# sum of conditionally independent Bernoulli variables on a bayesian network

I constructed a Bayesian network which presents the conditional independence among N random variables. Each random variable $X_i$ represents a Bernoulli random variable with an associated success probability $\pi$ (the probabilities, $\pi$`s, are the parameters of the Bayesian graph).

I'm interested in computing the sum $S=X_1+X_2+...+X_N$. This means I'm interested in computing the sum over the nodes of the Bayesian graph. I know that Bayesian graph enables us to compute the joint probability distribution $p(X_1,X_2,X_3,...,X_N)$ but not their sum.

Are there some references or ideas that deal with the same sum on a Bayesian graph.

Can some one tell whether we can use this link to compute the sum $S$.

• More specifically, I'm looking to find out the probability distribution of p(S=k) or its argmax. For independent bernoulli variables, this distribution is called poisson's binomial distribution. However, when there are conditional independence and dependence among variables (modeled using the bayesian graph), I don't know how to find out this distribution. – Fred Dec 29 '16 at 18:34
• why don't you just create a new deterministic node $S=\sum_i X_i$ in your network. Then you can sample from it using MCMC, Gibbs etc. Or even do exact inference on it. – YBE Dec 30 '16 at 0:10
• thanks YBE for your answer, but for now I don't see what you mean exactly. – Fred Dec 30 '16 at 2:07
• thanks YBE for your answer. Sorry, I didn't find the details how your suggestion works. Please note that I need to estimate/predict the value of S before getting its exact value from the experiment. Means, I will not wait the result of the trials in order to compute S. I want to quantify S given the success probabilities on the Bayesian graph. – Fred Dec 30 '16 at 2:15
• You have Bayes net with variables X1,..., XN right? What I am saying is introduce an additional node to your Bayes Net, which is the S you defined in your question. Then just using whatever inference routine you are utilizing; compute the joint posterior p(X1,...,XN,S). Then you can marginalize if you just want p(S). – YBE Dec 30 '16 at 2:23