# SBC Interpretation:bias of prior and bias of MCMC sampling are trade off

I implemented SBC for my model.
The following is the result. Please tell me is there any problem? In the MCMC simulation, the divergent transition occurred.

In my model, when I fit my model to data, then the strong assumption on variance of prior is not required. However, when I implement SBC, if I put large variances on prior, then it generated a very odd data, which cause the very odd sampling. So, in my model, to implement SBC, I have to carefully choose the prior. My opinion, SBC algorithm requires the suitable informative priors and SBC is strongly affected the selection of prior.

If I choose informative prior, then SBC tells me it is good MCMC sampling, so, sampling itself contains no bias, however, such informative prior itself contains bias. And if I do not assume informative prior, then the odd data are made by the noninoformative priors and MCMC sampling contains bias. So, I think bias of prior and bias of MCMC sampling is trade off.

R code

``````stanmodel <- stan_model("sbc.stan")

tttttt <- function( ww=-0.81,www =0.001, #0.001 can be change to 0.1 but 1 is too large to get sampling.
mm=0.65,mmm=0.001,
vv=5.31,vvv=0.001,
zz= 1.55,zzz=0.001 ){

output <- sbc(stanmodel, data = list(
www= www,
mmm= mmm,
vvv= vvv,
zzz =zzz,

ww=ww,
mm=mm,
vv=vv,
zz=zz,

N = 3, NL = 259, NI = 57,C=3,c=c(3,2,1)), M = 500, refresh = 0)
}

fit <- tttttt( ww=-0.81,www =0.001,
mm=0.65,mmm=0.001,
vv=5.31,vvv=0.001,
zz= 1.55,zzz=0.001 )

plot(fit, bins = 111)
``````

sbc.stan

``````   data{ // SBC

//This is not prior truth data, but somedata to run
int <lower=0>N; //This is exactly same as C
int <lower=0>NL; //Number of Binomial trials
int <lower=0>NI; //This is redandunt
int <lower=0>C; // Number of Confidence level
int <lower=0>c[N]; //Each component means confidence level

//Prior which shold be specified
real www;
real mmm;
real vvv;
real zzz;
real zz;
real ww;
real vv;
real mm;

}

transformed data {
int h[C];
int f[C];

real    w_ ;
real <lower=0>dz_[C-1] ;
real m_;
real <lower =0> v_;

real <lower=0,upper=1>p_[C];
real <lower=0>l_[C];
real  <lower=0>dl_[C];
real  z_[C];

real a_;
real <lower=0>b_;

w_ =  normal_rng (ww,www);
for(cd in 1:C-1) dz_[cd] = normal_rng (zz,zzz);
m_ = normal_rng (mm,mmm);
v_ = normal_rng (vv,vvv);

a_=m_/v_;
b_=1/v_;

for(cd in 1 : C-1) {   z_[1]=w_;
z_[cd+1] =z_[cd] +dz_[cd];
}

for(cd in 1 : C) {   if (cd==C) {
p_[cd] = 1 - Phi((z_[cd] - m_)/v_);
}else{
p_[cd] = Phi((z_[cd+1] - m_)/v_)- Phi( (z_[cd] -m_)/v_);

}
}

for(cd in 1 : C) {l_[cd] = (-1)*log(Phi(z_[cd]));     }
for(cd in 1:C){
if (cd==C) {dl_[cd]=fabs(l_[cd]-0);
}else{

dl_[cd]=fabs(l_[cd]-l_[cd+1]);
}
}

for(n in 1:N) {
h[n] = binomial_rng(NL, p_[c[n]]);
// fff[n] ~ poisson( l[c[n]]*NL);//Non-Chakraborty's model
f[n] = poisson_rng (dl_[c[n]]*NI);//Chakraborty's model //<-------very very very coution, not n but c[n] 2019 Jun 21
// fff[n] ~ poisson( l[c[n]]*NI);//Non-Chakraborty's model

}
}

parameters{
real w;
real <lower =0>dz[C-1];
real m;
real <lower=0>v;

}

transformed parameters {

real <lower=0,upper=1>p[C];
real <lower=0>l[C];
real <lower=0>dl[C];
real  z[C];

real a;
real b;

a=m/v;
b=1/v;

for(cd in 1 : C-1) {   z[1] = w;
z[cd+1] = z[cd] +dz[cd];
}

for(cd in 1 : C) {
if (cd==C) {        p[cd] = 1 - Phi((z[cd] -m)/v);
}else{
p[cd] = Phi((z[cd+1] -m)/v)- Phi((z[cd] -m)/v);

}
}

for(cd in 1 : C) {    l[cd] = (-1)*log(Phi(z[cd]));     }
for(cd in 1:C){
if (cd==C) {dl[cd] = fabs(l[cd]-0);
}else{

dl[cd] = fabs(l[cd]-l[cd+1]);
}
}
}

model{
for(n in 1:N) {
h[n]   ~ binomial(NL, p[c[n]]);
// fff[n] ~ poisson( l[c[n]]*NL);//Non-Chakraborty's model
f[n] ~ poisson(dl[c[n]]*NI);//Chakraborty's model //<-------very very very coution, not n but c[n] 2019 Jun 21
// fff[n] ~ poisson( l[c[n]]*NI);//Non-Chakraborty's model

}

// priors

w ~  normal(ww,www);
for(cd in 1:C-1) dz[cd] ~  normal(zz,zzz);
m ~ normal(mm,mmm);
v ~ normal(vv,vvv);

}

generated quantities { // these adhere to the conventions above
int h_[C];
int f_[C];
vector [1] pars_;
int ranks_[1] = {w > w_};
pars_[1] = w_;
h_ = h;
f_ = f;

}``````