Distributions of Fitted Values vs Input Values and Estimated Errors.

The true parameters are taken (or inferred) from the 3EG catalog assuming a power-law spectrum. See the description of the default parameterization in the Likelihood tutorial).

Error estimates for any single fit are given by taking the square root of the diagonal elements of the covariance matrix, which in turn is estimated as the inverse Hessian of the -log-likelihood. Note that the definition of the confidence interval corresponds to a specified change in the log-likelihood along a given direction in parameter space. The estimates obtained from the inverse Hessian are only accurate insofar as the likelihood surface at the local minimum can be accurately represented as a quadratic function of the model parameters. The error estimates reported below are averaged over each set of Monte Carlo trials.

Binned Analysis

Counts maps for the Crab_Pulsar, the three anticenter sources, and extragalactic diffuse tests are prepared for the following geometry:
E _min = 30, E _max = 3e5 MeV, 39 bins, logrithmically spaced
RA _center = 83 ^o, eighty 0.5 ^o pixels
Dec _center = 22 ^o, eighty 0.5 ^o pixels

A single point source: Crab_Pulsar

Three strong sources in the Galactic anticenter region: PKS 0528+134, Crab, Geminga

Here is a FITS image of the count map, summed over energies, used in these fits.

• PKS0528+134
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  13.65 +/- 1.68	14.37 +/- 1.70	-2.46 +/- 0.09	-2.55 +/- 0.13

• Crab
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  27.00 +/- 2.14	26.74 +/- 2.19	-2.19 +/- 0.05	-2.15 +/- 0.07

• Geminga
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  23.29 +/- 1.73	23.62 +/- 1.88	-1.66 +/- 0.02	-1.66 +/- 0.04

Unbinned Analysis

These analyses use this ROI file.

Crab_Pulsar

Markov Chain Monte Carlo (MCMC)

Here is a constrasting analysis. Using Markov Chain Monte Carlo, we sample the posterior distribution of the fit parameters for a single trial. (NB: Since the prior distributions of the fit parameters are assumed to be flat, the posterior distribution is the likelihood function.) An advantage of using the Markov Chain is that binning over any single parameter in the chain automatically marginalizes over the remaining parameters.

The blue curves are the best-fit parameter values and error estimates obtained from the inverse Hessian represented as Gaussian functions. The red curves are Gaussians fit to the histograms.

PKS 0528+134, Crab, & Geminga

• PKS0528p134
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  13.65 +/- 1.76	14.62 +/- 1.75	-2.46 +/- 0.11	-2.52 +/- 0.11

• Crab
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  27.00 +/- 2.02	28.10 +/- 2.03	-2.19 +/- 0.05	-2.22 +/- 0.06

• Geminga
  

  Prefactor		Index
  true +/- err. est.	MC	true +/- err. est.	MC
  23.29 +/- 1.68	24.38 +/- 1.74	-1.66 +/- 0.03	-1.68 +/- 0.03

MCMC results for a single trial

As before, the blue curves are the best-fit parameter values and error estimates obtained from the inverse Hessian represented as Gaussian functions; the red curves are Gaussians fit to the histograms.

Extragalactic Diffuse

The ROI file.