The algorithm works by implicitly constructing the joint probability distribution
induced by the Bayesian network, and then summing out attributes, therefore constructing a marginal distribution over the variables of observed.
where P([h.sub.k] | [h.sub.k+1]) is the conditional probability distribution of [h.sub.k] for the given [h.sub.k+1] state; P([h.sub.l-1], [h.sub.l]) is the joint probability distribution
of [h.sub.l-1] and [h.sub.l].
and, in general, the joint probability distribution
for any Bayesian Network, given nodes X = [X.sub.1], ..., [X.sub.n], is
To derive a joint probability distribution
of two or more variables that are dependent on each other, the most viable method is the copula method.
In that case, a complete knowledge of the joint probability distribution
of the observation conditioned on the target state is required in order to obtain the optimal results as shown in the following equation :
Then, the joint probability distribution
is given by applying probability multiplication and additive rules:
A Bayesian network represents a Joint Probability Distribution
(JPD) between its variables X1, ..., [X.sub.n] by means of the chain rule for BNs (2):
at each iteration, the value along a randomly selected dimension is updated according to the conditional distribution." Bayes' posterior joint probability distribution
is defined as the product of conditional distributions, and Gibbs sampling is said to work well in this case.
Representing the joint probability distribution
as a directed graphical network simplifies the computation of the joint distribution by expressing it as a product of the conditional probability distributions at every node using Bayes' rule.
One important property of BNs is their ability to represent the joint probability distribution
P([A.sub.1], ..., [A.sub.n]) for all the variables [A.sub.1], ..., [A.sub.n] in a compact form.
Therefore, for drought risk assessment, it is useful to construct a joint probability distribution
from these two variates and perform frequency analysis.
The joint probability distribution
in G describes the given knowledge base, and Bayesian network expresses the knowledge model of a problem domain.