Bayesian Reasoning and Machine Learning

We can now make a first connection between probability and graph theory. A belief network introduces structure into a probabilistic model by using graphs to represent independence assumptions among the variables. Probability operations such as marginalising and conditioning then correspond to simple operations on the graph, and details about the model can be ‘read’ from the graph. There is also a benefit in terms of computational efficiency. Belief networks cannot capture all possible relations among variables. However, they are natural for representing ‘causal’ relations, and they are a part of the family of graphical models we study further in Chapter 4.

The benefits of structure

It's tempting to think of feeding a mass of undigested data and probability distributions into a computer and getting back good predictions and useful insights in extremely complex environments. However, unfortunately, such a naive approach is likely to fail. The possible ways variables can interact is extremely large, so that without some sensible assumptions we are unlikely to make a useful model. Independently specifying all the entries of a table p(x 1, …, xN ) over binary variables xi takes O(2 N ) space, which is impractical for more than a handful of variables. This is clearly infeasible in many machine learning and related application areas where we need to deal with distributions on potentially hundreds if not millions of variables. Structure is also important for computational tractability of inferring quantities of interest.