In the documentation for Computing One Dimensional Integrals, an example is given for calling the integrator for a left-truncated normal distribution. The documentation makes clear that this is just an example, and that there are better ways to model left-truncated outcomes in Stan, but gives the code:

```
parameters {
real mu;
real<lower=0.0> sigma;
real left_limit;
}
model {
mu ~ normal(0, 1);
sigma ~ normal(0, 1);
left_limit ~ normal(0, 1);
target += normal_lpdf(y | mu, sigma);
target += log(integrate_1d(normal_density,
left_limit,
positive_infinity(),
{ mu, sigma }, x_r, x_i));
}
```

I understand this code as using the likelihood function:

p(y_i | \mu, \sigma, y_i > \text{left_limit}) = \mathcal{N}(y_i | \mu, \sigma) \cdot P(y_i > \text{left_limit} | \mu, \sigma)

But shouldn’t the likelihood function for a left-truncated normal be:

p(y_i | \mu, \sigma, y_i > \text{left_limit}) = \frac{\mathcal{N}(y_i | \mu, \sigma)}{P(y_i > \text{left_limit} | \mu, \sigma)}

Then that would mean the log of the integral in the above code should be subtracted, not added? Please forgive me if I misunderstood something obvious.