For a thorough study of the statistics that yields to our model comparison, see Vegetti and Koopmans, 2009 [25]. Here, I explain the relevant parts that I use in this work.
Bayesian statistics: Bayesian statistics is a framework that has been shown to be powerful to analyze over model parameters and, as in our case, to compare different models objectively. A good introduction to the required basics is given in section 7.4.5. of the book Astrophysical Applications of
Gravi-Figure 2.4: NGC2623 galaxy as used for doing the tests. Squared image with pixel values for the brightness from 0 to 1. [8]
tational Lensing, from the XXIV Canary Islands Winter School of Astrophysics [20]. Here, I introduce the necessary concepts to make our comparison between models, when applicable.
If we consider a set of data d and a model of parameters q, we can express their joint probability distribution, using conditional and marginal distributions, as
p(d, q) = p(q|d)p(d) = p(d|q)p(q) (2.6) from where we can deduce simply the Bayes’s Theorem, which is the base of all Bayesian statistics:
p(q|d) = p(d|q)p(q)
p(d) . (2.7)
In this equation, p(d|q) is the probability of the data given the model, also called Likelihood (L(d|q)); p(q) is the prior probability distribution for q; p(q|d) is the posterior probability distribution for the model given the data; and p(d) is the evidence, henceforward called Z. The evidence tells us the probability of a model to generate the data, including the effect of the prior.
Bayes factor: In order to compare models, the Bayes factor is used. It is supposed that there is no a priori knowledge for both models. I consider this because both models are going to be the same except for one change (parameter, psf, etc.). Therefore, the Bayes factor is the ratio of the evidence of both models.
It can be defined as:
f = Z(M1) Z(M2)
P (M1)
P (M2) ≈ Z(M1)
Z(M2) (2.8)
In equation 2.8, P are the priors and Z are the evidence for each model. Using the value of f to compare between two models, the comparison method that I use rely on the one provided by Kass and Raftery (1995)[19], which is based on Jeffrey’s scale (1961) [17]. Table 2.1 shows this scale which, even though it is only qualitative, can help us to decide between models.
log10f Relation between models 0 to 1/2 Not worth more than a bare mention
1/2 to 1 Substantial
1 to 2 Strong
> 2 Decisive
Table 2.1: Kass and Raftery [19] version of Jeffreys’ scale [17] for a qualitative indication of the meaning of the Bayes factor f .
Chapter 3
Experimental:
Understanding the code
Having introduced the statistical method that we are using, some small trials are done to understand how the program works. Physical aspects of gravitational lens theory (source position, lens axis ratio, etc.) are tested. The outcome is used to localize systematic effects in source reconstruction. The effect of different masks on the same data is used to understand the capability of the reconstruction method. Because of the use of a different mask for each trial, every data set is different from each other. Hence, comparison between models using the evidence can not be done. Comparison of the models with real gravi-tational lens systems is made sometimes to show the ability of the program to reproduce real results.
3.1 Default model
To study the differences that changing one parameter produces, a default model is needed. The values used for these specific forward and inverse problems are shown in Appendices 6 and 7, respectively. With these values, the mock lens and the reconstruction that is achieved can be seen in Figure 3.1.
This first plot shows the final result of the methods explained in Chapter 2.
At the bottom-left of the image, the true source is shown. This is a squared image of a galaxy, which, in this case, is Messier 83 (M83). The pixel brightness is rescaled to range values between 0 and 1. Above the true source, there is the data generated from it using a specific lens model, which shown in Appendix 6 for this case. Next to the true source, the reconstruction of the source in the adaptive grid is shown. Above the reconstruction, the lens image model that the reconstructed source generates. The last column shows the difference between the data and the data model, namely the residual; and lastly the errors are shown (calculated using eq. B22 from Suyu et al, 2006 [24]).
The source reconstruction reproduces the general contour and the position
Figure 3.1: Outcome of the program for source reconstruction with default values. True Source: a squared image of the source with pixel values between 0 and 1. Data: lens image produced by the true source with a defined lens model.
Reconstruction: source reconstruction in an adaptive-grid. Model: lens image produced by source reconstruction. Residual: the difference between data and model. Errors: errors in source reconstruction.
of the true source. However, all the details are lost. For instance, the “sub-structure” at the bottom-right of the source has disappeared. Hence, the re-construction using default values is not really good. For the residuals, our goal is to only see the noise. This would imply the residuals looking homogeneous, meaning that the data model does not exceedingly diverge from the actual data.
In the residuals from Figure 3.1 it can be seen that the top image of the lensed source shows both positive and negative residuals significantly exceeding the noise level. These structures reveal that the model needs to be improved to achieve a good source reconstruction. Chapter 4 is more focused on this.