5. Joint probabilities
As already discussed in Sect. 2.3one can consider two cases for the joint probability for a law to be valid in all the sample objects: The free parameters
The resulting values are presented in Table 4, normalised to the largest value. By far the most probable laws are the power law dependence on gas density - with the exponent being different for each galaxy - and the linear law . The other parameter-free laws can definitely be excluded. The one-parameter laws have factors which are at least one order of magnitude lower than those for the linear law, with and being by far more probable than the others. Of the remaining laws, can safely be excluded. Interestingly, the law does rather well; probably this is because deviations from the average, if concentrated near the centre, can be covered by this law very well by taking small positive or negative y. The most general law, with two parameters, is unlikely, but not the worst one.
Table 4. Joint Bayes factors, from the product of individual Bayes factors and from the joint likelihood density
One notices that the products usually give lower probabilities than the ones derived from the joint likelihood. This is because the two approaches have different sensitivities to a change of the size of the parameter space: In the first approach, the normalisation of the parameter priors enters in each individual Bayes factor. If e.g. for all twelve objects the main contributions to the mean likelihood were contained in the parameter space, doubling of its volume would result in a reduction of the product by a factor . The probabilities from the joint likelihood would be merely halved.
The confidence regions obtained from the joint likelihood are presented in Fig. 4, using a smaller parameter space in order to bring out the details. The standard factor for the conversion of measured CO surface brightness into H2 surface density (cf. next section) is assumed. Notice that such a region must not be an ellipse and may even consist of two disconnected regions. The tongue at the lower left is due to the increase of the density towards and caused by NGC 5457.
One obtains a region which is rather narrowly ( and ) confined around a nearly linear dependence of the SFR on gas density, and independent of radial distance. The best value for the gas exponent and the confidence interval for agree well with the findings of Kennicutt (1989). The linear law, , is well within this 90 per cent confidence region, and thus has one of the largest joint Bayes factors (cf. Table 4). From both approaches for the joint probability, we conclude that with the standard conversion factor there is no need to consider any explicit radial dependence of the SFR.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998