Set scale parameter for scale mixtures of normals

The default method for setting the scale parameter for function ebnm_normal_scale_mixture.

ebnm_scale_normalmix(
  x,
  s,
  mode = 0,
  min_K = 3,
  max_K = 300,
  KLdiv_target = 1/length(x)
)

Arguments

x

A vector of observations. Missing observations (NAs) are not allowed.

s

A vector of standard errors (or a scalar if all are equal). Standard errors may not be exactly zero, and missing standard errors are not allowed.

mode

A scalar specifying the mode of the prior \(g\).

min_K

The minimum number of components \(K\) to include in the finite mixture of normal distributions used to approximate the nonparametric family of scale mixtures of normals.

max_K

The maximum number of components \(K\) to include in the approximating mixture of normal distributions.

KLdiv_target

The desired bound \(\kappa\) on the KL-divergence from the solution obtained using the approximating mixture to the exact solution. More precisely, the scale parameter is set such that given the exact MLE $$\hat{g} := \mathrm{argmax}_{g \in G} L(g),$$ where \(G\) is the full nonparametric family, and given the MLE for the approximating family \(\tilde{G}\) $$\tilde{g} := \mathrm{argmax}_{g \in \tilde{G}} L(g),$$ we have that $$\mathrm{KL}(\hat{g} \ast N(0, s^2) \mid \tilde{g} \ast N(0, s^2)) \le \kappa,$$ where \(\ast \ N(0, s^2)\) denotes convolution with the normal error distribution (the derivation of the bound assumes homoskedastic observations). For details, see References below.

References

Jason Willwerscheid (2021). Empirical Bayes Matrix Factorization: Methods and Applications. University of Chicago, PhD dissertation.