Merge branch 'master' of https://github.com/smsharma/StrongLensing-In…

…ference

draft/lensing-lfi.tex

@@ -414,7 +414,7 @@ \section{Results}
 
 Several interesting features can already be seen in these results. The 95\% CL contours contain the true parameter point, with the overall likelihood surface being strongly correlated with the corresponding image. A smaller projected surface area of the lensed arc, resulting from a smaller host halo or a larger offset between the host and source centers, generally results in a flatter likelihood surface. This is expected, since a smaller host galaxy will contain relatively less substructure, and a smaller host or larger relative offset will result in a smaller effective arc area over which the substructure can imprint itself. The first column of Figure~\ref{fig:individual_predictions} shows an example of such a system. In contrast, the last columns show a system with a relatively massive host and a small offset, producing a symmetric image with a larger effective arc surface area over which the effects of substructure can be discerned. This results in a ``peakier'' inferred likelihood surface, corresponding to a higher sensitivity to $\fsub$ and $\beta$. The second and third columns of Figure~\ref{fig:individual_predictions} correspond to systems with a small, centered and a large, offset halo respectively, and show intermediate sensitivity to substructure properties.
 
-In the spirit of stacking multiple observations, we next consider a simultaneous analysis of multiple lensed images. As discussed in Section~\ref{sec:lfi-inference}, the product of the likelihood maps of the individual images defines the appropriate test statistic. For the purpose of high-level inference, the two-dimensional likelihood maps are hence a good alternatives to probabilistic catalogs, avoiding the complications of a prior dependence and of communicating a complicated transdimensional posterior. In the left panel of Figure~\ref{fig:expected_likelihood}, we show the expected log likelihood ratio surface per-image in the asymptotic limit, with the 1-D slice corresponding to $\beta = -0.9$ shown in the right panel. The 95\% CL expected exclusion limits for 5, 20, and 100 lenses are shown using the dotted, dashed, and solid lines respectively. The procedure can easily be extended to an arbitrarily large collection of lenses.
+In the spirit of stacking multiple observations, we next consider a simultaneous analysis of multiple lensed images. As discussed in Section~\ref{sec:lfi-inference}, the product of the likelihood maps of the individual images defines the appropriate test statistic. For the purpose of population-level inference, these two-dimensional likelihood maps are hence a good alternative way to define a probabilistic catalog over individual observations, avoiding the complications of prior dependence and of communicating a complicated trans-dimensional posterior. In the left panel of Figure~\ref{fig:expected_likelihood}, we show the expected log likelihood ratio surface per-image in the asymptotic limit, with the 1-D slice corresponding to $\beta = -0.9$ shown in the right panel. The 95\% CL expected exclusion limits for 5, 20, and 100 lenses are shown using the dotted, dashed, and solid lines respectively. The procedure can easily be extended to an arbitrarily large collection of lenses.
 
 We find that, at least within the simplifying assumptions of our simulator, an analysis of a few tens of lenses is already sensitive to the overall substructure abundance parameterized by $\fsub$. A larger observed lens sample provides a tighter constraint on substructure properties. Approximately 100 lens images are required to begin resolving $\beta$. The expected exclusion contours are centered around the true values, confirming that our inference methods yield an unbiased estimate of the underlying substructure properties. Note the ``banana'' shape of the expected exclusion limits, which approximately traces the total deflection contributed by substructure. We demonstrate this in Figure~\ref{fig:banana}, where we show a proxy for the total subhalo-induced deflection, $\sum_{\text{subhalos}} 4 \kappa_s r_s$, equal to the space-independent part of Equation~\eqref{eq:nfw_deflection}, and compare it to the expected exclusion limits. In our particular substructure scenario, this proxy can be shown to approximately scale like $\sum_{\text{subhalos}} m_s^{2/3}$. We note that this comparison is schematic, as the subtle effects of substructure over a wide range of masses cannot be quantified through a single number (here, the total deflection).
 
@@ -478,7 +478,7 @@ \section{Conclusions}
 
 Strong lensing offers a unique way to probe the properties and distribution of dark matter on sub-galactic scales through the subtle imprint of substructure on lensed arcs. The high dimensionality of the underlying latent space characterizing substructure poses a significant challenge, however. In this paper, we have introduced a novel simulation-based inference technique based on the ideas introduced in~\citet{Cranmer:2015bka, 1805.00013, 1805.00020, 1805.12244, Stoye:2018ovl} for inferring high-level population properties characterizing the distribution of substructure in an ensemble of galaxy-galaxy strong lenses and demonstrated its feasibility through proof-of-principle examples.
 
-Our results on simulated data demonstrate that this method, based on calibrated likelihood ratio estimators with a machine learning back end, offers a promising way to analyze extended-arc strong lensing images with the goal of inferring properties of dark matter substructure. Our proposed method offers several combined advantages over established techniques. In probing the collective effect of a large number of low-mass, sub-threshold subhalos it can offer sensitivity to the faint end of the subhalo mass function where deviations from the concordance \lcdm paradigm and the effects of new physics are most likely to be expressed. It can naturally be applied to perform fast, principled, and concurrent analyses of a large sample of strong lenses that share a common set of hyperparameters describing the underlying substructure population properties. In efficiently marginalizing out the individual subhalo properties and directly inferring the population-level parameters of interest we are able to sidestep the more expensive two-step procedure of characterizing individual subhalos or constructing probabilistic catalogs before using this to infer higher-level population parameters. Likelihood scans at the level of population-level parameters are thus a suitable alternative to probabilistic catalogs, avoiding a prior dependence as well as the logistical complexity of communicating a complicated trans-dimensional posterior. Furthermore, rigorous selection of lensing images out of a large sample is not necessary within our framework since images with a smaller effective arc area or low overall fidelity simply do not contribute significantly to the simultaneous substructure analysis, and non-detections are just as valuable as detections. Finally, our analysis is performed at the level of image data without incurring loss of information associated with dimensionality reduction.
+Our results on simulated data demonstrate that this method, based on calibrated likelihood ratio estimators with a machine learning back end, offers a promising way to analyze extended-arc strong lensing images with the goal of inferring properties of dark matter substructure. Our proposed method offers several combined advantages over established techniques. In probing the collective effect of a large number of low-mass, sub-threshold subhalos it can offer sensitivity to the faint end of the subhalo mass function where deviations from the concordance \lcdm paradigm and the effects of new physics are most likely to be expressed. It can naturally be applied to perform fast, principled, and concurrent analyses of a large sample of strong lenses that share a common set of hyperparameters describing the underlying substructure population properties. By efficiently marginalizing out the individual subhalo properties and directly inferring the population-level parameters of interest we are able to sidestep the more expensive two-step procedure of characterizing individual subhalos before using them to infer higher-level population parameters. Population-level likelihood scans for individual images are thus a suitable alternative to probabilistic catalogs over subhalos, avoiding both prior dependence as well as the logistical complexity of communicating a complicated trans-dimensional posterior. Furthermore, rigorous selection of lensing images out of a large sample is not necessary within our framework since images with a smaller effective arc area or low overall fidelity simply do not contribute significantly to the simultaneous substructure analysis, and non-detections are just as valuable as detections. Finally, our analysis is performed at the level of image data without incurring loss of information associated with dimensionality reduction.
 
 Although we have focused on a simple proof-of-principle example in this paper, extensions to more realistic scenarios---including more complicated descriptions of the host, source, and substructure populations---are easily admitted within our framework.  The flexibility of the proposed method allows for applications beyond substructure population inference as well. For example, a large lens sample can be used to perform cosmological parameter estimation while accounting for substructure effects and in particular to independently constrain the Hubble constant~\citep{1907.02533,1907.04869} through its dependence on the angular diameter distance scales in lensing systems. In the spirit of \citet{Alsing:2017var}, these techniques can also be  used to learn powerful summary statistics~\citep{1805.12244}.