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.
Best Answer
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)