# Sample size as function of needed HDI width

Hello, everybody.

I’ve done some exploratory studies about the weight perception error of fish consumer in my region. From these studies, I model mu 95% HDI width (HDIwid) as function of sample size (N) in an exponential relationship ( HDIwid = beta0 * N ^ beta1 ). Below is the Stan model (notice HDIwid=Y and N=X):

``````data {
int<lower=1> Ntotal;    // number of simulated samples
real Y[Ntotal];         // HDI width
int<lower=1> X[Ntotal]; // sample size (must be integer)
real Ysd;               // for y model
real beta0ct;           // intercept central tendency
real beta0sd;           // intercept standard deviation
real beta1ct;           // power central tendency
real beta1sd;           // power standard deviation
}
parameters {
real<lower=0> beta0 ; // intercept location
// unless beta1 is always integer, beta0>
real<upper=0> beta1 ; // power location
// unless sample size increase HDI width, beta1<0
real<lower=0> nu ;    // prediction normality (just a precaution against outliers)
// scale parameters seems of little use
}
model {
beta0 ~ normal( beta0ct , beta0sd ) ;
beta1 ~ normal( beta1ct , beta1sd ) ;
nu ~ exponential( 1/30.0 ) ;
for ( i in 1:Ntotal ) {
Y[i] ~ student_t( nu , beta0 * pow( X[i] , beta1 ) , Ysd ) ;
}
}
``````

If used directly, this model estimate the uncertainty of HDIwid for a given N. However, I think the inverse is more interesting for future studies: estimate the uncertainty of N for a needed HDIwid.

Question: What would be the Stan model for N ~ HDIwid?

Observations: Actually, I’m doing this by inverting the model coefficients ( beta1Inv = 1/beta1 ; beta0Inv = (1/beta0)^beta1Inv ; N = beta0Inv * HDIwid ^ beta1Inv ) outside Stan. However, since N is a count, I think its resulting uncertainty is incorrect. I tried to invert the above Stan code and use Poisson distribution for N, but it was a miserable fail.

That’s a strong prior on `nu` toward fat-tailed postriors.