\section{Isochrone Fitting}\label{sec:isoFit}
We fit pairs of isochrones to the HUGS data for NGC 2808 using \fidanka, as
described in \S \ref{sec:fidanka}. \textbf{As was mentioned in \S
\ref{sec:sub:fiducial}, the method used by \fidanka only consideres two filters --- in
the case of this work F275W and F814W --- and therefore might be unable to
distinguish between populations separated only in the higher-dimensional space
of a chromosome map. For further discussion of why we adopt this method, despite it limits, we refer the reader to \S \ref{sec:sub:numpops}.} Two isochrones, one for P1 and one for P2 are fit
simultaneously. These isochrones are constrained to have distance modulus,
$\mu$, and color excess, E(B-V) which agree to within 0.5\% and an ages which
agree to within 1\%. Moreover, we constrain the mixing length, $\alpha_{ML}$,
for any two isochrones in a set to be within 0.5 of one and other. For each 
isochrone set we optimize $\mu_{P1}$, $\mu_{P2}$, $E(B-V)_{P1}$, $E(B-V)_{P2}$, Age$_{P1}$, and $Age_{P2}$ in order to reduce the
$\chi^{2}$ distance ($\chi^{2} = \sum\sqrt{\Delta \text{color}^{2} + \Delta
\text{mag} ^{2}}$) between the fiducial lines and the isochrones. Because we
fit fiducial lines directly, we do not need to consider the binary population
fraction, $f_{bin}$, as a free parameter.

The best fit isochrones are shown in Figure \ref{fig:BestFitResults} and
optimized parameters for these are presented in Table \ref{tab:BestFitResults}.
{\bf The initial guess for the age of these populations was locked to 12 Gyr
and the initial extinction was locked to 0.5 mag. The initial guess for the
distance modulus was determined at run time using a dynamic time warping
algorithm to best align the morphologies of the fiducial line with the target
isochrone. This algorithm is explained in more detail in the \fidanka
documentation under the function called \texttt{guess\_mu}}. We find helium mass
fractions that are consistent with those identified in past literature
\citep[e.g.][]{Milone2015}. Note that our helium mass fraction grid has a
spacing of 0.03 between grid points and we are therefore unable to resolve
between certain proposed helium mass fractions for the younger sequence (for
example between 0.37 and 0.39). {\bf We also note that the best fit mixing
length parameters which we derive for P1 and P2 do not agree within
their uncertainties. This is not surprising, as the much higher mean molecular mass
of P2 --- when compared to P1, due to population P2's larger
helium mass fraction --- will result in a steeper adiabatic temperature
gradient}. 

\begin{figure*}
  \centering
  \includegraphics[width=0.9\textwidth]{BestFitResults.pdf}
  \caption{Best fit isochrone results for NGC 2808. The best fit P1 and P2
  models are shown as black lines. The following 50 best fit models are
  presented as grey lines. The dashed black line is fit to P1, while
  the solid black line is fit to P2.}
  \label{fig:BestFitResults}
\end{figure*}

\begin{table*}
  \centering
  \begin{tabular}{c | c c c c c c}
    \hline
    Population & Age & Distance Modulus & Extinction & Y & $\alpha_{ML}$ & $\chi^{2}_{\nu}$\\
    & [Gyr] & & [mag] & & &\\
    \hline
    \hline
    P1 & 12.996$^{+0.87}_{-0.64}$ & 15.021 & 0.54 & 0.24 & 2.050 & 0.021\\
    P2 & 13.061$^{+0.86}_{-0.69}$ & 15.007 & 0.537 & 0.39 & 1.600 & 0.033 \\
    \hline
  \end{tabular}
  \caption{Best fit parameters derived from fitting isochrones to the fiducual
  lines derived from the NCG 2808 photometry. The one sigma uncertainty
  reported on population age were determined from the 16th and 84th percentiles
  of the distribution of best fit isochrones ages.}
  \label{tab:BestFitResults}
\end{table*}


Past literature \citep[e.g. ][]{Milone2015, Milone2018} has found helium mass
fraction variation from the redmost to bluemost populations of $\sim 0.12$.
Here we find a helium mass fraction variation of 0.15 that, given the spacing
of the helium grid we use {\em is consistent with these past results}.
\textbf{The helium mass fractions we derive for P1 and P2 are consistent with
those of populations A and E in \cite{Milone2015}; however, populations B+C and
D in that study are more prominently separated in the F275W-F814W colorband.
The inferred helium mass fractions for P1 and P2 are not consistent  
with those reported for populations B+C and D.}

\subsection{The Number of Populations in NGC 2808}\label{sec:sub:numpops} In
order to estimate the number of populations that ideally fit the NGC 2808
F275W-F814W photometry without overfitting the data we make use of silhouette
analysis \citep[][and in a similar manner to how \citet{Valle2022} perform
their analysis of spectroscopic data]{ROUSSEEUW198753, Shahapure2020}. We find
the average silhouette score for the hypothesizes that there are two, three,
four, or five population in each magnitude bin. We preform this analysis over
all magnitude using routines built into the standard Python module
\texttt{sklearn}. Figure \ref{fig:clusterAn} (top) shows the silhouette
analysis results and that two populations fit the photometry most ideally. This
result is in line with what our BGMM model predicts for the majority of the
CMD.

\begin{figure}
  \centering
  \includegraphics[width=0.45\textwidth]{ClusterAnalysis-275-814.pdf}
  \includegraphics[width=0.45\textwidth]{ClusterAnalysis-336-438.pdf}
  \caption{Silhouette analysis for NGC 2808 F275W-F814W (top) and F336W-F438W
  (bottom) photometry. The Silhouette scores are an average of score for each
  magnitude bin. Scores have been normalized to indicate the most
  well-distinguished (+1) to least well-distinguished (-1) hypothesizes.}
  \label{fig:clusterAn}
\end{figure}

{\bf While we make use of a purely CMD-based approach in this work, other
literature has made use of chromosome maps. These consist of implicitly
verticalized pseudo colors. In the chromosome map for NGC 2808 there may be
evidence for more than two populations; further, the chromosome maps used
include information from additional filters (F336W and F438W) which we do not
use in our CMD approach. We preform the same analysis on the F336W-F438W CMD
using \fidanka as we do on the F275W-F814W CMD. While the clustering algorithm
does find a more strongly distinguished potential third population  using these
filters (Figure \ref{fig:clusterAn} bottom), the two population hypothesis remains strongly preferred. Moreover, the
process of transforming magnitude measurements into chromosome space results in
dramatically increased uncertainties for each star. We find a mean fractional
uncertainty for chromosome parameters --- $\Delta_{F275W,F814W}$ and
$\Delta_{CF275W,F336W,F438W}$ --- of $\approx1$ (Figure
\ref{fig:chromosomeError}) when starting with magnitude measurements having a
mean best-case (i.e. where the uncertainty is assumed to only be due to Poisson statistics)
fractional uncertainty of $\approx 0.0005$. Fractional uncertainties for
chromosome parameters were calculated via standard propagation of uncertainty.
Because of how \fidanka operates, i.e. resampling a probability distribution for
each star in order to identify clusters, we are unable to make statistically
meaningful statements from the chromosome map}

\begin{figure}
  \centering
  \includegraphics[width=0.45\textwidth]{ChromosomeMapFractionalErrorDist.pdf}
  \caption{\textbf{Fractional error distribution of $\Delta_{F275W,F814W}$
  (top) and $\Delta_{CF275W,F336W,F438W}$}. The vertical line near 0 in each
  figure indicates the mean fractional error of the magnitude measurements used
  to find the chromosome parameters.}
  \label{fig:chromosomeError}
\end{figure}



% \subsection{ACS-HUGS Photometric Zero Point Offset}
% The Hubble legacy archive photometry used in this work is calibrated to the
% Vega magnitude system. However, we have found that the photometry has a
% systematic offset of $\sim0.026$ magnitudes in the F814W band when
% compared to the same stars in the ACS survey (Figure \ref{fig:offset}). The
% exact cause of this offset is unknown, but it is likely due to a difference in
% the photometric zero point between the two surveys. A full correction of this
% offset would require a careful re-reduction of the HUGS photometry, which is
% beyond the scope of this work. We instead recognize a 0.02 inherent uncertainty
% in the inferred magnitude of any fit when comparing to the ACS survey. This
% uncertainty is small when compared to the uncertainty in the
% distance modulus and should not affect the conclusion of this
% paper. 

% \begin{figure*}
%   \centering
%   \includegraphics[width=0.90\textwidth]{src/figures/photometricOffset.pdf}
%   \caption{(left) CMD showing the photometric offset between the ACS and HUGS data for NGC 2808. CMDs have been randomly subsampled and colored by point density for clarity. (right) Mean difference between the color of the HUGS and ACS fiducual lines at the same magnitude. Note that the ACS data is systematically bluer than the HUGS data.}
%   \label{fig:offset}
% \end{figure*}

% The oberved photometric offset between ACS and HUGS reductions introduces a
% systematic uncertainity when comparing parameters derived from isochrone fits
% to ACS data vs those fit to HUGS data. Specifically, this offset introduces a
% $\sim 2 Gyr$ uncertainity when comparing ages between ACS and HUGS. Moreover,
% for two isochrone of the same age, only seperated by helium mass fraction, a
% shift of the main sequence turn off of is also expected. Figure \ref{fig:HeMO}
% shows this shift. Note a change in the helium mass fraction of a model by 0.03
% results in an approximate 0.08 magnitude shift to the main sequence turn off
% location. This means that the mean 0.026 magnitude offset we find in between
% ACS and HUGS data corresponds to an additional approaximate 0.01 uncertainity
% in the derived helium mass fraction when comparing between these two datasets. 

% \begin{figure}
%   \centering
%   \includegraphics[width=0.45\textwidth]{src/figures/HeliumMeanOffset.pdf}
%   \caption{Main sequence turn off magnitude offset from a guage helium mass fraction (Y=0.30 chosen). All main sequence turn off locations are measured at 12.3 Gyr}
%   \label{fig:HeMO}
% \end{figure}
