Loi de Wishart Inverse – Wikipedia

In probabilities and statistics theory, the Loi de Wishart Inverse , also called loi the wishapart inrseas , is a law of probability defined on all of the matrices defined positive with real coefficients.

A variable following a reverse Wishart law will be noted

${displaystyle mathbf {x} sim w^{-1} ({mathbf {psi}}, well)}$

and is defined by the law of its inverse matrix:

${displaystyle mathbf {X} ^{-1}}$

follows a law of Wishart

${Displastyle W ({mathbf {psi}^^{-1}, Nu)}$

.

The probability density of the reverse Wishart law is:

${Displaystyle {fraud {left | {mathbf {psi}} right |^{fraud {2}}}} {2^{fraud {nu p} {2}} gamma _ {p} ({fraud { {2}})}} left | mathbf {x} right |^{-{fra {nu +p +1} {2}}} {rm {e}}^{-{fraud {1} {2}}}}}}}} Operatorname {tr} ({mathbf {psi}} mathbf {x} ^{-1})}}}$

Or

${displaystyle mathbf {X} }$

And

${displaystyle {mathbf {Psi } }}$

are positive defined matrices

${displaystyle ptimes p}$

And

${displaystyle Gamma _{p}}$

is the multidimensional gamma function.

Law of the lawyer of a Wishart law [ modifier | Modifier and code ]

And

${displaystyle {mathbf {A} }sim W({mathbf {Sigma } },nu )}$

And

${displaystyle {mathbf {Sigma } }}$

is a matrix

${displaystyle ptimes p}$

, SO

${displaystyle mathbf {X} ={mathbf {A} }^{-1}}$

is the law of reverse Wishart:

${displaystyle mathbf {X} sim W^{-1}({mathbf {Sigma } }^{-1},nu )}$

^{[ 2 ]} .

Marginal and conditional laws [ modifier | Modifier and code ]

suppose that

${displaystyle {mathbf {a}} sim w^{-1} ({mathbf {psi}}, well)}$

is the law of reverse Wishart. Let’s separately partially in two matrices

${displaystyle {mathbf {A} }}$

And

${displaystyle {mathbf {Psi } }}$

:

${displaystyle {mathbf {a}} = {begin {bmatrix} mathbf {a} _ {11} & mathbf {a} _ {12} \ mathbf {a} _ {21} & mathbf {a} _ {22} end {bmatrix }} ,; {mathbf {psi}} = {begin {bmatrix} mathbf {psi} _ {11} & mathbf {psi} _ {12} \ mathbf {psi} _ {21} & mathbf {psi} _ {22} end {bmatrix}}}$

Or

${displaystyle {mathbf {A} _{ij}}}$

And

${displaystyle {mathbf {Psi } _{ij}}}$

are matrices

${displaystyle p_{i}times p_{j}}$

, so we get

1. 
   ${displaystyle {mathbf {A} _{11}}}$

   is independent of

   ${displaystyle {mathbf {A} }_{11}^{-1}{mathbf {A} }_{12}}$

   and of

   ${displaystyle {mathbf {A} }_{22cdot 1}}$

   , Or

   ${displaystyle {mathbf {a} _ {22cdot 1}} = {mathbf {a}} _ {22}-{mathbf {a}} _ {21} {mathbf {a}} _ {11}^{-1} {mathbf {a}} _ {12}}$

   is the complement of Schur of

   ${displaystyle {mathbf {A} _{11}}}$

   In

   ${displaystyle {mathbf {A} }}$

   ;

2. 
   ${displaystyle {mathbf {a} _ {11}} sim w^{ -1} ({mathbf {psi} _ {11}}, nu -p_ {2})}}$

   ;

3. 
   ${displaystyle {mathbf {a}} _ {11}^{-1} {mathbf {a}} _ {12} | {mathbf {a}} _ {22cdot 1} sim mn_ {p_ {1} times p_ {2 }} ({mathbf {psi}} _ {11}^{-1} {mathbf {psi}} _ {12}, {mathbf {a}} _ {22cdot 1} otimes {mathbf {psi}} _ {11 }^{-1})}$

   , Or

   ${displaystyle MN_{ptimes q}(cdot ,cdot )}$

   is the normal multidimensional law;

4. 
   ${displaystyle {mathbf {A} }_{22cdot 1}sim W^{-1}({mathbf {Psi } }_{22cdot 1},nu )}$

   

Moments [ modifier | Modifier and code ]

This section is based on the article [Press, 1982] ^{[ 3 ]} , after having repaumated the degree of freedom to be consisted with the definition of the density given above.

The average is ^{[ 2 ]} :

${displaystyle e (mathbf {x}) = {frac {mathbf {psi}} {nu -p -1}}.}$

The variance of each element of

${displaystyle mathbf {X} }$

East :

${Displaystyle OperatorName {Var} (x_ {IJ}) = {Frac {(Nu -P+1) PSI _ {IJ}^{2}+(Nu -P -1) PSI _ {II} PSI _ {JJ} } {(Nu -p) (Nu -P -1)^{2} (Nu -P -3)}}}$

a variance of the useful diagonal the same formula as above with

${displaystyle i=j}$

, which is simplified in:

$mm ssprively hope Feolley Mal Foy emp embeh) mpie m hölame m -De ) möto km kmb) mupe mupe hupm hupm hupm hupm h may demons$

A one-dimensional version of the reverse Wishart law is the reverse-gamma law. With

${displaystyle p=1}$

, that is to say one-dimensional,

${displaystyle alpha =nu /2}$

,

${displaystyle beta =mathbf {Psi } /2}$

And

${displaystyle x=mathbf {X} }$

, the probability density of the reverse Wishart law becomes

${displaystyle p(x|alpha ,beta )={frac {beta ^{alpha },x^{-alpha -1}exp(-beta /x)}{Gamma _{1}(alpha )}}.}$

that is, the inverse-Gamma law where

${displaystyle Gamma _{1}(cdot )}$

is the classic gamma function.

The law of reverse Wishart is a particular case of the multidimensional gamma law.