Non-centered parameterisation with boundaries


#1

Dear all,

I am trying to use non-centred parameterisation on a hierarchical model. My problem is that I’m not sure how to deal with boundaries for the parameter of the individual subject that is drawn from the group level. Let’s assume I want the parameter for my individual subject to be larger than 0. In the centered parameterisation I would code:

parameters{
real<lower=0> p_group_mu; // group level parameter in my model that only makes sense >0
real<lower=0> p_group_sd; // group level standard deviation
real<lower=0> p_sub[nsub]; //individual subject parameter
}

model{
p_sub ~ normal(p_group_mu,p_group_sd);
}

In the non-centred parameterisation, I am stuck with what boundaries to set for p_sub_raw or how otherwise to have the boundaries on p_sub as before:

parameters{
real<lower=0> p_group_mu;
real<lower=0> p_group_sd;
real<lower=0> p_sub_raw[nsub]; // ?? what boundaries to choose here??
}

transformed parameters{
real<lower=0> p_sub[nsub];
p_sub = p_group_mu + p_group_sd*p_sub_raw;
}

model{
p_sub_raw ~ normal(0,1);
}

One solution that I see is to have an extra transformation that has p_sub_raw without boundaries and in the transformed parameters I then use

p_sub = exp(p_group_mu + p_group_sd*p_sub_raw)

It does change the meaning of the hiearchy somewhat. Is this the only way?

Many thanks
Jacquie


#2

I don’t know about the only way but certainly the most common. This is what happens internally in the sampler anyway when you constrain parameters.


#3

but doesn’t it make a difference whether I say:
p_sub ~ normal(p_group_mu, p_group_sd)

vs.
p_sub_raw ~ normal(p_group_mu, p_group_sd)
and p_sub = exp(p_sub_raw)

it seems to me that i’'m making a different assumption about how the parameter follows a normal distribution from the group level? or does it internally become the same?


#4

Yes it does make a difference and if you don’t have intuition for it yet it’s great to work out by simulation what the implied prior is on p_sub in the second case. That’s also an important process to go through when you have multiple levels to the hierarchy.