D-Separation in AI

UNCERTAIN EVIDENCE

• The evidence may be uncertain in some cases. 
• If the evidence is uncertain in expressions like P(A | ɛ) then we cannot get an appropriate probability about the query node. 
• It is required to know whether the propositions represented by evident nodes are true or false. 
• If we arrange the evident nodes in such a way that they have child nodes which we are certain about then the requirement is achieved. 
• In case of uncertainty about the arm does not move by using a node M' the evident can be provided where M' represents the arm sensor tells that are moved. Depending on the reading we can known the proposition is true or false. 
• So, by using Bayes network we calculate P(ᆨL, ᆨB,ᆨM') instead of P(ᆨL | ᆨB, ᆨM). 
• In Fig. 3.5.1 node B is uncertain as we do not know whether the battery is charged or not. 
• Node G is child node of B so, the reliability of the gauge is represented by P(G | B). Now we have to compute P(ᆨL | ᆨG, ᆨN') instead of P(ᆨL, ᆨB, ᆨM). 
• For large networks the methods like' brute-force may result in worst-case. 

D—SEPARATION 

• Conditional independencies are mostly implied in Bayes network than just involving the parents of a node. 
• In the Fig. 3.5.1 M is conditionally independent of G if B is given. 
• G is an effect and it can given some information of cause B which in turn influences M. 
• If B is given, as B directly influences M, G do not tell anything more about M. 
• That is, B D-separates G and M. 
• Consider a figure. 3.7.1 which shows conditional independence via blocking nodes. 

Fig. 3.7.1 Conditional Independence via Blocking Nodes 

• Let two nodes A_i and A_j are conditional independent given a set of nodes ε which is represented as I(A_i A_j | ε) and let a node A_b exists in the path from A_i to A_j which posses anyone of the properties, given below, 
• 1) The node be in the set of evident nodes and both the arcs of the path lead out of A_b, this can be seen as A_b1 in the Fig. 3.7.1 where both arcs lead out and it belongs to ε.
• 2) The node belongs to set of evident nodes and one are on the path leads into A_b and another out of it. This is represented as A_b2 in Fig. 3.7.1.
• 3) The node should be present in ε and its descendants should not belong to ε, and both the arcs should be into A_b. This is represented by A_b3 in the Fig. 3.7.1. 
• If any of the above conditions are satisfied for a path them it is said that A_b blocks the path. 
• It has to noted that paths are undirected paths. 
• It is said that ε d-separates A_i and A_j if all the paths are blocked and they are said to be conditionally independent given ε.
• D-separation concept can be as applied to sets.
• Two set of nodes are D-separated by ε if every path between all nodes in the two sets are blocked if ε is given.
• Two sets are conditionally independent if ε is given and D-separated by ε.