In this chapter we have presented a new fitting algorithm (
PHI
) to perform 2D photometric decom- positions of galaxy images from a Bayesian perspective. The algorithm is implemented to run an adaptive MCMC for a prescribed amount of time, diagnose when adaptation is sufficient, and then run a conventional MCMC with an estimated covariance matrix to better explore the parameter space. Convergence diagnostics are also used to ensure a robust estimation of the target posterior probability distribution.Our approach offers a number of significant advantages for estimating surface brightness profile pa- rameters. Algorithms that use standard downhill optimisation techniques can have five commonly occurring factors which lead to failings in the fitting process: i) Local minima trapping, ii) unreal- istic solutions, iii) reversal of components, iv) indecisiveness to which model to use , and v) bad representation of the final errors.
PHI
addresses each problem as follows:I.
PHI
incorporates a triple layer approach. The first layer uses a blocked adaptive Metropolis algorithm to obtain an estimate of the scale for each parameter in the chain. The second layer uses an adaptive Metropolis algorithm with the purpose of estimating the target covariance matrix. We assume the proposed distribution can be described as multivariate normal distri- bution. The final level uses this calculated covariance matrix to quickly and effectively explore the parameter space reducing the chances of a local minima trap.II. We have implemented a number of priors that aim to allow the parameters to stay realistic and physical (i.e. positive in the case of the dimensions and intensities). These priors are better understood as boundary regions similar to the filtering process used in past work to remove non-physical parameter outcomes.
III. To prevent the reversal of components (i.e. the desired inner component profile switching to fit the outer and vice versa) we use a combination of priors. A Newton-Raphson algorithm
2.6. Summary
determines the crossing points in the total light profile and calculates the dominant component in the centre regions. This prior combination specifies that the bulges of galaxies are better modelled by a Sérsic profile and the discs are described by an exponential profile.
IV. Finally,
PHI
gives the full posterior probability distribution for a set of model parameters. This is a powerful description of the model uncertainties that can be used in further analysis of galaxy structures.For future studies a full Bayesian analysis of galaxy morphologies is essential in unlocking the re- maining unanswered questions about galaxy structures. With the addition of
PHI
into the array of 2D photometric decomposition toolbox we hope to improve our understanding of galaxy properties.3
A Bayesian Approach to 2D Photometric
Decompositions: Applications to synthetic & real
galaxies
3.1
Introduction
In the previous chapter we introduced the new 2D photometric decompositions code,
PHI
. As previ- ously stated, a maximum-likelihood analysis can erroneously imply correlations owing to the com- plexity of the parameter space when dealing with multi-component fits of galaxy images. The me- chanics of the Markov Chain Monte Carlo (MCMC; more specifically the Metropolis-Hastings algo- rithm) allows for a more rigorous exploration of this complex parameter space so as to overcome lo- cal minima. However, this has yet to be tested for 2D galaxy images being fit with multi-components using an MCMC algorithm. In order to better quantify the systematic and random uncertainties we have performed two tests: i) tests using synthetic galaxies and ii) using real galaxies.galaxies
vations as well as to assess biases in the estimated values. With a sample of computer generated galaxies one can gain information about systematic and random uncertainties in photometric de- compositions using various fitting procedures, while learning more about the fitting process as well. Previous studies have also utilised this idea for both single component (Häussler et al. 2007; van der Wel et al. 2012; Newman et al. 2012) and multi-component photometric decompositions (Méndez- Abreu et al. 2008a; Davari et al. 2014; Bruce et al. 2014).
Despite the advantages synthetic galaxies have, they still lack the complexity of real galaxies. Therefore before any scientific analyses can be done, we need to verify that the algorithm can repro- duce and match the parameter estimates given by other codes one the same images. Comparisons of this nature will help reveal differences between the codes as the images and systematics will be the same, and it will also highlight how different systematics will alter the fitting outcomes.
This section describes the use of a synthetic galaxy imager to test the robustness of