deletions | additions       diff --git a/Ultimately_we_are_interested_in__.tex b/Ultimately_we_are_interested_in__.tex  index eea0d48..576e834 100644  --- a/Ultimately_we_are_interested_in__.tex  +++ b/Ultimately_we_are_interested_in__.tex 
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Ultimately, we are interested in converting the density of observed parameters to the density of actual parameters, and Equation \ref{eqn:convert_from_actual_to_observed_density} provides a way to do that. The equation involves an integral over $b$, which, given $\Gamma_{\rm obs}$ and $P_{\rm obs}$, represents a fixed curve in $\Gamma_{\rm act}-P_{\rm act}$. In other words, $\Gamma_{\rm obs}$ and $P_{\rm obs}$ define a locus of points for $\Gamma_{\rm act}$ and $P_{\rm act}$, and the integral over $b$ involves traveling along the locus from the point $(\Gamma_{\rm act}, P_{\rm act}) = (\Gamma_{\rm obs}, P_{\rm obs})$ up to $(\Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}, P_{\rm max})$. In fact, any points $(\Gamma_{\rm obs}^\prime, P_{\rm obs}^\prime)$ satisfying Equation \ref{eqn:P_obs_Gamma_obs}, $(P_{\rm obs}^\prime\ \Gamma_{\rm obs}^{\prime 2}) = (\Gamma_{\rm obs}^2\ P_{\rm obs}^2)$, lies lie  on this locus of points. Thus, locus. Consequently,  the only difference between $\rho(\Gamma_{\rm obs}^\prime, P_{\rm obs}^\prime)$ and $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ is where on the track the integral starts -- the integrals for both end at the same point, $(\Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}, P_{\rm max})$. Each track is also  anchored at the point $(\Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm min} \right)^{1/2}, P_{\rm min})$. FigureFigure  \ref{fig:integration_path} illustrates the integration track, and we have taken $\Gamma_0 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm min} \right)^{1/2}$ and $\Gamma_1 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}$.\begin{eqnarray}  \label{eqn:convert_from_observed_to_actual_density}  \rho({\rm act}) &=& \frac{1}{2} b_{\rm max}^2\ f^{-1}\ b^{-1}\ \dfrac{d\rho({\rm obs})}{db}\\  &=& \frac{1}{2} b_{\rm max}^2\ f^{-1}\ b^{-1}\ \left[ \dfrac{d \Gamma_{\rm obs}}{db} \left( \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) + \dfrac{d P_{\rm obs}}{db} \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} \right) \right] \\  &=& b_{\rm max}^2\ f^{-1}\ \left[ \left( \dfrac{\Gamma_{\rm obs}}{2} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} + \left(P_{\rm obs} - P_{\rm min} \right) \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} \right]_{{\rm obs} \rightarrow {\rm act}},  \end{eqnarray}  where the following substitutions should be made: $\Gamma_{\rm obs} = $