e., , are assumed to occur earlier, which is sound when we assume the accumulating events to be irreversible.

TP is combined with probability raising to complete Suppes’ conditions for prima facie (see below). Its contribution is fundamental for model-selection, as we now elaborate.

First of all, recall that the model-selection problem for BNs is in general NP-HARD koller2009probabilistic, and that, as a result of the assessment of Suppes’ conditions (TP and PR), we constrain our search space to the networks with a given order. Because of time irreversibility, marginal distributions induce a total ordering on the , i.e., reflexing . Learning BNs given a fixed order – even partial koller2009probabilistic – of the variables bounds the cardinality of the parent set as follows

(8)

and, in general, it make inference easier than the general case koller2009probabilistic. Thus, by constraining Suppes’ conditions via ’s total ordering, we drop down the model-selection complexity. It should be noted that, after model-selection, the ordering among the variables that we practically have in the selected arcs set is in general partial; in the BN literature this is sometimes called poset.

Probability raising.

Besides TP, as a second constraint we further require that the arcs are consistent to the condition of PR: this leads to another relation . Probability raising is equivalent to constraining for positive statistical dependence loohuis2014inferring

(9)

thus we model all and only the positive dependant relations. Definition 1 is thus obtained by selecting those probability raising relations that are consistent with TP,

(10)

as the core of Suppes’ characterization of causation is relevance suppes1970probabilistic.

If reduces the search space of the possible valid structures for the network by setting a specific total order to the nodes, instead reduces the search space of the possible valid parameters of the network by requiring that the related conditional probability tables, i.e., , account only for positive statistical dependencies. It should be noted that these constraints affect the structure and the parameters of the model, but the likelihood-function is the same for BNs and SBCNs.

Network simplification, regularization and spurious causality.

Suppes’ criteria are known to be necessary but not sufficient to evaluate general causal claims suppes1970probabilistic. Even if we restrict to causal cumulative phenomena, the expressivity of probabilistic causality needs to be taken into account.

When dealing with small sample sized datasets (i.e., small ), many pairs of variables that satisfy Suppes’ condition may be spurious causes, i.e., false positive555An edge is spurious when it satisfies Definition 1, but it is not actually the true model edge. For instance, for a linear model , transitive edge is spurious. Small induces further spurious associations in the data, not necessarily related to particular topological structures.. False negatives should be few, and mostly due to noise in the data. Thus, it follows

  • we expect all the “statistically relevant” relations to be also prima facie ramazzotti2015capri;

  • we need to filter out spurious causality instances666A detailed account of these topics, the particular types of spurious structures and their interpretation for different types of models are available in loohuis2014inferring; ramazzotti2015capri; ramazzotti2016model., as we know that prima facie overfits .

A model-selection strategy which exploits a regularization schema seems thus the best approach to the task. Practically, this strategy will select simpler (i.e., sparse) models according to a penalized likelihood fit criterion – for this reason, it will filter out edges proportionally to how much the regularization is stringent. Also, it will rank spurious association according to a criterion that is consistent with Suppes’ intuition of causality, as likelihood relates to statistical (in)dependence. Alternatives based on likelihood itself, i.e., without regularization, do not seem viable to minimize the effect of likelihood’s overfit, that happens unless koller2009probabilistic. In fact, one must recall that due to statistical noise and sample size, exact statistical (in)dependence between pair of variables is never (or very unlikely) observed.

2.1 Modeling heterogeneous populations

Complex biological processes, e.g., proliferation, nutrition, apoptosis, are orchestrated by multiple co-operative networks of proteins and molecules. Therefore, different “mutants” can evade such control mechanisms in different ways. Mutations happen as a random process that is unrelated to the relative fitness advantage that they confer to a cell. As such, different cells will deviate from wildtype by exhibiting different mutational signatures during disease progression. This has an implication for many cumulative disease that arise from populations that are heterogeneous, both at the level of the single patient (intra-patient heterogeneity) or in the population of patients (inter-patient heterogeneity). Heterogeneity introduces significant challenges in designing effective treatment strategies, and major efforts are ongoing at deciphering its extent for many diseases such as cancer and HIVbeerenwinkel2007conjunctive; ramazzotti2015capri; caravagna2016picnic.

We now introduce a class of mathematical models that are suitable at modeling heterogenous progressions. These models are derived by augmenting BNs with logical formulas, and are called Monotonic Progression Networks (MPNs) farahani2013learning; korsunsky2014inference. MPNs represent the progression of events that accumulate monotonically777The events accumulate over time and when later events occur earlier events are observed as well. over time, where the conditions for any event to happen is described by a probabilistic version of the canonical boolean operators, i.e., conjunction (), inclusive disjunction (), and exclusive disjunction ().

Following farahani2013learning; korsunsky2014inference, we define one type of MPNs for each boolean operator: the conjunctive (CMPN), the disjunctive semi-monotonic (DMPN), and the exclusive disjunction (XMPN). The operator associated with each network type refers to the logical relation among the parents that eventually lead to the common effect to occur.

Definition 4 (Monotonic Progression Networks, farahani2013learning; korsunsky2014inference)

MPNs are BNs that, for and , satisfy the conditions shown in Table 1 for each .

(11)
(12)
(13)
Table 1: Definitions for CMPN, DMPN, and XMPN.

Here, represents the conditional probability of any “effect” to follow its preceding “cause” and models the probability of any noisy observation – that is the observation of a sample where the effects are observed without their causes. Note that the above inequalities define, for each type of MPN, specific constraints to the induced distributions. These are sometimes termed, according to the probabilistic logical relations, noisy-AND, noisy-OR and noisy-XOR networks pearl2014probabilistic; korsunsky2014inference.

Model-selection with heterogeneous populations.

When dealing with heterogeneous populations, the task of model selection, and, more in general, any statistical analysis, is non-trivial. One of the main reason for this state of affairs is the emergence of statistical paradoxes such as Simpson’s paradox yule1903notes; simpson1951interpretation. This phonomenon refers to the fact that sometimes, associations among dichotomous variables, which are similar within subgroups of a population, e.g., females and males, change their statistical trend if the individuals of the subgroups are pooled together. Let us know recall a famous example to this regard. The admissions of the University of Berkeley for the fall of showed that men applying were much more likely than women to be admitted with a difference that was unlikely to be due to chance. But, when looking at the individual departments separately, it emerged that out of were indead biased in favor of women, while only presented a slighly bias against them. The reason for this inconsistency was due to the fact that women tended to apply to competitive departments which had low rates of admissions while men tended to apply to less-competitive departments with high rates of admissions, leading to an apparent bias toward them in the overall population bickel1975sex.

Similar situations may arise in cancer when different populations of cancer samples are mixed. As an example, let us consider an hypothetical progression leading to the alteration of gene . Let us now assume that the alterations of this gene may be due to the previous alterations of either gene or gene exclusively. If this was the case, then we would expect a significant pattern of selective advantage from any of its causes to if we were able to stratify the patients accordingly to either alteration or , but we may lose these associations when looking at all the patients together.

In ramazzotti2015capri, the notion of progression pattern is introduced to describe this situation, defined as a boolean relation among all the genes, members of the parent set of any node as the ones defined by MPNs. To this extent, the authors extend Suppes’ definition of prima facie causality in order to account for such patterns rather than for relations among atomic events as for Definition 1. Also, they claim that general MPNs can be learned in polynomial time provided that the dataset given as input is lifted ramazzotti2015capri with a Bernoulli variable per causal relation representing the logical formula involving any parent set.

Following ramazzotti2015capri; ramazzotti2016model, we now consider any formula in conjunctive normal form (CNF)

where each is a disjunctive clause over a set of literals and each literal represents an event (a Boolean variable) or its negation. By following analogous arguments as the ones used before, we can extend Definition 1 as follows.

Definition 5 (CNF probabilistic causation, ramazzotti2015capri; ramazzotti2016model)

For any CNF formula and , occurring respectively at times and , under the mild assumptions that , is a prima facie cause of if

(14)

Given these premises, we can now define the Extended Suppes-Bayes Causal Networks, an extension of SBCNs which allows to model heterogeneity as defined probabilistically by MPNs.

Definition 6 (Extended Suppes-Bayes Causal Network)

A BN is an Extended SBCN if and only if, for any edge , Suppes’ generalized conditions (Definition 5) hold, that is

(15)

3 Evaluation on simulated data

We now evaluate the performance of the inference of Suppes-Bayes Causal Network on simulated data, with specific attention on the impact of the constraints based on Suppes’ probabilistic causation on the overall performance. All the simulations are performed with the following settings.

We consider different topological structures: the first two where any node has at the most one predecessor, i.e., trees, forests, and the others where we set a limit of predecessors and, hence, we consider directed acyclic graphs with a single source and conjunctive parents, directed acyclic graphs with multiple sources and conjunctive parents, directed acyclic graphs with a single source and disjunctive parents, directed acyclic graphs with multiple sources and disjunctive parents. For each of these configurations, we generate random structures.

Moreover, we consider different sample sizes (, , and samples) and noise levels (i.e., probability of a random entry for the observation of any node in a sample) from to with step . Furthermore, we repeat the above settings for networks of and nodes. Any configuration is then sampled times independently, for a total of more than million distinct simulated datasets.

Finally, the inference of the structure of the SBCN is performed using the algorithm proposed in ramazzotti2015capri and the performance is assessed in terms of: , and with and being the true and false positive (we define as positive any arc that is present in the network) and and being the true and false negative (we define negative any arc that is not present in the network). All these measures are values in with results close to indicators of good performance.

In Figures 1, 2 and 3 we show the performance of the inference on simulated datasets of samples and networks of nodes in terms of accurancy, sensitivity and specificity for different settings which we discuss in details in the next Paragraphs.

Suppes’ prima facie conditions are necessary but not sufficient

We first discuss the performance by applying only the prima facie criteria and we evaluate the obtained prima facie network in terms of accurancy, sensitivity and specificity on simulated datasets of samples and networks of nodes (see Figures 1, 2 and 3). As expected, the sensitivity is much higher than the specificity implying the significant impact of false positives rather than false negatives for the networks of the prima facie arcs. This result is indeed expected being Suppes’ criteria mostly capable of removing some of the arcs which do not represent valid causal relations rather than asses the exact set of valid arcs. Interestingly, the false negatives are still limited even when we consider DMPN, i.e., when we do not have guarantees for the algoritm of ramazzotti2015capri to converge. The same simulations with different sample sizes (, and samples) and on networks of nodes present a similar trend (results not shown here).

Figure 1: Performance of the inference on simulated datasets of samples and networks of nodes in terms of accurancy for the considered topological structures. The axis refers to the performance while the axis represent the different noise levels.
Figure 2: Performance of the inference on simulated datasets of samples and networks of nodes in terms of sensitivity for the considered topological structures. The axis refers to the performance while the axis represent the different noise levels.
Figure 3: Performance of the inference on simulated datasets of samples and networks of nodes in terms of specificity for the considered topological structures. The axis refers to the performance while the axis represent the different noise levels.

The likelihood score overfits the data

In Figures 1, 2 and 3 we also show the performance of the inference by likelihood fit (without any regularizator) on the prima facie network in terms of accurancy, sensitivity and specificity on simulated datasets of samples and networks of nodes. Once again, in general, sensitivity is much higher than specificity implying also in this case a significant impact of false positives rather than false negatives for the inferred networks. These results make explicit the need for a regularization heuristic when dealing with real (not infinite) sample sized datasets as discussed in the next paragraph. Another interesting consideration comes from the observation that the prima facie networks and the networks inferred via likelihood fit without regularization seem to converge to the same performance as the noise level increases. This is due to the fact that, in general, the prima facie constraints are very conservative in the sense that false positives are admitted as long as false negatives are limited. When the noise level increases, the positive dependencies among nodes are generally reduced and, hence, less arcs pass the prima facie cut for positive dependency. Also in this case, the same simulations with different sample sizes (, and samples) and on networks of nodes present a similar trend (results not shown here).

Model-selection with different regularization strategies

We now investigate the role of different regularizations on the performance. In particular, we consider two commonly used regularizations: the Bayesian information criterion (BIC) schwarz1978estimating and the Akaike information criterion (AIC) akaike1998information.

Although BIC and AIC are both scores based on maximum likelihood estimation and a penalization term to reduce overfitting, yet with distinct approaches, they produce significantly different behaviors. More specifically, BIC assumes the existence of one true statistical model which is generating the data while AIC aims at finding the best approximating model to the unknown data generating process. As such, BIC may likely underfit, whereas, conversely, AIC might overfit888Thus, BIC tends to make a trade-off between the likelihood and model complexity with the aim of inferring the statistical model which generates the data. This makes it useful when the purpose is to detect the best model describing the data. Instead, asymptotically, minimizing AIC is equivalent to minimizing the cross validation value stone1977asymptotic. It is this property that makes the AIC score useful in model selection when the purpose is prediction. Overall, the choise of the regularizator tunes the level of sparsity of the retrieved SBCN and, yet, the confidence of the inferred arcs..

The performance on simulated datasets are shown in Figure 1, 2, 3. In general, the performance is improved in all the settings with both regularizators, as they succeed in shrinking toward sparse networks.

Furthermore, we observe that the performance obtained by SBCNs is still good even when we consider simulated data generated by DMPN. Although in this case we do not have any guarantee of convergence, in practice the algorithm seems efficient in approximating the generative model. In conclusion, without any further input, SBCNs can model CMPNs and, yet, depict the more significant arcs of DMPNs. To infer XMPN, the dataset needs to be lifted ramazzotti2015capri.

The same simulations with different sample sizes (, and samples) and on networks of nodes present a similar trend (results not shown here).

4 Application to HIV genetic data

We now present an example of application of our framework on HIV genomic data. In particular we study drug resistance in patients under antiretroviral therapy and we select a set of amino acid alterations in the HIV genome to be depicted in the resulting graphical model, namely , , , , , , , where, as an example, the genomic event describes a mutation from lysine () to arginine () at position 20 of the protease.

In this study, we consider datasets from the Stanford HIV Drug Resistance Database (see, rhee2003human) for protease inhibitors, ritonavir () and indinavir (). The first dataset consists of samples (see Figure 4) and the second of samples (see Figure 4).

Figure 4: Mutations detected in the genome for HIV patients under ritonavir (top), and under indinavir (bottom). Each black rectangle denotes the presence of a mutation in the gene annotated to the right of the plot; percentages correspond to marginal probabilities.

We then infer a model on these datasets by both Bayesian Network and Suppes-Bayes Causal Network. We show the results in Figures 5 where each node represent a mutation and the scores on the arcs measure the confidence in the found relation by non-parametric bootstrap.

Figure 5: HIV progression of patients under ritonavir or indinavir, (Figure 4) described as a Bayesian Network or as a Suppes-Bayes Causal Network. Edges are annotated with non parametric bootstrap scores.

In this case, it is interesting to observe that the set of dependency relations (i.e., any pair of nodes connected by an arc, without considering its direction) depicted both by SBCNs and BNs are very similar, with the main different being in the direction of some connection. This difference is expected and can be attributed to the constrain of temporal priority adopted in the SBCNs. Furthermore, we also observe that most of the found relations in the SBCN are more confidence (i.e., higher bootstrap score) than the one depicted in the related BN, leading us to observe a higher statistical confidence in the models inferred by SBCNs.

5 Conclusions

In this work we investigated the properties of a constrained version of Bayesian network, named SBCN, which is particularly sound in modeling the dynamics of system driven by the monotonic accumulation of events, thanks to encoded poset based on Suppes’ theory of probabilistic causation. In particular, we showed how SBCNs can, in general, describe different types of MPN, which makes them capable of characterizing a broad range of cumulative phenomena not limited to cancer evolution and HIV drug resistance.

Besides, we investigated the influence of Suppes’ poset on the inference performance with cross-sectional synthetic datasets. In particular, we showed that Suppes’ constraints are effective in defining a partially order set accounting for accumulating events, with very few false negatives, yet many false positives. To overcome this limitation, we explored the role of two maximum likelihood regularization parameters, i.e., BIC and AIC, the former being more suitable to test previously conjectured hypotheses and the latter to predict novel hypotheses.

Finally, we showed on a dataset of HIV genomic data how SBCN can be effectively adopted to model cumulative phenomena, with results presenting a higher statistical significance compared to standard BNs.

Acknowledgements.
This work has been partially supported by grants from the SysBioNet project, a MIUR initiative for the Italian Roadmap of European Strategy Forum on Research Infrastructures (ESFRI).

References

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This is a comment super asjknd jkasnjk adsnkj
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