GRBs as standard candles: There is no "circularity problem" (and there
  never was)
  The 2002 discovery of the "Amati Relation" of GRB spectra created the
possibility that this and other correlations of GRB phenomenology might be used
to make GRBs into standard candles. One recurring apparent difficulty with this
program has been that some of the primary observational quantities to be fit as
"data" - the isotropic-equivalent prompt energy $E_{iso}$ and the
collimation-corrected "total" prompt energy energy $E_{\gamma}$ - depend for
their construction on the very cosmological models that they are supposed to
help constrain. This is the so-called "circularity problem" of standard candle
GRBs. This paper is intended to point out that the circularity problem is not
in fact a problem at all, except to the extent that it amounts to a
self-inflicted wound. It arises essentially because of an unfortunate choice of
data variables, such as $E_{iso}$, which are unnecessarily model-dependent. If,
instead, the empirical correlations of GRB phenomenology which are formulated
in source-variables are {\it mapped to the primitive observational variables}
(such as fluence) and compared to the observations in that space, then all
circularity disappears. I also indicate here a set of procedures for encoding
high-dimensional empirical correlations in a "Gaussian Tube" smeared model that
includes both the correlation and its intrinsic scatter, and how that
source-variable model may easily be mapped to the space of primitive
observables and fashioned into a likelihood. I discuss the projections of such
Gaussian tubes into sub-spaces, which may be used to incorporate data from GRB
events that may lack some element of the data (for example, GRBs without
ascertained jet-break times). In this way, a large set of inhomogeneously
observed GRBs may be assimilated into a single analysis, so long as each
possesses at least two correlated data attributes.
