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\section{Time Delay Estimation Metrics}  \label{sec:motivation}  The primary application of time delay lenses is cosmography: inferring cosmological parameters from the measurement of distance in the universe. The gravitational lens ``time delay distance'' \citep[e.g.][]{Suy++2012} is primarily sensitive to the Hubble constant, $H_0$. While we expect the LSST lens sample to also provide interesting constraints on other cosmological parameters, notably the curvature density and dark energy equation of state, as a first approximation we can quantify cosmographic accuracy via $H_0$. Certainly, an accurate measurement of $H_0$ would be a highly valuable addition to a joint analysis, provided it was precise to around 0.2\% \citep[and hence competitive with Stage IV BAO experiments, for example][]{Weinberg, SuyuWhitePaper}. example][]{Weinberg,SuyuWhitePaper}.  The analysis we wish to emulate therefore is a joint inference of $H_0$ given a sample of $N$ observed strong lenses, each providing (for simplicity) a single measured time delay $\Delta t_k$. This $k^{\rm th}$ measurement is encoded in a contribution to the joint likelihood, which when written as a function of all the independently-obtained data $\mathbf{\Delta t}$ is the probability distribution  \begin{equation}