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Given the following joint normal PDF of $X \in \mathcal{R}^K$ and $Y,Z \in \mathcal{R}$

$p(\begin{bmatrix} X \\\ Y \\\ Z \end{bmatrix}) = \mathcal{N}(\begin{bmatrix} \mu_X \\\ \mu_Y \\\ \mu_Z \end{bmatrix},\begin{bmatrix} \Sigma_{XX} \ \Sigma_{XY} \ \Sigma_{XZ} \\\ \Sigma_{YX} \ \Sigma_{YY} \ \Sigma_{YZ} \\\ \Sigma_{ZX} \ \Sigma_{ZY} \ \Sigma_{ZZ} \end{bmatrix})$

How can we derive the closed form expression for the following PDF?

$P(X|A)$ (or equivalently $P(X|A^2)$)

where, $A = \sqrt{Y^2+Z^2}$

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Duplicate of stats.stackexchange.com/q/27413/6633 –  Dilip Sarwate Apr 30 '12 at 21:22
The closed form is quite unlikely. One is first to consider $X|(Y,Z)$, then represent $(Y,Z) = A (\cos(\Phi), \sin(\Phi))$ and then average out $\Phi$. I would expect appearance of some matrix Bessel functions. Has this example its origin in wireless communication theory, by chance? –  Sasha Apr 30 '12 at 21:37
Thanks for the hints. Actually, the question is originated from speech enhancement domain. –  smo Apr 30 '12 at 21:58
smo: Any luck with the answer below? –  Did May 7 '12 at 12:12

1 Answer 1

In theory, this is straightforward: the $(K+2)$-dimensional vector $(X,Y,Z)$ is jointly normal, hence there exists some deterministic $x$, $\eta$ and $\zeta$ in $\mathbb R^K$ and a centered $K$-dimensional normal random vector $T$, with variance-covariance matrix $C$, independent of $(Y,Z)$, such that conditionally on $(Y,Z)$, $X$ is distributed as $$ U=x+Y\eta+Z\zeta+T. $$ Thus, conditionally on $A^2=Y^2+Z^2$, $X$ is distributed as $x+T+V$, where $V$ follows the conditional distribution of $Y\eta+Z\zeta$ conditionally on $A^2$.

First step: To find $(x,\eta,\zeta,C)$, one solves the system $$ (1)\ \mathrm E(U)=\mathrm E(X),\quad (2)\ \mathrm E(UY)=\mathrm E(XY),\quad (3)\ \mathrm E(UZ)=\mathrm E(XZ), $$ and $$ (4)\ \mathrm{var}(Y\eta+Z\zeta)+C=\mathrm{var}(X). $$ These four equations are equivalent to $(1)\ x+\mu_Y\eta+\mu_Z\zeta=\mu_X$, $$ (2)\ x\mu_Y+\Sigma_{YY}\eta+\Sigma_{YZ}\zeta=\Sigma_{XY},\qquad (3)\ x\mu_Z+\Sigma_{YZ}\eta+\Sigma_{ZZ}\zeta=\Sigma_{XZ}, $$ and $$ (4)\ \Sigma_{YY}\eta\eta^*+\Sigma_{YZ}(\eta\zeta^*+\zeta\eta^*)+\Sigma_{ZZ}\zeta\zeta^*+C=\Sigma_{XX}, $$ which, in the general case, determine uniquely the $K$-dimensional vectors $x$, $\eta$ and $\zeta$ and the $K\times K$ matrix $C$. (Note that $\mu_Y$, $\mu_Z$, $\Sigma_{YY}$, $\Sigma_{YZ}=\Sigma_{ZY}$ and $\Sigma_{ZZ}$ are real numbers while $x$, $\eta$, $\zeta$, $\Sigma_{XY}$ and $\Sigma_{XZ}$ are $K$-dimensional vectors and $\Sigma_{XX}$, $\eta\eta^*$, $\eta\zeta^*$, $\zeta\eta^*$, $\zeta\zeta^*$ and $C$ are $K\times K$ matrices.)

Second step: Basically, a two dimensional gaussian vector $(Y,Z)$ with prescribed distribution is given and one looks for the distribution of the vector combination $Y\eta+Z\zeta$ conditionally on $A^2=Y^2+Z^2$, for possibly any $\eta$ and $\zeta$ in $\mathbb R^K$.

There exists some i.i.d. standard gaussian vector $(Y_0,Z_0)$ and a $2\times2$ matrix $B$ such that $(Y,Z)=M+(Y_0,Z_0)B$, where $M=(\mu_Y,\mu_Z)$ and $B^*B$ is the variance-covariance matrix of $(Y,Z)^*$. Thus, $A^2=a^2$ corresponds to an ellipse in the $(Y_0,Z_0)$ plane and one seeks the distribution of some ($K$-dimensional) affine combination of $(Y_0,Z_0)$ conditionally on the fact that $(Y_0,Z_0)$ is on this ellipse.

I doubt that some convenient formula exists in the general case, but maybe some more specific hypotheses are made.

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