Density function for a multivariate Bernoulli-like distribution

Question

I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$. Suppose I have observed that $\sum_i X_i = n$. In this case I do not want to treat them as independent Bernoulli random variables. Instead, I would like something like the multinomial:

$P(X_1=x_1, \ldots, X_k=x_k) = f(x_1, \ldots, x_k; n, p_1, \ldots, p_k) = \frac{n!}{x_1! \cdots x_k!} \prod_{i=1}^k p_i^{x_i}$

but instead of the $x_i$ being nonnegative integers, I want them restricted to be either 0 or 1. I have been trying to see if the multivariate hypergeometric is appropriate, but I'm not sure.

Thanks in advance for any advice.

Answer 1

You need to specify distribution over your random vector $\mathbf{X}$. If individual components are binary valued, and you only care about positive distributions, it can be written in the following form

$$P(\mathbf{x})=\exp(\theta_0+\theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_{12} x_1 x_2 + \ldots +\theta_{1\ldots k}x_1 \cdots x_k)$$

Now the task is determining the distribution of $P(\mathbf{X}|X_1+\ldots+X_k=n)$, this distribution is related to hypergeometric distribution, described in Percy Diaconis "Algebraic algorithms for sampling from conditional distributions" (equations 1.1-1.4)