Free Access
Volume 547, November 2012
Article Number A117
Number of page(s) 12
Section Cosmology (including clusters of galaxies)
Published online 09 November 2012

Online material

Appendix A: Model for computing the predicted prior of the mass-richness scaling including the weak-lensing selection function

Equations (12) and (17) to (29) are almost literally translated into JAGS (Plummer 2008), Poisson, normal, and uniform distributions become dpois, dnorm, dunif, respectively. JAGS5, following BUGS (Spiegelhalter et al. 1995), uses precisions, prec = 1/σ2, in place of variances σ2. The only complication comes from sampling from a distribution unavailable in JAGS, a truncated Schechter function. This is achieved by exploiting the property that a Poisson(φ) observation of zero has a likelihood e − φ. Conseguently, if our observed data are a set of 0’s, and φ [i]  is set to  − log ℒ [i] , we obtain the correct likelihood contribution. The quantity λ [i]  should always be greater than 0, because it is a Poisson mean, and we may accordingly need to add a suitable constant, C, to ensure that it is positive. The quantity lg10tot.norm normalises the integral of the obslgM200 likelihood to one. The model (set of equations) reads in JAGS:

preclgM200 <- 1./(errlgM200^2)
# normaliz
lg10tot.norm <-0.386165-3.92996*obsz-0.247050*obsz^2-2.55814*obsz^3-5.26633*obsz^4
# dummy variable for zero-trick, to sample from a distribution not available in JAGS
for (i in 1:length(obslgM200)) {
 dummy[i] <-0
intrscat ~ dnorm(0.25,1/0.03/0.03)
prec.intrscat <- 1/intrscat^2
alpha ~ dnorm(0.08,1/0.04/0.04)
beta ~ dnorm(0.47,1/0.12/0.12)
gamma ~dt(0,1,1)
csi ~dt(0,1,1)
for (i in 1:length(obsn200))  { 
    # modelling lgM200 
    # dummy prior, requested by JAGS, to be modified later 
    lgM200[i] ~ dunif(13.9891+1.04936*obsz[i]+0.488881*obsz[i]^2,16) 
    # modelling a truncated schechter 
    lnnumerator[i] <- -(10^(0.4*(lgM200[i]-12.6+(obsz[i]-0.3)))) 
    # its integral, from the starting point of the integration (S/N=5) 
    loglike[i] <- -lnnumerator[i]+lg10tot.norm[i]*log(10)+C 
    # sampling from an unavailable distribution 
    dummy[i] ~ dpois(loglike[i]) 
    obslgM200[i] ~ dnorm(lgM200[i],preclgM200[i]) 
    # modelling n200, z and relations 
    obsn200[i] ~ dpois(pow(10, lgn200[i])) 
    obsz[i] ~ dnorm(z[i],pow(0.02,-2)) 
    # modelling mass -n200 relation allowing for evolution 
    lgn200m[i] <- alpha+1.5 +beta*(lgM200[i]-14.5)+ gamma*(log(1+z[i])) 
    lgn200[i] ~ dnorm(lgn200m[i], prec.intrscat.z[i]) 
    prec.intrscat.z[i] <- 1/( 1/prec.intrscat-1+(1+z[i])^(2*csi)) 

To adopt a Student t–distribution with ten degrees of freedom dt to model the intrinsic scatter (Sect. 4), it suffices to replace the line starting by lgn200[i] with

 lgn200[i] ~ dt(lgn200m[i], prec.intrscat.z[i],10)

© ESO, 2012

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