### Table 4. Joint Bayes factors, from the product of individual Bayes factors and from the joint likelihood density SFR-law product joint likelihood

1997

"... In PAGE 8: ... This is obtained by taking the product of the likelihood densities k(x; y) to get the density distribution for the joint mean like- lihood. The resulting values are presented in Table4 , normalised to the largest value. By far the most probable laws are the power law dependence on gas density { with the exponent being di erent for each galaxy { and the linear law / g.... In PAGE 8: ... The linear law, / g, is well within this 90 per cent con dence region, and thus has one of the largest joint Bayes factors (cf. Table4 ). From both ap- proaches for the joint probability, we conclude that with the standard conversion factor there is no need to consider... In PAGE 11: ...O-H2 prescription of Arimoto et al. (1996). The last column shows the relative increase in h i as compared with the stan- dard case X = 3:68 Galaxy x y h i =h i(X=3:68) Milky Way 2:24 ?3:02 10:6 NGC 4254 1:40 ?0:30 0:6 NGC 4303 2:59 ?1:50 1:2 NGC 4321 0:20 0:54 0:9 NGC 5194 1:56 ?0:31 1:0 NGC 5457 0:38 0:04 2:3 NGC 6946 1:56 ?0:30 0:9 Table 6. As Table4 , but using the metallicity-dependent CO-conversion recipe SFR-law product joint likelihood g 0.24 g2 1:4 10?12 g=r 3:0 10?17 gx 1 0.... ..."

### Table 1. Probability Density functions .

"... In PAGE 96: ...one dimension less and rk = akn; bij = aij ? anj However, the probability distributions are not the same for the elements of A in the replicator equation and for the o -diagonal elements of B in the Lotka-Volterra model. Table1 lists the density functions used in this contribution. The probability for the quadratic form xAx and detA to have a certain sign is clearly 1=2.... ..."

### Table 3: Importance of the density factor

### Table 2: Expected number of cells overlapped by the unit intersection ball. hypercube H only on average. For simplicity, we return to the case of m = 2, although the results can be extended to any m. We show that Eq. (37) is valid, to within constant factors, even without this assumption. We only require that the probability density that the nearest neighbor ball has intersection volume be independent of the location of x. This weaker assumption is satis ed in the limit for large dimension, for points uniformly distributed in the unit hypercube. In this case, it is easily veri ed that the probability that the nearest neighbor ball has intersection volume , has the Poisson density, exp[? ]. Consider an ensemble of hypercubes with points distributed randomly and uniformly in their interior, with unit mean density. We assume that, by averaging over query points, N is the same for every member of the ensemble, even though for a particular query point x it is possible for N(x) to have signi cant uctuations from one member to another in the ensemble. For the ensemble average of N, the expression in Eq. (37) for N can be replaced by

1995

"... In PAGE 20: ... (29) can be carried out from here for any value of m. The results are given in Table2 , which shows the expected number of cells overlapped by the unit intersection ball, for di erent values of m. We now consider possible modi cations to our result, Eq.... ..."

Cited by 33

### Table 2: Expected number of cells overlapped by the unit intersection ball. hypercube H only on average. For simplicity, we return to the case of m = 2, although the results can be extended to any m. We show that Eq. (37) is valid, to within constant factors, even without this assumption. We only require that the probability density that the nearest neighbor ball has intersection volume be independent of the location of x. This weaker assumption is satis ed in the limit for large dimension, for points uniformly distributed in the unit hypercube. In this case, it is easily veri ed that the probability that the nearest neighbor ball has intersection volume , has the Poisson density, exp[? ]. Consider an ensemble of hypercubes with points distributed randomly and uniformly in their interior, with unit mean density. We assume that, by averaging over query points, N is the same for every member of the ensemble, even though for a particular query point x it is possible for N(x) to have signi cant uctuations from one member to another in the ensemble. For the ensemble average of N, the expression in Eq. (37) for N can be replaced by

"... In PAGE 20: ... (29) can be carried out from here for any value of m. The results are given in Table2 , which shows the expected number of cells overlapped by the unit intersection ball, for di erent values of m. We now consider possible modi cations to our result, Eq.... ..."