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I'm looking for a simple, symmetric multivariate distribution for $N$ Bernoulli variables with the following properties:

  • Each individual variable takes on values 1 or 0

  • Fix a subset of $M$ variables. Let $P_K$ be the probability that conditional on $K$ of these $M$ variables taking on value 1, the $N$th variable is 1. Want: $P_K$ increasing in $K$.

  • A parameterized functional form with a single parameter, so that at the extremes, we arrive at the case of perfect correlation and at independence.

Hopefully this is not too simple a question to ask on mathoverflow, but I've had a tough time coming up with a simple distribution that does this. Many thanks.

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    $\begingroup$ Take a multivariate Gaussian distribution on $(y_1,..,y_N)$ with a constant pairwise correlation $\rho$. Pick some threshold $T$ and set $x_i = 1$ if and only if $Y_i \geq T$ (otherwise zero). The $x_i$'s are correlated Bernoulli r.v.s. with a distribution parametrized by $\rho$ which satisfies your conditions. $\endgroup$
    – Or Zuk
    Commented Apr 27, 2011 at 21:50
  • $\begingroup$ Possibly related: math.stackexchange.com/questions/52137/… $\endgroup$
    – Chill2Macht
    Commented Apr 10, 2023 at 22:42

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Gibbs measures provide a solution. Introduce the energy $H(x)$ of a configuration $x=(x_k)_{1\le k\le N}$ in $S=\{0,1\}^N$ as $$ H(x)=\sum_kx_k+\sum_{k\ne\ell}x_kx_\ell, $$ and, for every parameter $a>0$, define a probability $P_a$ on $S$ by $$ P_a(x)=a^{H(x)}/Z_a,\qquad Z_a=\sum_{x\in S}a^{H(x)}. $$ Special values are $P_1$ uniform on $S$ and $P_\infty$ the Dirac mass at the configuration all ones. For every $a>1$, the random variables $x_k$ are positively correlated due to the $x_kx_\ell$ terms in the energy $H(x)$ hence the properties you ask are satisfied.

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