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I am trying to figure out what I need to assume in a project work I am writing.

Suppose I have $N$ $0-1$ Bernoulli trials $\{B_i\}$ with success probabilities $p_i$. I know that they are positively correlated since I know their conditional distribution given another Bernoulli variable $X$ (and given my particular parameters this indeed happens). We have $B_i|X=^d Be(q_i)$ where $q_i$ is computable/known. I know that in general, the joint distribution of an $N$-dimensional Bernoulli vector is determined by the "moments" $E(B_iB_j...B_k)$ (see Multivariate normal and multivariate Bernoulli, Joint distribution of dependent Bernoulli Random variables). However, one should think that knowing the conditional distribution is quite a bit stronger than just knowing the second moment. Is there any way I can extract more information from conditional distribution above about the joint distribution? If not, what sort of "information" would I require to compute these mixed moments?

Any help or reference would be much appreciated!

EDIT: Would it for instance suffice to know the joint distribution of $\{B_i\}$ conditional on $X$?

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Winston
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  • I may not following your question correctly, just pointing out one thing. For example in a bivariate Bernoulli, you have $2^2 = 4$ support points, so in general you need $4$ conditions to determine them. If you are given the two marginal distributions, and the sum of them must equal to $1$, so you only need $1$ more condition, e.g. given the cross moment to completely determine it. – BGM Sep 30 '16 at 06:52
  • I think you are interpreting it correctly, sort of at least. This is precisely the point that is being made in the question I linked to after my comment on the moments. The problem is that I don't really know the moments, but I can know other things which might lead me to them (this is an aspect of the model I am working on). – Winston Sep 30 '16 at 07:01
  • So maybe you can clarify what do you want (the joint distribution of ...?) and what information do you have. – BGM Sep 30 '16 at 14:12

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