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## The Availability of Evidence

The construction of a Q-DAG requires the identification of query and evidence variables. This may give an incorrect impression that we must know up front which variables are observed and which are not. This could be problematic in (1) applications where one may lose a sensor reading, thus changing the status of a variable from being observed to being unobserved; and (2) applications where some variable may be expensive to observe, leading to an on-line decision on whether to observe it or not (using some value-of-information computation). Figure: A belief network and its corresponding Q-DAG in which variable is declared to be both query and evidence.

Both of these situations can be dealt with in a Q-DAG framework. First, as we mentioned earlier, Q-DAGs allow us to handle missing evidence through the use of the notation which denotes an unknown value of a variable. Therefore, Q-DAGs can handle missing sensor readings. Second, a variable can be declared to be both query and evidence. This means that we can incorporate evidence about this variable when it is available, and also compute the probability distribution of the variable in case evidence is not available. Figure [*] depicts a Q-DAG in which variable is declared to be a query variable, while variable is declared to be both an evidence and a query variable (both variables have and as their values). In this case, we have two ESNs for variable and also two query nodes (see Figure [*]). This Q-DAG can be used in two ways:

1. To compute the probability distributions of variables and when no evidence is available about . Under this situation, the values of and are set to 1, and we have 2. To compute the probability of variable when evidence is available about . For example, suppose that we observe to be . The value of will then be set to 0 and the value of will be set to 1, and we have Utility_Of_Observing

The ability to declare a variable as both an evidence and a query variable seems to be essential in applications where (1) a decision may need to be made on whether to collect evidence about some variable ; and (2) making the decision requires knowing the probability distribution of variable . For example, suppose that we are using the following formula [PearlPearl1988, Page 313,] to compute the utility of observing variable : where is the utility for the decision maker of finding that variable has value . Suppose that and . We can use the Q-DAG to compute the probability distribution of and use it to evaluate : which leads us to observe variable . Observing , we find that its value is . We can then accommodate this evidence into the Q-DAG and continue with our analysis.

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