## 5. Joint probabilitiesAs 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 - are allowed to vary from galaxy to galaxy. This is done by
taking the
*product*of the individual Bayes factors from Table 2 for all objects. - vary for all galaxies simultaneously. This is obtained by taking
the product of the likelihood densities to get
the density distribution for the
*joint*mean*likelihood*.
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
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 H 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 © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |