Fig. 1

Illustrative Bayesian network describing causal relationships between five variables, with their associated conditional probability tables. The values of each variable have been discretized into low (l), medium (m) and high (h) bins. The notation P(B = l|A) refers to the probability of B being in the low value bin, conditional upon the value of A (which itself can be l, m or h). Note that with one parent only (i.e. the case for nodes B and C), both row and column probabilities sum to 1, whereas with multiple inputs (i.e. in the case of D), only the rows sum to 1. The nodes are colored green (high), white (medium) and red (low) to illustrate in silico perturbation where A falls into a high concentration bin (probability 0.9) and E falls into a low concentration bin (probability 0.8). The implications of this are demonstrated through the conditional probability tables associated with each downstream node wherein having D in a given concentration state is dictated by the particular combination of states of its parents B and C as dictated by their corresponding overall (marginal) probabilities (the entries as captured in the conditional probability tables do not change on intervention), as summarized in the histogram attached to each node. Note that the overall marginal probability of D being in a particular state (which changes under intervention) is the sum as summarized in the histogram attached to each node. Readers more familiar with Bayesian networks will note that the sub-network structures (B → A, A → C, B → D, C → D) and (C → A, A → B, B → D, C → D) are both Markov equivalent with the present one (A → B, A → C, B → D, C → D). When we condition on A, the other nodes B and C become independent no matter whether A is a tail-to-head or tail-to-tail intermediate node (this scenario differs from that seen with head-to-head node D). In other words, these three sub-networks specify the same independent assumptions belonging to the same equivalence class and the true causal network can possibly be any one of the sub-network solutions. However, causal networks based on observation alone (i.e. without intervention, which is an important tool for inferring causality) can be still partially constructed. In the present study, prior knowledge seed network edge directionalities were assigned by a modified depth-first search algorithm which helped to choose the sub-network from an equivalence class containing more than one Markov equivalent member as suggested here (see later)