Practicalities

• https://github.com/JohannesBuchner/BXA
• uses MultiNest (Feroz&Hobson 2008)
• via PyMultiNest

Fitting in PyXspec

from xspec import *
Fit.statMethod = 'cstat'
Plot.xAxis = 'keV'
s = Spectrum('example-file.fak')
s.notice("0.2-8.0")

m = Model("pow")
# set parameters range : val, delta, min, bottom, top, max
m.powerlaw.norm.values = ",,1e-10,1e-10,1e1,1e1" # 10^-10 .. 10
m.powerlaw.PhoIndex.values = ",,1,1,5,5"         #      1 .. 3

m.fit()

Analysis in PyXspec

from xspec import *
Fit.statMethod = 'cstat'
Plot.xAxis = 'keV'
s = Spectrum('example-file.fak')
s.ignore("**"); s.notice("0.2-8.0")

m = Model("pow")
# set parameters range : val, delta, min, bottom, top, max
m.powerlaw.norm.values = ",,1e-10,1e-10,1e1,1e1" # 10^-10 .. 10
m.powerlaw.PhoIndex.values = ",,1,1,5,5"         #      1 .. 3

import bxa.xspec as bxa
transformations = [
	bxa.create_uniform_prior_for( m, m.powerlaw.PhoIndex),
	bxa.create_jeffreys_prior_for(m, m.powerlaw.norm)       ]

bxa.standard_analysis(transformations, 
	outputfiles_basename = 'simplest-',)

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 $\hat{L}$

Nested Sampling

draw randomly uniformly 200 points
always remove least likely point, replace with a new draw of higher likelihood


converges to maximum likelihood, stops when flat


how to draw new points efficiently?
MultiNest does it via clustering and ellipses

Nested Sampling with MultiNest

explores the problem in

• high dimensions (3-20)
• handles multiple maxima
• handles peculiar shapes
• runs efficiently to convergence
  (typically 10000-40000 points)

measures and describes shapes (like MCMC)

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