Matrix F-distribution

Matrix $F$
Matrix F
Notation	ℱ(𝜳,ν,δ)
Parameters	𝜳>0, p×p scale matrix (pos. def.); ν>p−1 degrees of freedom (real); δ>0 degrees of freedom (real)
Support	𝐗 is p × p positive definite matrix
PDF	Γp(ν+δ+p−12)Γp(ν2)Γp(δ+p−12)\|𝜳\|ν2 \|𝐗\|ν−p−12\|𝐈p+𝐗𝜳−1\|−ν+δ+p−12 Γp is the multivariate gamma function; 𝐈p is the p × p identity matrix;
Mean	νδ−2𝜳, for δ>2.
Variance	see below

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]

Density

The probability density function of the matrix $F$ distribution is:

$f_{𝐗} (𝐗; 𝜳, ν, δ) = \frac{Γ_{p} (\frac{ν + δ + p - 1}{2})}{Γ_{p} (\frac{ν}{2}) Γ_{p} (\frac{δ + p - 1}{2}) | 𝜳 |^{\frac{ν}{2}}} | 𝐗 |^{\frac{ν - p - 1}{2}} | 𝐈_{p} + 𝐗 𝜳^{- 1} |^{- \frac{ν + δ + p - 1}{2}}$

where $𝐗$ and $𝜳$ are $p \times p$ positive definite matrices, $| \cdot |$ is the determinant, Γ_p(⋅) is the multivariate gamma function, and $𝐈_{p}$ is the p × p identity matrix.

Properties

Construction of the distribution

The standard matrix F distribution, with an identity scale matrix $𝐈_{p}$ , was originally derived by.^[1] When considering independent distributions,

$𝜱_{1} \sim 𝒲 (𝐈_{p}, ν)$ and $𝜱_{2} \sim 𝒲 (𝐈_{p}, δ + k - 1)$ , and define $𝐗 = {𝜱_{2}}^{- 1 / 2} 𝜱_{1} {𝜱_{2}}^{- 1 / 2}$ , then $𝐗 \sim ℱ (𝐈_{p}, ν, δ)$ .

If $𝐗 | 𝜱 \sim 𝒲^{- 1} (𝜱, δ + p - 1)$ and $𝜱 \sim 𝒲 (𝜳, ν)$ , then, after integrating out $𝜱$ , $𝐗$ has a matrix F-distribution, i.e.,

$f_{𝐗 | 𝜱, ν, δ} (𝐗) = \int f_{𝐗 | 𝜱, δ + p - 1} (𝐗) f_{𝜱 | 𝜳, ν} (𝜱) d 𝜱 .$
This construction is useful to construct a semi-conjugate prior for a covariance matrix.^[3]

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

Marginal distributions from a matrix F distributed matrix

Suppose $𝐀 \sim F (𝜳, ν, δ)$ has a matrix F distribution. Partition the matrices $𝐀$ and $𝜳$ conformably with each other

𝐀 = [\begin{matrix} 𝐀_{11} & 𝐀_{12} \\ 𝐀_{21} & 𝐀_{22} \end{matrix}], 𝜳 = [\begin{matrix} 𝜳_{11} & 𝜳_{12} \\ 𝜳_{21} & 𝜳_{22} \end{matrix}]

where $𝐀_{i j}$ and $𝜳_{i j}$ are $p_{i} \times p_{j}$ matrices, then we have $𝐀_{11} \sim F (𝜳_{11}, ν, δ)$ .

Moments

Let $X \sim F (𝜳, ν, δ)$ .

The mean is given by: $E (𝐗) = \frac{ν}{δ - 2} 𝜳 .$

The (co)variance of elements of $𝐗$ are given by:^[3]

cov (X_{i j}, X_{m l}) = Ψ_{i j} Ψ_{m l} \frac{2 ν^{2} + 2 ν (δ - 2)}{(δ - 1) (δ - 2)^{2} (δ - 4)} + (Ψ_{i l} Ψ_{j m} + Ψ_{i m} Ψ_{j l}) (\frac{2 ν + ν^{2} (δ - 2) + ν (δ - 2)}{(δ - 1) (δ - 2)^{2} (δ - 4)} + \frac{ν}{(δ - 2)^{2}}) .

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 $p = 1$ (i.e. univariate) and $𝜳 = 1$ , and $x = 𝐗$ , the probability density function of the matrix F distribution becomes the univariate (unscaled) F distribution:
$f_{x ∣ ν, δ} (x) = B {(\frac{ν}{2}, \frac{δ}{2})}^{- 1} {(\frac{ν}{δ})}^{ν / 2} x^{ν / 2 - 1} {(1 + \frac{ν}{δ} x)}^{- (ν + δ) / 2},$

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

References

1 2 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.
↑ 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.
1 2 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.
1 2 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.
↑ 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.
↑ 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.
↑ 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.

[olkinrubin1964-1] 1 2 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.

[dawid1981-2] 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.

[mulderpericchi2018-3] 1 2 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.

[williamsmulder2020-4] 1 2 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.

[tan1969-5] 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.

[perez2017-6] 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.

[gelman2006-7] 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.

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