Strongly Informative Priors and Standardizing Transformations?

Modeling
2

So you have (and I’ll make up a set of strongly informed priors):

data{
  int n ;
  vector[n] x ;
  vector[n] y ;
}
parameters{
  vector[n] z ;
  real z_mu ;
  real<lower=0> z_sigma ;
}
model{
  z_mu ~ normal(10,.1) ; // strongly-informed prior here
  z_sigma ~ weibull(2,3) ; // strongly-informed prior here
  z ~ normal(z_mu,z_sigma) ; // structural "prior"
  y ~ f(x,z) ; // where f is presumably defined in a user-defined functions block
}

So you could focus just on getting the hyperparameters for z such that the sampler is seeing quantities in the unit normal realm. For a normal prior on the location hyperparameter z_mu the analytically obvious transform is a shift and scale, but for priors on the scale hyperparameter z_sigma you might have to play with what scaling gets the PDF into the right range (here I would, in R, run hist(rweibull(1e5,2,3)*q,br=100), varying q until I see the resulting histogram places values mostly in the 0-1 range).

Now, since z_mu is unbounded and the transform we want to achieve a generally-unit-scale domain involves just a shift and scale, you can use the wonderful offset=...,multiplier=... syntax. The same is coming for bounded variables soon, but in the meantime we have to do things by hand:

...
parameters{
  vector[n] z ;
  real<offset=10,multiplier=.1> z_mu ; 
  real<lower=0> z_sigma_ ; //note suffix to denote this is to be transformed (just a personal syntax of mine)
}
transformed parameters{
  real<lower=0> z_sigma  = z_sigma_* (3/.8) ;
}
model{
  z_mu ~ normal(10,.1) ; // same prior as before 
  z_sigma_ ~ weibull(2,.8) ; // with TP, implies z_sigma ~ weibull(2,3) 
  z ~ normal(z_mu,z_sigma) ; // structural "prior"
  y ~ f(x,z) ; // where f is presumably defined in a user-defined functions block
}

Now, z is also a parameter, so we probably want the sampler to see a quantity on the unit scale associated with it. I don’t think the offset/multiplier syntax accepts transformed parameters as values, otherwise we could do simply:

...
parameters{
  vector[n]<offset=z_mu,multiplier=z_sigma> z ;
  ...
}

and we’d be done. But that would involve a bunch of look-ahead by the parser to discern that z_sigma is a TP that gets its value from z_sigma_, etc so as I say I don’t think it’s currently supported. Instead we can do things manually:

...
parameters{
  vector[n] z_ ;
  real<offset=10,multiplier=.1> z_mu ; 
  real<lower=0> z_sigma_ ; 
}
transformed parameters{
  real<lower=0> z_sigma  = z_sigma_* (3/.8) ;
  vector[n] z = z*z_sigma + z_mu ;
}
model{
  z_mu ~ normal(10,.1) ; // same prior as before 
  z_sigma_ ~ weibull(2,.8) ; // with TP, implies z_sigma ~ weibull(2,3) 
  z_ ~ std_normal() ; // structural "prior"
  y ~ f(x,z) ; // where f is presumably defined in a user-defined functions block
}

And that’s really it for this model. Sometimes you’ll see folks scaling y and/or x to unit-normal, but that’s usually motivated by either a desire to get the parameters into unit normal (which we’ve done). Sometimes you’ll also see folks shifting y and/or x and that’s usually about putting the intercept of a regression model into the unit normal range (plus I think it can reduce correlation in the posterior in some multiple regression contexts?).