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Table 1 Bayesian networks from [1]

From: A pedagogical walkthrough of computational modeling and simulation of Wnt signaling pathway using static causal models in MATLAB

Bayesian networks. In reverse engineering methods for control networks [10] there exist many methods that help in the construction of the networks from the datasets as well as give the ability to infer causal relations between components of the system. A widely known architecture among these methods is the Bayesian network (BN). These networks can be used for causal reasoning or diagnostic reasoning or both. It has been shown through reasoning and examples in [11] that the probabilistic inference mechanism applied via Bayesian networks are analogous to the structural equation modeling in path analysis problems. Initial works on BNs in [12, 13] suggest that the networks only need a relatively small amount of marginal probabilities for nodes that have no incoming arcs and a set of conditional probabilities for each node having one or more incoming arcs. The nodes form the driving components of a network and the arcs define the interactive influences that drive a particular process. Under these assumptions of influences the joint probability distribution of the whole network or a part of it can be obtained via a special factorization that uses the concept of direct influence and through dependence rules that define d-connectivity/separability as mentioned in [14] and [15]. This is illustrated through a simple example in [11].
The Bayesian networks work by estimating the posterior probability of the model given the dataset. This estimation is usually referred to as the Bayesian score of the model conditioned on the dataset. Mathematically, let \(\mathcal {S}\) represent the model given the data \(\mathcal {D}\) and ξ is the background knowledge. Then according to the Bayes Theorem [16]:
Thus the Bayesian score is computed by evaluating the posterior distribution \(\mathcal {P}(\mathcal {S}|\mathcal {D},\xi)\) which is proportional to the prior distribution of the model \(\mathcal {P}(\mathcal {S}|\xi)\) and the likelihood of the data given the model \(\mathcal {P}(\mathcal {D}|\mathcal {S},\xi)\). It must be noted that the background knowledge is assumed to be independent of the data. Next, since the evaluation of probabilities require multiplications a simpler way is to take logarithmic scores which boils down to addition. Thus, the estimation takes the form
Finally, the likelihood of the function can be evaluated by averaging over all possible local conditional distributions parameterized by θ i s that depict the conditioning of parents. This is equated via
Work on biological systems that make use of Bayesian networks can also be found in [1721]. Bayesian networks are good in generating network structures and testing a targeted hypothesis which confine the experimenter to derive causal inferences [22]. But a major disadvantage of the Bayesian networks is that they rely heavily on the conditional probability distributions which require good sampling of datasets and are computationally intensive. On the other hand, these networks are quite robust to the existence of the unobserved variables and accommodate noisy datasets. They also have the ability to combine heterogeneous datasets that incorporate different modalities. In this work, simple static Bayesian network models have been developed with an aim to show how (a) incorporation of heterogeneous data can be done to increase prediction accuracy of test samples, (b) prior biological knowledge can be embedded to model biological phenomena behind the Wnt pathway in colorectal cancer, (c) to test the hypothesis regarding direct correspondence of active state of β-catenin-based transcription complex and the state of the test sample via segregation of nodes in the directed acyclic graphs of the proposed models, and (d) inferences can be made regarding the hidden biological relationships between a particular gene and the β-catenin transcription complex. This work uses Matlab-implemented BN toolbox from [4].