Expected Fisher information of unconstrained \(\boldsymbol{p}\)

For the multivariate-normal mean–covariance map of the DeCovarT convolution, \(\boldsymbol{y}\sim\mathcal{N}_{G}(\boldsymbol{\mu}\boldsymbol{p}, \boldsymbol{\Sigma}(\boldsymbol{p}))\) with \(\boldsymbol{\Sigma}(\boldsymbol{p})=\sum_j p_j^{2}\boldsymbol{\Sigma}_j\) and precision \(\boldsymbol{\Omega}(\boldsymbol{p})=\boldsymbol{\Sigma}(\boldsymbol{p})^{-1}\), the expected Fisher information has entries $$ I(\boldsymbol{p})_{jk} = \boldsymbol{\mu}_{\cdot j}^{\top} \boldsymbol{\Omega}(\boldsymbol{p}) \boldsymbol{\mu}_{\cdot k} + 2 p_j p_k\, \mathrm{tr}\bigl( \boldsymbol{\Omega}(\boldsymbol{p})\boldsymbol{\Sigma}_j \boldsymbol{\Omega}(\boldsymbol{p})\boldsymbol{\Sigma}_k \bigr). $$ The first summand is the mean contribution (an \(\boldsymbol{\Omega}\)-inner product of signature columns); the second is the covariance contribution of the quadratic map \(\boldsymbol{p}\mapsto\boldsymbol{\Sigma}(\boldsymbol{p})\). See the multivariate-normal formula on https://en.wikipedia.org/wiki/Fisher_information#Multivariate_normal_distribution.

Usage

expected_fisher_unconstrained(p, mean_signature_matrix, Sigma)

Arguments

p

Numeric proportions on the open simplex.

mean_signature_matrix

Mean signature \(\boldsymbol{\mu}\) (\(G\times J\)).

Sigma

Cell-type covariances \(G\times G\times J\).

Value

Symmetric \(J\times J\) expected Fisher information matrix \(I(\boldsymbol{p})\).

See also

vcov_ilr_delta(), vcov.decovart_fit(), confint.decovart_fit(), .inner_product()

Examples

# Two cell types, bivariate Gaussians: I(p) is 2 x 2.
p <- c(0.6, 0.4)
mu <- cbind(c(0, 0), c(3, 0))
Sigma <- array(c(diag(2), diag(2)), dim = c(2, 2, 2))
I <- expected_fisher_unconstrained(p, mu, Sigma)
I
#>          [,1]      [,2]
#> [1,] 5.325444  3.550296
#> [2,] 3.550296 19.674556