deletions | additions       diff --git a/Time Delay Estimation Metrics.tex b/Time Delay Estimation Metrics.tex  index 337f11f..9587624 100644  --- a/Time Delay Estimation Metrics.tex  +++ b/Time Delay Estimation Metrics.tex 
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{\rm Pr}(\Delta t_k|H_0) = \exp \left[ -\frac{(\Delta t_k - \alpha_k / H_0)}{2(\sigma_k^2 + \sigma_0^2)} \right].  \label{eq:gaussian}  \end{equation}  Here, we have used the general relation that the predicted time delay is inversely proportional to the Hubble constant. Indeed, for a simulated lens whose true time delay $\Delta t_k^*$ is known, we can see that $\alpha_k$ must be equal to the product $(\Delta t_k^* H_0^*)$, where $H_0^*$ is the true value of the Hubble constant (used in the simulation). $H_0$ is the parameter being inferred: how different it is from the true value is of great interest to us. The denominator of the exponent contains two terms, that express the combined uncertainty due to the time delay estimation, $\sigma_k$, but also the uncertainty in the lens model $\sigma_0$  that would have been used to predict the time delay, $\sigma_0$. delay.  \citet{Suy++2013a}, for example, find this to be approximately 5-6\%, and due to two approximately equal 4\% contributions, from the lens model, and the weak lensing effects of mass along the line of sight to the lens. In a large sample, we expect the environment uncertainty to be somewhat lower, as we sample lines of sight that are less over-dense than the systems so far studied. Conservatively, we take $\sigma_0$ to be 5\% per lens. In practice, the probability for the measured time delay given the light curve data will not be Gaussian. However, for simplicity we can still use Equation~\ref{eq:gaussian} as an approximation, by asking for measurements of time delays to be reported as $\Delta t_k \pm \sigma_k$, and then interpreting these two numbers as above.  
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B = \frac{\left( \hat{H_0} - H_0^* \right)}{\sigma_{H_0}}.  \end{equation}  Values of $P$ and $B$ can be computed for any contributed likelihood function, and used to compare the associated measurement algorithms.   Focusing on the likelihood for $H_0$ allows us to do two things: first, derive well-defined targets for the analysis teams to aim for, and second, weight the different lens systems in approximately the right way given our focus on cosmology.