I am aware that doing operations on lpdf is not the standard way to proceed (and that must be a trivial reason to this), but I am curious why multiplying the lpdf by a constant is invariant

```
x = rnorm(100, 4, 3);
stan(model_code="
data{ int N;
real x[N];}
parameters{
real mu;
real<lower=0> sigma;
}
model{
target += normal_lpdf(x | mu, sigma) * 10000;
}
",
data =list(N = N, x = x))
```

```
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
mu 3.69 0.00 0.00 3.69 3.69 3.69 3.70 3.70 981 1
sigma 3.07 0.00 0.00 3.07 3.07 3.07 3.07 3.08 4557 1
lp__ -2541121.85 0.03 0.97 -2541124.40 -2541122.24 -2541121.55 -2541121.14 -2541120.88 1496 1
Samples were drawn using NUTS(diag_e) at Thu Sep 05 22:06:04 2019.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
```

While dividing by a constant leads to failure

```
stan(model_code="
data{ int N;
real x[N];}
parameters{
real mu;
real<lower=0> sigma;
}
model{
target += normal_lpdf(x | mu, sigma) / 10000;
}
",
data =list(N = N, x = x))
```

```
mean se_mean sd 2.5% 25% 50% 75% 97.5%
mu -5.309213e+12 5.435324e+12 9.125369e+12 -2.718509e+13 -6.702003e+12 -3.620251e+10 5.901567e+10 1.286364e+11
sigma 1.137980e+305 NaN Inf 1.823595e+20 7.674246e+84 9.643890e+158 2.100146e+232 2.716923e+299
lp__ -9.200000e-01 0.000000e+00 0.000000e+00 -9.200000e-01 -9.200000e-01 -9.200000e-01 -9.200000e-01 -9.200000e-01
n_eff Rhat
mu 3 2.47
sigma NaN NaN
lp__ 3503 1.00
Samples were drawn using NUTS(diag_e) at Thu Sep 05 21:55:43 2019.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
```