Log-likelihood of one single-cell bulk sample

Gaussian log-likelihood of \(\boldsymbol{y}\sim\mathcal{N}(c\tilde{\boldsymbol{\mu}}\boldsymbol{p}, \phi\boldsymbol{S}_W(\boldsymbol{p}))\) with \(\boldsymbol{S}_W(\boldsymbol{p})=\sum_j p_j\boldsymbol{\Sigma}_{W,j}\). The scales \(c\) and \(\phi\) are held fixed. Up to \(-\tfrac{G}{2}\log(2\pi)\), $$ \ell = -\tfrac{G}{2}\log\phi -\tfrac{1}{2}\log\det\boldsymbol{S}_W(\boldsymbol{p}) -\tfrac{1}{2\phi} \boldsymbol{r}^{\top}\boldsymbol{S}_W(\boldsymbol{p})^{-1}\boldsymbol{r}, $$ \(\boldsymbol{r}=\boldsymbol{y}-c\tilde{\boldsymbol{\mu}}\boldsymbol{p}\). The population objective loglik_multivariate() is a different model: its covariance is \(\sum_j p_j^2\boldsymbol{\Sigma}_j\).

Usage

loglik_single_cell(
  p,
  y,
  mean_signature_matrix,
  Sigma,
  mean_scale,
  variance_scale
)

Arguments

p

Composition of length \(J\). It does not have to sum to 1 (numerical derivatives step off the simplex).

y

Bulk vector of length \(G\).

mean_signature_matrix

Reference mean, \(G\times J\).

Sigma

Cell-level covariances, \(G\times G\times J\).

mean_scale

Positive plug-in \(c=an\).

variance_scale

Positive plug-in \(\phi=a^2 n\).

Value

Scalar log-likelihood.

Examples

mu <- matrix(c(2, 0.2, 0.2, 2, 0.2, 3, 0.2, 3), nrow = 4)
sigma <- array(rep(diag(0.05, 4), 2), c(4, 4, 2))
p <- c(0.7, 0.3)
y <- 20 * drop(mu %*% p)
loglik_single_cell(p, y, mu, sigma, 20, 4)
#> [1] 3.218876