Anyone any experience with Bayesian inference within a network, things like
markov random fields?
here is my problem:
Given a Graph G = (V,E) with vertices V connected by edges in E, the network is undirected and cyclic, my not be connected generally has a square lattice structure but not rigidly and may have chains (a sub-graph of vertices may resemble v1-v2-v3-v4-v5.)
At each vertex there is up to n Boolean variables, e.g. if n = 4 then the variables can be labeled a,b,c, d. At each node a could be true or false, b true or false, etc, etc. I record the probability that each variable is true, e.g. a= 0.7 means there is a 70% chance a is true, etc. There are also some conditions, e.g. if a is true then c must be false and if c is true a must be false, etc.
There may be multiple different evidence variables for each of the boolean variables, e.g. one bit of information suggest variable a is true at a particular vertex with p=0.75 confidence, another bit of information suggests a is true with p=0.87 confidence, and the prior probability of a being true may be even odds (p=0.5). How can I incorporate the change in probabilities as each variable gains more information in its value?
The complexity comes from propagating probabilities to neighboring nodes. E.g., if a is true at 1 vertex, the probability of a being true increases at all neighbor vertices, and in general their is a propagation factor so nodes 2 hops away (i.e, 2 edges and one intermediate node) will increase/decrease probability at a lower rate, perhaps 1/2^d where d is the distance metric equal to minimum number edges between the 2 vertices.
I am having difficulties framing in this is a proper Bayesian inference model to correctly propagate and update probabilities. I have a naive model that may work but I want a more formal framework to follow. I believe the problem is related to domains like markov random fields
markov random fields?
here is my problem:
Given a Graph G = (V,E) with vertices V connected by edges in E, the network is undirected and cyclic, my not be connected generally has a square lattice structure but not rigidly and may have chains (a sub-graph of vertices may resemble v1-v2-v3-v4-v5.)
At each vertex there is up to n Boolean variables, e.g. if n = 4 then the variables can be labeled a,b,c, d. At each node a could be true or false, b true or false, etc, etc. I record the probability that each variable is true, e.g. a= 0.7 means there is a 70% chance a is true, etc. There are also some conditions, e.g. if a is true then c must be false and if c is true a must be false, etc.
There may be multiple different evidence variables for each of the boolean variables, e.g. one bit of information suggest variable a is true at a particular vertex with p=0.75 confidence, another bit of information suggests a is true with p=0.87 confidence, and the prior probability of a being true may be even odds (p=0.5). How can I incorporate the change in probabilities as each variable gains more information in its value?
The complexity comes from propagating probabilities to neighboring nodes. E.g., if a is true at 1 vertex, the probability of a being true increases at all neighbor vertices, and in general their is a propagation factor so nodes 2 hops away (i.e, 2 edges and one intermediate node) will increase/decrease probability at a lower rate, perhaps 1/2^d where d is the distance metric equal to minimum number edges between the 2 vertices.
I am having difficulties framing in this is a proper Bayesian inference model to correctly propagate and update probabilities. I have a naive model that may work but I want a more formal framework to follow. I believe the problem is related to domains like markov random fields