deletions | additions       diff --git a/Ultimately_though_we_re_interested__.tex b/Ultimately_though_we_re_interested__.tex  index aad7fd3..1b3fffb 100644  --- a/Ultimately_though_we_re_interested__.tex  +++ b/Ultimately_though_we_re_interested__.tex 
...
Ultimately, though, we're we are  interested in working from converting  the density of  observeddistribution back  to the underlying distribution. density of actual parameters.  Fortunately, the combined recovery bias and distortion expression Equation \ref{eqn:convert_from_actual_to_observed_density}  provides a simple way of working from the observed  to do this:  \begin{equation}\label{eqn:n-Pact_from_n-Pobs}  n(P_{\rm act}) = -P_{\rm act}^{-1} \left( \dfrac{P_{\rm max} - the actual distribution of physical parameters. Differentiating with respect to $b(\Gamma_{\rm obs},  P_{\rm min}}{P_{\rm min}} \right) \left( \frac{d}{dP_{\rm obs}} \right)_{P_{\rm obs} = obs})$ and then solving for $\rho(\Gamma_{\rm act}(b),  P_{\rm act}} \bigg[ n(P_{\rm obs})\ 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}^2 \bigg], obs}}{db}  \end{equation}  where $\left( \frac{d}{dP_{\rm obs}} \right)_{P_{\rm obs} = P_{\rm act}}$ means to calculate the $P_{\rm obs}$-derivative of the expression in square brackets and then replace $P_{\rm obs}$ with $P_{\rm act}$.  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.