In Bayes networks it is assumed that directed acyclic graphing makes for simple calculations where there are conditional dependencies between the states. I do not understand what calculations are made easier, would someone please give an example for this? I understand a Markov Net and a Bayes Net can be called a zeroth order Markov Net with no hidden layer. Would someone explain this?
Bayes Networks represent probability distributions in a very simple manner. That being said, all models apply simplifying assumptions to the problem, so in that sense they do end up simplifying assumptions.
- If you really understand the joint distribution of all of the variables you're interested in you're golden. You can then compute conditionals, marginals (their densities, statistics, means, medians). So this is a reasonable goal.
- Each node in a Bayes Network represents a random variable conditionally distributed on it's parents.
- Because of the chain rule, and the fact that the Bayes Network is acyclic, the joint distribution's density can be computed as a product of the individual nodes' densities.
- Causation flows along arrows, and correlation travels in both directions. Then there are important ideas like conditional independence, d-separation, inference.
I think your last sentence should say that Bayes Networks and Markov Random Fields are both examples of Artificial Neural Networks with no hidden layer.
EDIT: Also, I think your title should be changed, Bayes networks are also directed, where as MRF's are undirected.
1$\begingroup$ To understand the computational simplification, I suggest you study a simple algorithm like the forward algorithm for Hidden Markov models This turns a sum that could involve an exponentially large number of terms into something linear, just as a consequence of the independencies implicit in the model. $\endgroup$ Oct 14, 2016 at 16:51
You ask "I am not getting what calculcations it makes easier, can anyone please give an example for this". I cannot give a specific example, as I do not really use either Bayes Nets or Markov Random Fields (at least not explicitly).
However I believe the classical practical difference in calculations is belief propagation. This algorithm can be done exactly for DAGs, but the "loopy" variant is only approximate for graphs with cycles. My understanding, which is crude and may be wrong, is that the basic version essentially proceeds through the network in topological order, so all calculations that can affect a node have been completed prior to reaching that node.)