# Analysing the results

At the end of the run, Lensed will output a summary of the statistics and a table for the parameter values. The table columns give the individual marginal mean values, standard deviations (sigma), maximum likelihood (ML) and maximum a posteriori (MAP) parameters.

summary

log-evidence: -5123.3227 ± 0.4202
max log-like: -5061.6840
min chi²/dof: 1.0136

parameters

parameter         mean       sigma          ML         MAP
----------------------------------------------------------
x_L            50.5049      0.0118     50.5107     50.5001
y_L            50.4907      0.0118     50.4901     50.5116
r_L            17.4325      0.0062     17.4299     17.4340
q_L             0.7491      0.0016      0.7487      0.7518
\theta_L       45.0175      0.0821     44.9905     44.9403
x_S            49.0356      0.0082     49.0414     49.0461
y_S            51.7509      0.0074     51.7502     51.7606
r_S             3.8515      0.1866      3.8357      3.8413
mag_S          -3.0694      0.0365     -3.0691     -3.0547
n_S             3.1509      0.0818      3.1441      3.1506
q_S             0.8921      0.0135      0.8868      0.8795
\theta_S       31.4591      4.0208     31.0952     28.3584


This table is meant to provide a quick overview of the results, but often does not tell the whole story of the parameter space. Lensed can further provide a number of output files which contain the full posterior distribution. These files can be analysed with statistical tools from third parties, such as GetDist.

## The posterior distribution

A number of output files for the posterior distribution are created when Lensed runs with the output = true configuration option:

• <root>.txt: The chains file, compatible with GetDist. Each row contains one sample from the posterior. The (2 + number of parameters) columns contain the sample probability, -2*loglikehood, and parameter values. The sample probability is the sample prior mass multiplied by the likelihood, and normalized by the evidence.
• <root>post_equal_weights.dat: This file contains equally weighted samples from the posterior. Each row is thus a randomly drawn sample from the posterior distribution. The columns are the parameter values, followed by their likelihood value.
• <root>stats.dat: This file contains a human-readable summary of the posterior distribution. Contains the log-evidence, parameter mean values and standard deviations, together with their maximum likelihood and maximum a posteriori. If there are multiple modes, the statistics are given globally and for each of the individual modes.
• <root>post_separate.dat: This file is only created if the posterior is multi-model. It contains the posterior samples for each mode, separated by 2 blank lines. The format is the same as for the <root>.txt file.
• <root>summary.txt: This file contains (number of modes + 1) rows. The first row contains the global statistics for the posterior. Each subsequent row is a single mode, with the following (number of parameters * 4 + 2) values: mean, standard deviation, maximum likelihood, and maximum a posteriori for each parameter, together with the local log-evidence and maximum log-likelihood values of the mode. If importance nested sampling is used, the first row contains the INS log-evidence estimate at the end.

## Analysis with GetDist

GetDist is a tool for analysing and plotting posterior distributions obtained from MCMC chains. It was written by Antony Lewis and turned into a standalone Python package, after being part of the CosmoMC suite.

Installing GetDist is straightforward using the Python package manager pip:

$pip install GetDist  For more information on how to install GetDist, and a list of requirements, see the pages of the Python package and the project on GitHub. Once GetDist is installed, it can be used to analyse the output of Lensed right away. The syntax is $ GetDist.py [paramfile] [root]


where either the [paramfile] or the [root] can be omitted. If no parameter file is given, default values are used. If no root is given, it must have been specified in the parameter file.

A sample parameter file for GetDist is available at extras/getdist.ini in the Lensed root directory. It contains sensible defaults for analysing the results of Lensed. Using this file, one of the provided examples can be analysed (from the examples folder) using

examples$GetDist.py ../extras/getdist.ini chains/test_sersic_bulge-  The output of the analysis can be found in the plots/ subfolder. Besides analysing the posterior and finding confidence intervals and summary statistics, GetDist can be used to create scripts for plotting one-dimensional marginal distributions, as well as two-, three-dimensional, and triangle plots for the correlations between parameters. Note that GetDist does not create the plots directly. It is necessary to run the Python files at plots/<root>*.py, for example $ python plots/test_sersic_bulge-_tri.py


Below there are a number of sample plots from GetDist.

Sample plots from GetDist
One-dimensional marginal distribution
Two-dimensional correlations with confidence intervals
Two-dimensional correlations
Triangle plot