%es.tex:A Symbolic Computation Approach to a Problem involving Multivariate Poission distributions
%%a Plain TeX file by Eduardo Sontag Doron Zeilberger (x pages)

%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
  0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
  height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\1{{\overline{1}}}
\def\2{{\overline{2}}}
\parindent=0pt
\overfullrule=0in
\def\Tilde{\char126\relax}
\def\frac#1#2{{#1 \over #2}}
%\headline={\rm  \ifodd\pageno  \RightHead  \else  \LeftHead  \fi}
%\def\RightHead{\centerline{
%Title
%}}
%\def\LeftHead{ \centerline{Doron Zeilberger}}
%end macros
\bf
\centerline
{
A Symbolic Computation Approach to a Problem involving Multivariate Poission Distributions
}

(Draft, do not distribute!)

\rm
\bigskip
\centerline{ {\it
Eduardo SONTAG{$^1$} and
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt [sontag, zeilberg]  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg} .
First version: xx.x, 200x.
Accompanied by Maple package {\eighttt MVPOISSON}
downloadable from Zeilberger's website.
Supported in part by the NSF.
}
}


{\bf Introduction}

Eduardo, please write it.

{\bf The Problem}

Suppose that we have $n$ independent Poisson random variables, $X_i$ ($i=1 \dots n$) with parameters
$\lambda_i$ respectively. In other words
$$
Pr(X_1=k_1, \dots , X_n=k_n)=
{{\lambda_1^{k_1}} \over {k_1!}}{{\lambda_2^{k_2}} \over {k_2!}} \dots  {{\lambda_n^{k_n}} \over {k_n!}}
$$

Suppose that we can't observe the $X_i$'s directly, but only a certain number, $m$, of linear combinations
of them:
$$
Y_j=\sum_{i=1}^{n} a_{ij} X_j \quad , \quad (j=1 , \dots , m )\quad ,
$$
where $(a_{ij})$ is a certain $m \times n$ matrix with non-negative coefficients.

We are interested in the following questions:

{\bf 1.} Can one compute (fast!), for any given vector $(b_1, \dots, b_m)$ (possibly with large coordinates), the
probability
$$
F(b_1, \dots, b_m):= Pr (Y_1=b_1, \dots, Y_m=b_m) \quad
$$


{\bf 2.} Can one compute (fast!), for any given vectors $(b_1, \dots, b_m)$, (possibly with large coordinates) the
conditional expectation
$$
G_i(b_1, \dots, b_m):= E[X_i \,\, \Bigl \vert \, \, Y_1=b_1, \dots , Y_m=b_m]
$$



{\bf 3.} More generally, can one compute (fast!), 
the higher moments
$$
G_i^{(r)} (b_1, \dots, b_m):= E[X_i^r \,\, \Bigl \vert \,\, Y_1=b_1, \dots , Y_m=b_m], (r \geq 2) \quad .
$$
(which would immediately allow us to compute the moments about the mean.)


{\bf The generating function}

Fix a matrix $A=(a_{ij})$ ($1 \leq i \leq m$,  $1 \leq j \leq n$). Let $f(z_1, \dots, z_m)$ be
the (multivariable) generating function of $F(b_1, \dots, b_m)$, in other words
$$
f(z_1, \dots, z_m)=\sum_{b_1 \geq 0, \dots, b_m \geq 0} F(b_1, \dots, b_m) z_1^{b_1} \cdots z_m^{b_m} \quad.
$$
We have:
$$
f(z_1, \dots, z_m)=\sum_{b_1 \geq 0, \dots, b_m \geq 0} F(b_1, \dots, b_m) z_1^{b_1} \cdots z_m^{b_m} =
$$
$$
=\sum_{b_1 \geq 0, \dots, b_m \geq 0} \left (
\sum_{
{{k_1, \dots, k_n \geq 0}
\atop
{a_{11} k_1 + \dots + a_{1n} k_n=b_1 , \dots , a_{m1} k_1 + \dots + a_{mn} k_n=b_m} }
}
{{\lambda_1^{k_1}} \over {k_1!}}{{\lambda_2^{k_2}} \over {k_2!}} \dots  {{\lambda_n^{k_n}} \over {k_n!}}
\right )
z_1^{b_1} \cdots z_m^{b_m} 
$$
By changing the order of summation, this equals
$$
\sum_{k_1 \geq 0, \dots, k_n \geq 0} 
{{\lambda_1^{k_1}} \over {k_1!}}{{\lambda_2^{k_2}} \over {k_2!}} \dots  {{\lambda_n^{k_n}} \over {k_n!}}
z_1^{a_{11}k_1+ \dots + a_{1n} k_n} \cdots z_m^{a_{m1} k_1 + \dots + a_{mn} k_n}
$$
$$
=\sum_{k_1 \geq 0, \dots, k_n \geq 0} 
{{(\lambda_1 z_1^{a_{11}} z_2^{a_{21}} \cdots z_m^{a_{m1}})^{k_1}} \over {k_1!}}
\cdots
{{(\lambda_n z_1^{a_{1n}} z_2^{a_{2n}} \cdots z_m^{a_{mn}})^{k_n}} \over {k_n!}}
$$
$$
=
\left ( \sum_{k_1 \geq 0}  {{(\lambda_1 z_1^{a_{11}} z_2^{a_{21}} \cdots z_m^{a_{m1}})^{k_1}} \over {k_1!}} \right )
\cdots
\left ( \sum_{k_n \geq 0}  {{(\lambda_n z_1^{a_{1n}} z_2^{a_{2n}} \cdots z_m^{a_{mn}})^{k_n}} \over {k_n!}} \right )
$$
$$
=exp ( \lambda_1 z_1^{a_{11}} z_2^{a_{21}} \cdots z_m^{a_{m1}} )
\cdots
exp ( \lambda_n z_1^{a_{1n}} z_2^{a_{2n}} \cdots z_m^{a_{mn}} )
$$
$$
=exp \left ( \lambda_1 z_1^{a_{11}} z_2^{a_{21}} \cdots z_m^{a_{m1}} + \cdots +
\lambda_n z_1^{a_{1n}} z_2^{a_{21}} \cdots z_m^{a_{mn}} \right ) \quad .
$$

We have just derived

{\bf Theorem 1}: The generating function $f(z_1, \dots, z_m)$ is given by
$$
f(z)= exp \left ( \sum_{j}^{n} \lambda_j \prod_{i=1}^{m} z_i^{a_{ij}} \right ) \quad .
$$


So $P(b_1, \dots, b_m)$ is the coefficient of $z_1^{b_1} \cdots z_m^{b_m}$ in the multivariable
Taylor expansion about the origin of $f(z_1, \dots, z_m)$.

To get the factorial moments of $X_i$, $E[X_i^{(r)}]$ we apply the operator 
$\lambda_i^r ({{\partial} \over {\partial \lambda_i}})^r$ to $P(b_1, \dots, b_m)$ and divide
by $P(b_1, \dots , b_m)$. By doing the same for the generating function, we see that
the numerator is $\lambda_i^r$ times the coefficient of
$(\prod_{i=1}^{m} z_i^{a_{ij}})^r$, In other words we have

{\bf Theorem 2}: The factorial moments $E[X_j^{(r)}]$ are given in terms of the
$P(b_1, \dots , b_m)$ by
$$
{{P(b_1-ra_{1j},b_2- ra_{2j}, \dots, b_m-ra_{mj}) } \over
{P(b_1, \dots, b_m)}}
$$

So everything depends on a fast computation of the coefficients of $f(z_1, \dots , z_m)$.


\end





