  Bayesian Estimation Applied to Multiple Species (BEAMS) is implemented in the
BEAMS with Bias Corrections (BBC) framework to produce a redshift-binned Hubble
diagram (HD) for Type Ia supernovae (SNe Ia). BBC corrects for selection
effects and non-SNIa contamination, and systematic uncertainties are described
by a covariance matrix with dimension matching the number of BBC redshift bins.
For spectroscopically confirmed SN Ia samples, a recent "Binning is Sinning"
article (BHS21, arxiv:2012.05900) showed that an unbinned HD and covariance
matrix reduces the systematic uncertainty by a factor of ~1.5 compared to the
binned approach. Here we extend their analysis to obtain an unbinned HD for a
photometrically identified sample processed with BBC. To test this new method,
we simulate and analyze 50 samples corresponding to the Dark Energy Survey
(DES) witha low-redshift anchor; the simulation includes SNe Ia, and
contaminants from core-collapse SNe and peculiar SNe Ia. The analysis includes
systematic uncertainties for calibration, and measures the dark energy equation
of state parameter (w). Compared to a redshift-binned HD, the unbinned HD with
nearly 2000 events results in a smaller systematic uncertainty, in qualitative
agreement with BHS21, and averaging results among the 50 samples we find no
evidence for a w-bias. To reduce computation time for fitting an unbinned HD
with large samples, we propose an HD-rebinning method that defines the HD in
bins of redshift, color, and stretch; the rebinned HD results in similar
uncertainty as the unbinned case, and shows no evidence for a w-bias.
