🚀 go-pugleaf

Dear people:

    I'd like to know the exact form of distribution function of
multinomial with marginal restrictions.  I consider the restrictions
are linear on the frequencies,  so the distribution reduces to
composite of several multinomial ones,  if the restrictions are
unoverlapped.  But I consider the following one:

    Let n_ij , i=1,...,a, j=1,...,b be two-way frequecies and p_ij be
multinomial parameters.  If there is only one restriction such as
\sum_{ij} n_ij = n, then the distribution functions becomes
P(n_ij's|n) = n! \prod_{ij} p_{ij}^{n_{ij}} / n_{ij}!.  But we
consider the following restrictions: \sum_{i} n_{ij} = m_j, \sum_{j}
n_{ij} = l_i, for i=1,...,a, j=1,...,b.  Then, what is the conditional
distribution P(n_ij's|m_j's, l_i's) ?

    I know that if i and j are independent, that is, p_{ij} = p_{i.} *
p_{.j}, then the conditional distribution can be written in explicit
from.  But I'd like to know the explicit form for the general case.

If someone knows anything about those problmes,  please let me know.

        Thank you,


 //\\\
  @ @     Hidetoshi SHIMODAIRA    <shimo@bcl.t.u-tokyo.ac.jp>
   O      Dept. of Math. Eng. & Info. Physics, Univ. of Tokyo