deletions | additions       diff --git a/Ultimately_we_are_interested_in__.tex b/Ultimately_we_are_interested_in__.tex  index 576e834..6ccb307 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. Figure \ref{fig:integration_path} illustrates the integration track involved in Equation \ref{fig:integration_path}, and we define the end points of the integration track as $\Gamma_0 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm min} \right)^{1/2}$ and $\Gamma_1 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}$.  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}, $(P_{\rm obs}^\prime\ \Gamma_{\rm obs}^{\prime 2}) = (\Gamma_{\rm obs}^2\ P_{\rm obs}^2)$, lie on this 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 $(\Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm min} \right)^{1/2}, P_{\rm min})$.Figure \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 $\Gamma_1 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}$.