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What is the distribution of a sum of $n$ Bernoulli random variables, each having success probability $p$, where each pair is correlated with correlation coefficient $\rho$?

$$Y = \sum_{i=1}^n X_i$$ $$ X_i \sim \mathsf{Bernoulli}(p),\:\:\:\operatorname{corr}(X_i, X_j) = \rho$$

If $\rho=0$, then it is obviously a binomial distribution. Is there a closed-form expression for the probability mass function when $\rho > 0$?

Is it perhaps a beta-binomial? I cannot convince myself either way.

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    $\begingroup$ You might find the thread at stats.stackexchange.com/questions/318759/… to be illuminating. $\endgroup$ – whuber Aug 21 '18 at 0:44
  • $\begingroup$ @whuber Thanks, that's interesting. +1 to your answers in the linked threads. However, I did not find an answer to my question there. From the paper quoted in the BIM's answer below, it seems that my question does not have a unique answer, i.e. the distribution of $Y$ is not uniquely specified by $n$, $p$, and $\rho$... $\endgroup$ – amoeba Aug 21 '18 at 9:17
  • $\begingroup$ That's right: it looks like there is no unique answer. You need to stipulate more about the distribution. $\endgroup$ – whuber Aug 21 '18 at 13:58
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Have you seen this paper: Kadane, 2016, Sums of Possibly Associated Bernoulli Variables: The Conway-Maxwell-Binomial Distribution?

In this paper, you can see that the conditions assumed in your question i.e. having $n$ marginally Binomial r.v. with the same probability of success, $p$, and the same pairwise correlation, $\rho$, between all pairs does not fully specify the distribution of the sum of those random variables.

To be more specific, in Section 2.3 of the paper, the author has assumed "zero higher order additive interaction (Darroch, 1974)":

enter image description here

where $P\{W = k\} = P\{\sum_{i=0}^{m} X_i = k\}$. The model is also called correlated binomial model.

Here is also a brief summary of the first sections of the paper that you may find helpful for modeling the sum:

Proposition 1,2 and 3 provide reasoning for not using correlation as a measure of dependence and to model the sum without assuming a marginal distribution. section 2.1 and 2.2 are distribution models that have these two characteristics. They have some notion of dependence but it is not necessary the correlation. They also allow for symmetric dependence. (Proposition 1 states that correlation cannot be used as a measure of dependence as it is bounded below by $-1/(m-1)$ based on the conditions stated in the proposition).

Section 3 is the proposed model of the author to directly model the sums using a notion of dependence that allows both for positive and negative association.

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    $\begingroup$ +1, thanks a lot! Looking at this paper, it seems that my question does not have a unique answer. Meaning that the distribution of the $\sum X_i$ is not fully specified by $p$ and $\rho$. Is that right? Section 2.3 that you quoted provides one possible answer but e.g. section 2.4 seems to provide another possible distribution? I'm not quite sure about 2.1, 2.2, and 3. Are they also providing possible answers? $\endgroup$ – amoeba Aug 21 '18 at 9:12
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    $\begingroup$ (I guess it's not surprising because e.g. a sum of $m$ standard Gaussians with fixed pairwise correlations is also not fully specified... it's only specified if one assumes that this is a multivariate Gaussian...) $\endgroup$ – amoeba Aug 21 '18 at 9:15
  • $\begingroup$ Yes. this is exactly right. The proposed model in section 2.3 has the additional assumption of zero higher-order additive interaction which is found in (Darroch, 1974). I am not sure how they go about it in Section 2.4. $\endgroup$ – BIM Aug 21 '18 at 16:01
  • $\begingroup$ Proposition 1,2 and 3 provide reasoning for not using correlation as a measure of dependence and to model the sum without assuming a marginal distribution. section 2.1 and 2.2 are distribution models that have these two characteristics. They have some notion of dependence but it is not necessary the correlation. They also allow for symmetric dependence.(Proposition 1 states that correlation cannot be used as a measure o dependence as it is bounded below by -1/(m-1) based on the conditions stated in the proposition) $\endgroup$ – BIM Aug 21 '18 at 16:01
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    $\begingroup$ Could you update your answer with some of this information, in particular to say that my conditions are not enough to specify the distribution of $Y$ so different distributions would be consistent with my requirements (marginally Binomial with the same success probability + the same pairwise correlation between all pairs)? Then I'd accept. $\endgroup$ – amoeba Aug 22 '18 at 10:44

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