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where N is the total number of data points, d$_{i}$ is the experimental data value, m$_{i}$ is the mean value predicted by the model and σ$_{i}$ is the error, scale-modified as described below. This expression is applied to each data set, and the total likelihood is given by the product of the individual likelihoods. 

For the data values d$_{i}$ , we use the H β curve, the LOS dispersion, stellar rotation curve and the R − band SB profile from B89 and the extended H I curve from Begeman (1987). We ran several different MCMC chains (described in the next section) making use of different data sets. 

The model values m$_{i}$ are calculated as follows. For the gas, we assume that the observed H β /H I rotation curves follow circular orbits so that the model gas rotation curve can be calculated directly from the potential. For the stellar rotation curve and dispersion profile, we sample the disc and bulge DFs along the line of sight at each data point from B89, appropriately weighted by the M/L ratios. Finally, for the surface brightness we sample the DFs along elliptical annuli around the inclined galaxy centre at an axis ratio determined by the inclination angle (i.e., q = cos i = 0 . 28) to determine the elliptically averaged density and thus the surface brightness in mag arcsec − $^{2}$. We emphasize that in no cases have we corrected the observed profiles for inclination; all inclination effects are accounted for by rotating the model galaxy and sampling along the line of sight. 

For each dataset used in a particular MCMC run, a single noise parameter is imple- mented. This consists of an extra term added in quadrature to each error bar: σ 2 i = σ 2 0 $_{i}$+ f $^{2}$, where σ$_{0}$$_{i}$ is the original experimental error and f is the noise parameter, taken as constant for all points in a given data set (so there is a total of four scale factors). This approach allows us to account for effects that cannot be captured by our models - for example, that of spiral structure on the light profile when the original models are axisymmetric. Noise parameters use a Jeffreys prior: 

<img src='content_image/371669.jpg'>

(Gregory 2005) where f$_{min}$ and f$_{max}$ refer to the minimum and maximum values of the noise parameters adopted for the MCMC run. Otherwise, P ( M ) is uniform for all parameters. 

