content_image/355078.jpg
input set with R − band M/L ratios for the bulge and disc. We assume that the M/L ratios are constant with radius, and use the same value for the surface brightness and the stellar kinematic fits. Although the band pass over which the stellar observations were taken are not identical to the R − band, we do not expect the error introduced by this simplification to be significant. 

Thus the parameters that we modify are: the five halo parameters; the exponential disc mass, scale height, scale length, and two Kormendy parameters; all bulge parameters; the central radial dispersion and dispersion profile scale length; and the disc and bulge M/L ratios. In addition, each data set has an associated noise parameter that is allowed to vary (see § 4). We assume the halo to be nonrotating and truncate the disc at 5 kpc (well outside the measured data). The galaxy inclination i is also fixed at 74 $^{◦}$. A list of the relevant parameters that are fit, as well as calculated output quantities, is presented in Table 1. The units used by GalactICS are such that G = 1, but the masses have been converted back to 10 $^{9}$M ⊙ for simplicity. 

## 4.1. Bayes’ Theorem and MCMC 

## 4. Statistical Approach 

The core idea of this study is to derive the joint posterior probability distribution func- tion (PDF) of the galaxy parameters using Bayes’ theorem and MCMC. Bayes’ theorem provides a way to infer the posterior probability of a hypothesis, given some kind of evidence (such as observational data), using the prior probabilities of the evidence and model and a likelihood function (such as a χ 2 function). The joint posterior probability distribution function (PDF) of the model parameters is written as P ( M | D ), where M is the set of model input parameters and D is the set of observational data. Bayes’ theorem may be written as 

<img src='content_image/355078.jpg'>

where P ( M ) is the prior probability of the input parameters, P ( D | M ) is the likelihood of the data given a specific set of model parameters and P ( D ) is the prior probability of the data and functions as a normalization. 

MCMC, in turn, provides a method of surveying the parameter space that rapidly converges to the posterior probability distribution of the input parameters. Our MCMC technique explores the parameter space by way of the Metropolis-Hastings algorithm. Es- sentially, a random walk is constructed in parameter space by sampling a new set of model parameters M ∗ at each step of the chain, and P ( M $^{∗}$| D ) is calculated using the likelihood 

