# Inverse Function or 1/(*)

What is the difference between using 1/(a parameter) to denote another parameter and using the inverse function?

E.G.

hb ← ’
data
{
//Number of strata
int<lower=2> K;

vector<lower=0>[K] n_e; //sample sizes
vector[K] y_bar_e; //sample means
vector<lower=0> [K]S_e; // sample variances

//pre-specified constants
real k_0;
real b_01;
real b_02;
real b_03;
real b_04;
real<lower=0> v_0;
}

parameters
{
//level 1
vector[K] theta_ek;
real<lower=0.001> sigma2_e;

//level 2
vector[K] mu_k;
vector[K] tau2_k_i;

//level 3
real mu;
real<lower=0> phi2_i;
real<lower=0> eta2_i;
}

model
{
// priors
eta2_i ~ gamma(b_03, b_04);
phi2_i ~ gamma(b_01, b_02);
mu_k ~ normal(2,sqrt(inv(phi2_i)));
tau2_k_i ~ gamma(v_0 0.5,v_0 0.5*inv(eta2_i));

theta_ek ~ normal(mu_k,sqrt(inv(tau2_k_i)));
sigma2_e ~ inv_gamma(.01,.01);

// likelihood
for (i in 1:K)
{
// Sample Means
target += normal_lpdf(y_bar_e[i]|theta_ek[i], sqrt(sigma2_e inv(n_e[i])));
//Sample Variances
target += gamma_lpdf(S_e[i]|n_e[i]
.5-.5,(n_e[i]-1)* inv(2.0*sigma2_e));

}
}

If I remember correctly I think `inv(x)` might be implemented to be a bit more numerically stable than `1/x` but otherwise I think it’s the same.

2 Likes

Thank you Jonah! That makes sense

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I dug into this a little as I was interested to know how this might work, as far as I can see the `inv` function just calculates `1.0 / x` though. Am I looking in the right places?

This is the “prim” version, which is called directly only if inv() is used with data/transformed data.

With parameters this is called for inv() math/inv.hpp at 92075708b1d1796eb82e3b284cd11e544433518e · stan-dev/math · GitHub

And with / this is called: math/operator_division.hpp at 92075708b1d1796eb82e3b284cd11e544433518e · stan-dev/math · GitHub

I think there is a slight difference in the gradient there, I would have to write it out to make sure.

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Is it safe to say that inverse is faster with parameters because the gradient is written out? Though it seems really marginal as the number of ops for autodiff is maybe a couple more.

Edit: Ah yes, division operator also has the gradient. Anyway one can always profile the two to check.