I am trying to perform a multivariate analysis with three responses variables where each response variable has its own formula. But the three formulas shared parameters. I want to perform a Bayesian inference that results in one posterior distribution for each parameter. However, using brms I am obtaining one posterior distribution for each parameter for each response variable. How can I perform the bayesian inference obtaining just one posterior distribution?

I tried with:

for_tot<-bf(Vo~((VmaxG*K4*K6*GTP^2)+(VmaxdG*K2*K4*GTP*dGTP))/((K1*K2*K4*K6)+(K2*K4*K6*GTP)+(K4*K6*GTP^2)+(K2*K4*K5*dGTP)+(K2*K5*dGTP^2)+(K2*K4*GTP*dGTP)),
VmaxG+VmaxdG+K1+K2+K4+K5+K6~1,
nl=TRUE)
for_cyc<-bf(Vo_cyc~((VmaxG*K4

*K6*GTP^2))/((K1

*K2*K4

*K6)+(K2*K4

*K6*GTP)+(K4

*K6*GTP^2)+(K2

*K4*K5

*dGTP)+(K2*K5

*dGTP^2)+(K2*K4

*GTP*dGTP)),

VmaxG+K1+K2+K4+K5+K6~1,

nl=TRUE)

for_lin<-bf(Vo_lin~((VmaxdG

*K2*K4

*GTP*dGTP))/((K1

*K2*K4

*K6)+(K2*K4

*K6*GTP)+(K4

*K6*GTP^2)+(K2

*K4*K5

*dGTP)+(K2*K5

*dGTP^2)+(K2*K4

*GTP*dGTP)),

VmaxdG+K1+K2+K4+K5+K6~1,

nl=TRUE)

form_two<-mvbf(for_cyc,for_lin,for_tot)

But this results in three distributions by parameters.

Thank you very much in advance for your help,