Probing the AGN geometry
with Bayesian techniques

Santiago de Chile, May 2015

Johannes Buchner (MPE$\rightarrow$PUC)

in collaboration with A. Georgakakis, K. Nandra, L. Hsu, C. Rangel, M. Brightman, A. Merloni and M. Salvato

Buchner et al. 2014 - arxiv:1402.0004

About me, Johannes Buchner

• Studied Astronomy & Computer Science in Vienna
  • developed MCMC software for stellar pulsation analysis
• Master: Computer Science / Machine Learning in NZ, at a Radio Astronomy institute
  • developed PyMultiNest -- Python nested sampling package
• Doctorate: Astronomy @ Max Planck institute for extr.Physics
  1. Bayesian X-ray spectral analysis with Bayesian model comparison of physically motivated models (today)
  2. Population analysis of Active Galaxies (MCMC/Stan)
  3. published a nested sampling paper in Statistics&Computing (Springer)

Astronomical problem

• Black holes in the centers of all massive galaxies
• matter falls in, spirals, heats, creates radiation. drawing
• $E = M \cdot c^2$
• luminosity $\rightarrow$ mass inflow

Observables

X-ray telescopes produce spectra via Poisson process. $$\cal{L}=\prod_i \lambda_i^{k_i} \cdot \exp\{-\lambda_i\}/k_i!$$

Example - Source 179 z=0.605, 2485 counts

powerlaw

Spectral models

Bayesian inference

$$ p(\theta|D,M) = {p(\theta|M)p(D|\theta,M)\over p(D|\theta,M)} $$ parameter posterior probability distributions. assume $\int p(\theta|D,M)d\theta = 1 \rightarrow \int p(\theta|M)p(D|\theta,M)=p(D|\theta,M)=Z $ closed world assumption within defined parameter space of model M.

$$p(M|D)={p(M)p(D|M)\over p(D)}$$ assume $\sum p(M|D)=1 \rightarrow p(D)=\sum p(D|M)p(M)$ closed world assumption among models.

Bayesian inference

• can distinguish between physical effects (models) and constrain physical parameters consistently
• models with large predictive diversity are punished
• (Py)MultiNest makes it easy to compute Z and posterior distribution for up to 40 parameters, for arbitrary models
• now becoming more common practice in Exoplanet searches, Cosmology, etc.

Parameters: of torus+scattering

• Normalisation ~ luminosity
• Column density $N_H={10}^{20-26}\text{cm}^{-2}$
• Soft scattering normalisation: $f_{scat} < 10\%$
• Photon index: $\Gamma$
• Redshift

Parameters: of torus+scattering

• Uniform: log Normalisation ~ log Luminosity
• Uniform: Column density $\log N_H/\text{cm}^{-2} = {20-26}$
• Uniform: Soft scattering normalisation: $\log f_{scat}={10}^{-10 ... -1}$
• Photon index: informed, from nearby objects, $\Gamma \sim 1.95 \pm 0.15$ (Nandra&Pounds 1994)
• Redshift: informed probability distribution, from other observations.

unobscured powerlaw

absorbed with tiny blob in the LOS

torus+scattering

torus+pexmon+scattering

wabs+pexmon+scattering

Model comparison results

Physically motivated models

Models to compare

Model comparison results

Model comparison

• Can distinguish some models, others are equally probable
• Have made use of
  • local AGN information $\leftrightarrow$ $\Gamma$
  • other $\lambda$ information $\leftrightarrow$ $z$
• Have derived parameter distributions and $Z$ simultaneously
• incorporating uncertainty in $\Gamma$, $z$
• Can tell that it does not know

How can geometries be distinguished?

• How to increase strength of data?
• Combine many objects
• Traditionally: stacking
  • lose information in averaging
  • averaging of different redshifts?
• Here, multiply Z to combine information $$Z_1 = \prod_\mu Z_1^\mu$$

Results for the full CDFS sample

Conclusions

• Possible geometries:
  • Absorbed powerlaw necessary
  • Soft scattering powerlaw detected
  • Additional Compton-reflection detected
  • Torus model preferred (also in CT AGN)
• first use of nested sampling for model comparison & parameter estimation in X-ray astronomy
• Buchner et al. 2014 - arxiv:1402.0004
• coming next: population analysis of the $L$, $N_H$ distribution of Active Galactic Nuclei with non-parametric Bayesian analysis.

Likelihood ratio tests / F-test

• only if special case
• not at borders (feature detection)
• only if $n\rightarrow\infty$
• do not compute it, interpretation is wrong; critisize papers mis-interpreting the statistic

Pragmatic viewpoint

distinguish two models via data can use any statistic

does not have to be probabilistic

• can be counts in the 6keV bin
• determine discriminating threshold via simulations
  • false association rate ($B\rightarrow A$, $A\rightarrow B$)
  • correct association rate ($A\rightarrow A$, $B\rightarrow B$)

Comparison $\hat{L}$ vs. $Z$

red: falsely choose powerlaw, for wabs input

red: falsely choose wabs, for powerlaw input

$Z$ more effective than $\hat{L}$

Connecting

BXA: Bayesian X-ray Analysis

• MultiNest for Sherpa / (Py)Xspec
• installed on ds42 under

  /utils/bxa

• documentation: github.com/JohannesBuchner/BXA

Analysis

Likelihood value evaluated "everywhere"

• ML analysis: find confidence intervals
  • defined via: how often the estimator (maximum) gives the right answer
  • different for different estimators - property of the method
• credible intervals
  • defined via: prob. that the true value is inside this range rather than outside is x%.

results coincide for some choice of prior (usually "flat").

How?

just make histogram of 1/2 columns

contains all correlations

Error propagation: example

• set $N_H$ = 0
• compute intrinsic flux
• compute luminosity
  using $z$ and flux

incorporates uncertainty in $z$ and the parameters!

• just do your calculation with every value instead of one

model inadequacy

• Is the model right?
• Where is the model wrong?
• Systematic effects?
• Discover new physics beyond the model

Common route: residuals

good fit if straight line

Q-Q plot primer

Generate ideas for new models

Summary

• (see 5.1, Appendix 3)
  Parameter estimation:
  explore multiple maxima
  general solution with nested sampling
  
• Model comparison:
  Likelihood ratio is less effective than Z ratios
  (see 5.2, Appendix 2)
  
  computed by nested sampling; has right interpretation
• (see 5.3, Appendix 1)
  Model discovery:
  Q-Q plots + model comparison

arxiv:1402.0004 github.com/JohannesBuchner/BXA