How do Bayes nets simplify things?

Question

I recently came across bayes networks. I read that they help in reducing the dimensionality of the joint probability distribution of n random variables (let them be boolean).

In General
P(X1,X2,X3,.....Xn) = P(X1)*P(X2|X1)*P(X3|X1,X2)*.......*P(Xn|X1,X2,X3,.....Xn-1)

And Bayes network helps in simplifying the formula as it contains information about what variables actually depend on each other. This much I get.

What I don't get is how is it reducing the computations required to compute the probabilities? Basically I don't understand the concept of dimensionality in context of joint distributions.

Answer 1

Ok i found the answer. It doesn't actually reduce the computations. It just reduces the space complexity. This is a stub from wikipedia:

Using a Bayesian network can save considerable amounts of memory, if the dependencies i the joint distribution are sparse. For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for 2^{10} = 1024 values. If the local distributions of no variable depends on more than 3 parent variables, the Bayesian network representation only needs to store at most 10*2^3 = 80 values. One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distribution.