Matrix F-distribution

In statistics, the matrix F distribution (or matrix variate F distribution) is a matrix variate generalization of the F distribution which is defined on real-valued positive-definite matrices. In Bayesian statistics it can be used as the semi conjugate prior for the covariance matrix or precision matrix of multivariate normal distributions, and related distributions ^[1][2][3][4].

Matrix ${\displaystyle F}$
Notation	${\displaystyle {\mathcal {F}}({\mathbf {\Psi } },\nu ,\delta )}$
Parameters	${\displaystyle \mathbf {\Psi } >0}$, ${\displaystyle p\times p}$ scale matrix (pos. def.) ${\displaystyle \nu >p-1}$ degrees of freedom (real) ${\displaystyle \delta >0}$ degrees of freedom (real)
Support	${\displaystyle \mathbf {X} }$ is p × p positive definite matrix
PDF	${\displaystyle {\frac {\Gamma _{p}\left({\frac {\nu +\delta +p-1}{2}}\right)}{\Gamma _{p}\left({\frac {\nu }{2}}\right)\Gamma _{k}\left({\frac {\delta +p-1}{2}}\right)|\mathbf {\Psi } |^{\frac {\nu }{2}}}}~|{\mathbf {X} }|^{\frac {\nu -p-1}{2}}|{\textbf {I}}_{p}+{\mathbf {X} }\mathbf {\Psi } ^{-1}|^{-{\frac {\nu +\delta +p-1}{2}}}}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function ${\displaystyle {\textbf {I}}_{p}}$ is the p × p identity matrix
Mean	${\displaystyle {\tfrac {\nu }{\delta -2}}\mathbf {\Psi } }$, for ${\displaystyle \delta >2.}$
Variance	see below

Density

The probability density function of the matrix ${\displaystyle F}$  distribution is:

${\displaystyle f_{\mathbf {X} }({\mathbf {X} };{\mathbf {\Psi } },\nu ,\delta )={\frac {\Gamma _{p}\left({\frac {\nu +\delta +p-1}{2}}\right)}{\Gamma _{p}\left({\frac {\nu }{2}}\right)\Gamma _{k}\left({\frac {\delta +p-1}{2}}\right)|\mathbf {\Psi } |^{\frac {\nu }{2}}}}~|{\mathbf {X} }|^{\frac {\nu -p-1}{2}}|{\textbf {I}}_{p}+{\mathbf {X} }\mathbf {\Psi } ^{-1}|^{-{\frac {\nu +\delta +p-1}{2}}}}$ 

where ${\displaystyle \mathbf {X} }$  and ${\displaystyle {\mathbf {\Psi } }}$  are ${\displaystyle p\times p}$  positive definite matrices, ${\displaystyle |\cdot |}$  is the determinant, Γ_p(·) is the multivariate gamma function, and ${\displaystyle {\textbf {I}}_{p}}$  is the p × p identity matrix.

Properties

Construction of the distribution

• The standard matrix F distribution, with an identity scale matrix ${\displaystyle \mathbf {I} _{p}}$ , was originally derived by ^[1]. When considering independent distributions, ${\displaystyle {\mathbf {\Phi } _{1}}\sim {\mathcal {W}}({\mathbf {I} _{p}},\nu )}$  and ${\displaystyle {\mathbf {\Phi } _{2}}\sim {\mathcal {W}}({\mathbf {I} _{p}},\delta +k-1)}$ , and define ${\displaystyle \mathbf {X} ={\mathbf {\Phi } _{2}}^{-1/2}{\mathbf {\Phi } _{1}}{\mathbf {\Phi } _{2}}^{-1/2}}$ , then ${\displaystyle \mathbf {X} \sim {\mathcal {F}}({\mathbf {I} _{p}},\nu ,\delta )}$ .

• If ${\displaystyle {\mathbf {X} }|\mathbf {\Phi } \sim {\mathcal {W}}^{-1}({\mathbf {\Phi } },\delta +p-1)}$  and ${\displaystyle {\mathbf {\Phi } }\sim {\mathcal {W}}({\mathbf {\Psi } },\nu )}$ , then, after integrating out ${\displaystyle \mathbf {\Phi } }$ , ${\displaystyle \mathbf {X} }$  has a matrix F-distribution, i.e.,
  ${\displaystyle f_{\mathbf {X} |\mathbf {\Phi } ,\nu ,\delta }(\mathbf {X} )=\int f_{\mathbf {X} |\mathbf {\Phi } ,\delta +p-1}(\mathbf {X} )f_{\mathbf {\Phi } |\mathbf {\Psi } ,\nu }(\mathbf {\Phi } )d\mathbf {\Phi } .}$ 
  This construction is useful to construct a semi-conjugate prior for a covariance matrix^[3].

• If ${\displaystyle {\mathbf {X} }|\mathbf {\Phi } \sim {\mathcal {W}}({\mathbf {\Phi } },\nu )}$  and ${\displaystyle {\mathbf {\Phi } }\sim {\mathcal {W}}^{-1}({\mathbf {\Psi } },\delta +p-1)}$ , then, after integrating out ${\displaystyle \mathbf {\Phi } }$ , ${\displaystyle \mathbf {X} }$  has a matrix F-distribution, i.e.,
  ${\displaystyle f_{\mathbf {X} |\mathbf {\Psi } ,\nu ,\delta }(\mathbf {X} )=\int f_{\mathbf {X} |\mathbf {\Phi } ,\nu }(\mathbf {X} )f_{\mathbf {\Phi } |\mathbf {\Psi } ,\delta +p-1}(\mathbf {\Phi } )d\mathbf {\Phi } .}$ 
  This construction is useful to construct a semi-conjugate prior for a precision matrix^[4].

Marginal distributions from a matrix F distributed matrix

Suppose ${\displaystyle {\mathbf {A} }\sim F({\mathbf {\Psi } },\nu ,\delta )}$  has a matrix F distribution. Partition the matrices ${\displaystyle {\mathbf {A} }}$  and ${\displaystyle {\mathbf {\Psi } }}$  conformably with each other

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

where ${\displaystyle {\mathbf {A} _{ij}}}$  and ${\displaystyle {\mathbf {\Psi } _{ij}}}$  are ${\displaystyle p_{i}\times p_{j}}$  matrices, then we have ${\displaystyle {\mathbf {A} _{11}}\sim F({\mathbf {\Psi } _{11}},\nu ,\delta )}$ .

Moments

Let ${\displaystyle X\sim F({\mathbf {\Psi } },\nu ,\delta )}$ .

The mean is given by: ${\displaystyle E(\mathbf {X} )={\frac {\nu }{\delta -2}}\mathbf {\Psi } .}$ 

The (co)variance of elements of ${\displaystyle \mathbf {X} }$  are given by^[3]:

${\displaystyle \operatorname {cov} (X_{ij},X_{ml})=\Psi _{ij}\Psi _{ml}{\tfrac {2\nu ^{2}+2\nu (\delta -2)}{(\delta -1)(\delta -2)^{2}(\delta -4)}}+(\Psi _{il}\Psi _{jm}+\Psi _{im}\Psi _{jl})\left({\tfrac {2\nu +\nu ^{2}(\delta -2)+\nu (\delta -2)}{(\delta -1)(\delta -2)^{2}(\delta -4)}}+{\tfrac {\nu }{(\delta -2)^{2}}}\right).}$ 

Related distributions

• The matrix F-distribution has also been termed the multivariate beta II distribution^[5]. See also ^[6], for a univariate version.

• A univariate version of the matrix F distribution is the F-distribution. With ${\displaystyle p=1}$  (i.e. univariate) and ${\displaystyle \mathbf {\Psi } =1}$ , and ${\displaystyle x=\mathbf {X} }$ , the probability density function of the matrix F distribution becomes the univariate (unscaled) F distribution:
  ${\displaystyle f_{x\mid \nu ,\delta }(x)=\operatorname {B} \left({\tfrac {\nu }{2}},{\tfrac {\delta }{2}}\right)^{-1}\left({\tfrac {\nu }{\delta }}\right)^{\nu /2}x^{\nu /2-1}\left(1+{\tfrac {\nu }{\delta }}\,x\right)^{-(\nu +\delta )/2},}$ 

• In the univariate case, with ${\displaystyle p=1}$  and ${\displaystyle x=\mathbf {X} }$ , and when setting ${\displaystyle \nu =1}$ , then ${\displaystyle {\sqrt {x}}}$  follows a half t distribution with scale parameter ${\displaystyle {\sqrt {\psi }}}$  and degrees of freedom ${\displaystyle \delta }$ . The half t distribution is a common prior for standard deviations^[7].

See also

• Inverse matrix gamma distribution
• Matrix normal distribution
• Wishart distribution
• Inverse Wishart distribution
• Complex inverse Wishart distribution

References

1. Olkin, Ingram; Rubin, Herman (1964-03-01). "Multivariate Beta Distributions and Independence Properties of the Wishart Distribution". The Annals of Mathematical Statistics. 35 (1): 261–269. doi:10.1214/aoms/1177703748. ISSN 0003-4851.
2. Dawid, A. P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika. 68 (1): 265–274. doi:10.1093/biomet/68.1.265. ISSN 0006-3444.
3. Mulder, Joris; Pericchi, Luis Raúl (2018-12-01). "The Matrix-F Prior for Estimating and Testing Covariance Matrices". Bayesian Analysis. 13 (4). doi:10.1214/17-BA1092. ISSN 1936-0975. S2CID 126398943.
4. Williams, Donald R.; Mulder, Joris (2020-12-01). "Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints". Journal of Mathematical Psychology. 99: 102441. doi:10.1016/j.jmp.2020.102441. S2CID 225019695.
5. Tan, W. Y. (1969-03-01). "Note on the Multivariate and the Generalized Multivariate Beta Distributions". Journal of the American Statistical Association. 64 (325): 230–241. doi:10.1080/01621459.1969.10500966. ISSN 0162-1459.
6. Pérez, María-Eglée; Pericchi, Luis Raúl; Ramírez, Isabel Cristina (2017-09-01). "The Scaled Beta2 Distribution as a Robust Prior for Scales". Bayesian Analysis. 12 (3). doi:10.1214/16-BA1015. ISSN 1936-0975.
7. Gelman, Andrew (2006-09-01). "Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)". Bayesian Analysis. 1 (3). doi:10.1214/06-BA117A. ISSN 1936-0975.