ICL from the ELBO for PLN-PCA and ZIP-PLN-PCA models

Integrated Completed Likelihood: the BIC minus the entropy of the variational distribution of the latent variables (higher is better): $$\mathrm{ICL} = \mathrm{BIC} - \mathcal{H}(W) - \mathcal{H}(U).$$

• \(\mathcal{H}(W)\): entropy of the Gaussian variational distribution of \(W\), \(\frac{1}{2}\sum_{i,k} \log \sigma^2_{ik} + \frac{nq}{2}\{1 + \log(2\pi)\}\), where \(\sigma^2 = S\) for the ZIP-PLN-PCA model (S is a variance) and \(\sigma^2 = S^2\) for the PLN-PCA model (S is a standard deviation). Zero when q = 0.

• \(\mathcal{H}(U)\): entropy of the Bernoulli variational distribution of \(U\), \(-\sum_{ij} \{\xi_{ij}\log\xi_{ij} + (1-\xi_{ij})\log(1-\xi_{ij})\}\) (ZIP-PLN-PCA model only).

Usage

ICL(Y, X, fit, q, model = c("ZIPLNPCA", "PLNPCA"))

Arguments

Y

Numeric n x p count matrix (NAs allowed).

X

Design matrix, (n*p) x d (or n x d for a rowwise PLN-PCA design).

fit

Fitted object containing $elbo (output of Miss.ZIPLNPCA or Miss.PLNPCA).

q

Integer, size of the latent space.

model

"ZIPLNPCA" (default) or "PLNPCA".

Value

A single numeric value, the ICL score.

See also

BIC