deletions | additions       diff --git a/Ultimately_we_are_interested_in__.tex b/Ultimately_we_are_interested_in__.tex  index 1b3fffb..223c068 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 to the density of actual parameters. Fortunately, Equation \ref{eqn:convert_from_actual_to_observed_density} provides a simple way of working from the observed to the actual distribution of physical parameters. Differentiating with respect to $b(\Gamma_{\rm obs}, P_{\rm obs})$ and then solving for $\rho(\Gamma_{\rm act}(b), P_{\rm act}(b)) act}(b))$  gives \begin{equation}  \label{eqn:convert_from_observed_to_actual_density}  \rho(\Gamma_{\rm act}(b), P_{\rm act}(b)) = \frac{1}{2} b_{\rm max}^2 f(b) \dfrac{d\rho(\Gamma_{\rm obs}, P_{\rm obs}}{db}  \end{equation}  It's clear that plugging Equation \ref{eqn:n-Pobs_from_uniform_n-Pact} into Equation \ref{eqn:n-Pact_from_n-Pobs} recovers the original uniform distribution. On the surface, though, Equation \ref{eqn:n-Pact_from_n-Pobs} suggests the strange result that for the power-law distribution $n(P_{\rm obs}) \sim P_{\rm obs}^{-2}$, similar to that reported in \citet{Jackson_2015}, the underlying distribution $n(P_{\rm obs}) = 0$. However, the approach here assumes that $P_{\rm obs}$-values only span a finite range, which is violated by the simple $P_{\rm obs}^{-2}$ power-law. Instead, a form similar to Equation \ref{eqn:n-Pobs_from_uniform_n-Pact} can be used to describe a power-law, with the 2 replaced by the desired index.