Estimating differential entropy using recursive copula splitting

A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of the marginal distributions and the joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. Numerical examples demonstrate that the method is accurate for distributions with compact and non-compact supports, which is imperative when the support is not known or of mixed type (in different dimensions). At high dimensions (larger than 20), our method is not only more accurate, but also significantly more efficient than existing approaches

Editing Anti-DNA B Cells by Vλx

Receptor editing is performed by replacement of Vκ genes that contribute to autoreactivity. In addition, the Cκ locus can be deleted by Vκ rearrangement to intronic or 3′ of Cκ RS sequences (also referred to as κ deletion elements). B cells that delete the Cκ can then express λ light chains. However, the λ locus, either of man or mouse, does not allow V gene replacement. Nor does it appear to be deleted. Therefore, editing of autoreactive λ B cells may require alternative pathways. We have found that in anti-DNA heavy chain transgenic mice (tgs) VH3H9/56R, B cells that express anti-DNA receptors comprised of λ1 in association with an anti-DNA heavy chain often coexpress a κ chain that prevents DNA binding. We speculate that such isotypically included cells may have low anti-DNA receptor densities, a feature that may lead to self-tolerance. Here we describe a mechanism of preventing DNA binding by expression of a rarely used member of the Vλ family, Vλx. The λx B cells of the tgs also express CD25 and may represent B cells that have exhausted light chain editing possibilities