# Prior on Mean of Lognormal Distribution

I’m trying to create a simple model assuming data are distributed according to a lognormal distribution, but I want to put a prior on the mean of the lognormal rather than directly on the parameters of the lognormal. I created the mean as a transformed parameter and when using the ~ notation for the prior on the mean I got the warning:

Left-hand side of sampling statement (~) may contain a non-linear transform of a parameter or local variable.
If it does, you need to include a target += statement with the log absolute determinant of the Jacobian of the transform.

My question is, does simply changing from the ~ notation to target += fix the problem? The warning goes away, but I’m not sure if what I’m doing in appropriate.

``````data{
int<lower=0> n;
real<lower=0> y[n];
}
parameters{
real mu;
real<lower=0> sigma;
}
transformed parameters{
real<lower=0> mean;
mean = exp(mu + sigma^2/2);
}
model {
target += normal_lpdf(mean | 10, 2);  // as opposed to mean ~ normal(10,2);
y ~ lognormal(mu, sigma);
}
``````

I’m not sure what’s Stan’s behavior when renaming existing functions, but since `mean` is a function that computes the mean of an array and is on the left hand side of the `~`, do you still get the error if you name the variable something else?

Switching the notation doesn’t change anything. In both cases, you are incrementing the target by either `normal_lpdf(mean | 10, 2)` or `normal_lupdf(mean | 10, 2)`. If your intention is that the prior pushforward distribution for `mean` should be a normal distribution, then you have two options. Either you can follow your current notation and include a Jacobian adjustment (see Jacobian adjustments explained simply), or you can declare `mean` as a parameter and derive `mu` as the transformed parameter, in which case no jacobian adjustment will be necessary.

Thank you, that’s exactly what I was looking for. I wasn’t familiar with this, and in looking this stuff up I felt like I was only seeing these transformations in the context of the response, so wasn’t sure if I needed to do anything given I was transforming a parameter and not my response.

I think your article was very helpful, but just to ensure I got what you were stating I have these two implementations to ensure that the prior pushforward distribution for `mean` is a normal distribution.

Option 1: Include Jacobian adjustment (adjusting with respect to `mu` and `sigma` since my transformation is nonlinear in both).

``````data{
int<lower=0> n;
real<lower=0> y[n];
}
parameters{
real mu;
real<lower=0> sigma;
}
transformed parameters{
real<lower=0> mean;
mean = exp(mu + sigma^2/2);
}
model {
mean ~ normal_lpdf(10, 2);
target += log( exp(mu + sigma^2/2) );          //adjustment, partial with respect to mu
target += log( exp(mu + sigma^2/2) * sigma );  //adjustment, partial with respect to sigma
y ~ lognormal(mu, sigma);
}
``````

Option 2: Declare mean as a parameter and derive `mu` as the transformed parameter

``````data{
int<lower=0> n;
real<lower=0> y[n];
}
parameters{
real mean;
real<lower=0> sigma;
}
transformed parameters{
real<lower=0> mu;
mu = log(mean) - sigma^2/2;
}
model {
mean ~ normal_lpdf(10, 2);
y ~ lognormal(mu, sigma);
}
``````

I simulated data from these and the first implementation looks correct, the prior on `mean` appears to be distributed as I wanted to imply.

The second option doesn’t seem to be correct. I assume I would need to do a Jacobian adjustment there but can’t seem to get that correct.

I pretty sure trying to find a Jacobian is fruitless as the transformation from `mu` and `sigma` to the mean is not a 1-to-1 function.

The approach I use in my paper is to define a loss function to induce a posterior which has the same density as your “expert opinion”. In this case it is best to treat `E_mean` (the mean of the log normal distribution) as a parameter with `mu` a function of `E_mean` and `sigma` (which should be uniform). You can either put a improper uniform prior on E_mean and update with a “loss function” or set E_mean to have the density of your expert opinion (a Normal distribution), in which case there is no-loss function required.

Clearly the second approach is more intuitive, however, the loss framework is more general as it can include information when the observable outcome is not a straightforward function of the other parameters.

Loss approach

``````model_test2<- "

parameters{
real<lower=0> E_mean; //don't use the name "mean" as it is a function
real<lower=0> sigma;
}
transformed parameters{
real mu;
mu = log(E_mean) - sigma^2/2;
}
model {
// sigma can even be an improper uniform (should be non-informative)
sigma ~ uniform(0,10);
target += normal_lpdf(E_mean | 10, 2); //(negative) Loss function
}

"
``````

Hierarchical approach

``````model_test<- "

parameters{
real<lower=0> E_mean;
real<lower=0> sigma;
}
transformed parameters{
real mu;
mu = log(E_mean) - sigma^2/2;
}
model {
sigma ~uniform(0,10);
E_mean ~ normal(10, 2);
}

"
``````
1 Like

Thank you, I didn’t think about the use of `mean` in writing this post. I didn’t actually use `mean` in production so I didn’t get any error or anything related to that when trying to compile and execute my actual code. But that’s a good observation I didn’t consider in preparing my post.

MCMT_.5_1.pdf (865.1 KB)
Thanks. This was helpful and appears to provide the solutions I was looking for. I uploaded a document showing 5 different implementations I tried and the two you suggested appear to replicate the prior distribution I desired whereas the other implementations do not appear to. In the “prior modeling” of the loss and hierarchical approaches you suggested I get lots of divergent transitions but I don’t think I care about that. I really just wanted to make sure the model I specified matched what I desired and thought I was specifying, and sampling the priors suggest that I do get what I desired. When I incorporated some simulated data to include the likelihood contribution to the posterior I don’t get the divergent transitions.