## Bayesian Belief Networks

To succinctly express the probabilistic relations among a set of variables, it may be useful to factor the joint distribution of all variables into a product of 'marginal' and 'conditional distributions'. This can be done graphically using a directed acyclic graph in which each random variable is a node, and arrows connecting the nodes represent direct conditional dependences. The placement of these arrows is usually based on knowledge of the causal relations among the variables, with the tail of the arrow being placed at the causal 'parent' and the head at the 'child'. The joint distribution, p(xi,..., xn), then factors into the product of terms of the form p(xi|parents of Xi) for i = 1,..., n

Any two nodes that are not in a parent/child relationship are conditionally independent given the values of their parents. The combination of the graphical model and the composite probabilistic relations is called a Bayesian belief network (BBN). Nodes in a BBN do not have to represent random variables (they can be parameters, latent variables, or concepts), and probabilities do not have to represent frequencies (they can be degrees of belief, weight of evidence, or subjective assessments) -this is what is 'Bayesian' about BBNs.

The conditional distributions in a BBN can have any form and can either be specified according to prior domain knowledge (including process-based submodels or expert opinon) or be learned automatically from data. Once the relations are specified, prediction consists of fixing the values of one or more nodes (to either single values or distributions) and performing the probability calculations necessary to determine the resulting conditional distributions of the rest of the nodes in the network. This propagation can occur in either the up-arrow or down-arrow direction, the former being a process of causal 'inference' and the latter a process of 'prediction'. Of course, all inferences and predictions are in the form of full probability distributions, so uncertainty in relations is always accounted for.

In ecology, BBNs are useful because the predictive link that we want to model is often a complex causal chain, the entirety of which rarely falls within a single, coordinated research project. BBNs allow this causal chain to be factored into an articulated sequence of conditional relationships, each of which can then be quantified independently using an approach suitable for the type and scale of information available. Often these approaches are Bayesian implementations of the individual statistical methods discussed in other parts of this article. For this reason, BBNs can be viewed as a kind of integrative tool for predictive modeling.

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