## Bayesian evidence for galaxy model selection

Bayesian evidence is the integral of likelihood times prior over prior volume. As well-known, simple harmonic mean approximation almost always fails due to small likelihood outliers, which causes large bias in Bayesian evidence. Laplace approximation seems better if posterior is nicely approximated to multivariate Gaussian. But this is not the case often in real world. Martin proposed new algorithms to deal with this problem by subdividing posterior volume and excluding small likelihood outliers in calculating the Bayesian evidence. This gives robust estimate of Bayesian evidence for simple models.

However it is very difficult to tell whether a galaxy image data supports bulge-disk model or single Sersic model since these models are intrinsically degenerated. Bulge-disk model usually has larger likelihood than single Sersic model because it has more degree of freedom. However bulge-disk model has large volume of parameter space. If a single Sersic galaxy is modelled by bulge-disk and single Sersic profile. Bulge-disk model should have smaller Bayesian evidence than single Sersic model even though it has larger likelihood because large volume of parameter space (prior) gives a penalty. This is Occam’s razor. However it is very difficult to correctly and fully sample prior space with finite length of MCMC if the model is poorly constrained and has strong degeneracy.

Therefore it is really necessary to carefully choose or parametrize model parameter to avoid this strong degeneracy and obtain robust estimate of Bayesian evidence. I have run GALPHAT on the ensemble of galaxies composed of one and two-component and compared different algorithms to estimate the Bayesian evidence to see if we can practically use the Bayesian evidence ratio to distinguish galaxy models.