Final answer:
To specify a Bayesian network, one needs to determine the conditional probability distribution for each node based on its parents. The total number of probabilities required is the sum of the product of the number of values for each node by the number of configurations of its parents, minus 1 for each set of parents because of the normalization constraint.
Step-by-step explanation:
The question asks about the specifics of defining a Bayesian network and how many probabilities one needs to specify for it. A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). For each node (representing a variable) in a Bayesian network, one must specify the conditional probability distribution (CPD) that quantifies the effect of the parents on the node. The total number of probabilities needed explicitly depends on the number of parent nodes each node has. The full specification requires enough probabilities to cover every combination of parent states for every node, minus 1 for each CPD, since the last probability can be derived given that probabilities must sum to 1.
For example, if a node has no parents (it is a root node), only 1 probability is needed (the marginal probability of the node), and it can have multiple values if the variable is not binary. If a node has one binary parent, you need to specify 2 probabilities: one for each state of the parent node. If it has two binary parents, you'd need to specify 4 probabilities, and so on. The formula to calculate the totality of needed probabilities for the whole network is:
Total probabilities = ∑ (|Values(parents(i))| - 1) * |Values(Xi)|, where 'Xi' is the i-th variable and 'parents(i)' is the set of parents for that variable.
Bayesian networks incorporate prior knowledge and update this knowledge as new data becomes available, applying Bayes' theorem. They are particularly useful because they can model uncertainty and make inferences even with incomplete data, and they allow for the integration of observed data with expert knowledge or previous findings, characterized as prior probabilities.