Statistical bias is an inevitable factor in most measurements.  In many cases, bias transforms can be employed to counter the effect and produce (asymptotically) unbiased estimators.  The most common such transform is the size bias.  An infinitesimal version called zero bias was introduced by Goldstein and Reinert in 1997, and has become a powerful tool in Stein's method (for Gaussian and Poisson approximation).

In this talk, I will discuss recent work (arxiv.org/2403.19860) on free probability analogs of bias transforms.  I will discuss existence and regularity of free zero bias, and somewhat surprising connections to the theory of (freely) infinitely divisible laws, giving a new proof of the free Levy--Khintchine formula in the process.  I will also discuss connections between size bias and a new class: positively freely infinitely-divisible laws, and a new kind of free Levy--Khintchine formula.

Finally, time permitting, I will discuss our ongoing work developing Stein's method in free probability, using free zero bias to prove sharp quantitative free central limit theorems even for some systems with long range correlations.

This is joint work with Larry Goldstein.