Correlated Bernoulli random variables I have about $50$ Bernoulli random variables $X_i$ whose joint distribution is unknown, but I can generate a sample of size on the order of $10^4$.
They are not independent, but I think the dependence is rather weak.
  I would like to get some idea of ${\rm Pr}({\rm all}\ X_i = 1)$, which I expect to be
considerably smaller than $10^{-4}$, maybe around $10^{-6}$ (so it's unlikely that the sample will
contain outcomes with all $X_i = 1$).  If I assumed they were independent, I could estimate their means and multiply these estimates.  What can I do that
takes correlations into account?
 A: There are a large variety of lower-dimensional structures/models for dependence among high-dimensional tables. 


*

*One possibility might be to fit log-linear models that only go up to some low order of interaction (say to third or fourth order, perhaps). 

*Another possibility might be something akin to PCA link1 link2. 

*If there's any kind of ordering to the variables you might consider something akin to a D-vine copula. 

*you might have a model containing a number of latent variables with which each of the binary variables is correlated.

*You could consider each variable to be a mixture of an independent component and some (relatively) simple dependent component. 

*There are many other possibilities. No doubt including many in use that I haven't even heard of.
We don't have the information on which to suggest which kind of lower dimensional model might be suitable for your data ... you potentially can have enough data to do some splitting into several subsets (not necessarily of equal sizes) and then to do some modelling on one  subsets (the data snooping part), fitting on a second subset and validation on a third (or to do cross validation, etc).
More details (with what kind of structure there might be, for example) might help as far as giving suggestions.
