Indentifying skewed normal distribution prior in Stan

Hi all,

I am planning to assign a skewed normal distribution prior to one of the parameters \theta in my model. I know the mean \mu and standard deviation \sigma of that parameter and want to apply a weakly informative prior on the scale hyperparameter in the skewed normal distribution and let the model estimate it automatically.

The model I want to construct is \theta \sim skewednormal(\xi, \omega, \alpha), with known values of \xi and \alpha and assign a positive uniform prior on \omega.

However, given the three-hyperparameter in the skewed normal distribution, is it possible for me to know the exact value of hyperparameter \xi and \alpha based only on mean \mu and standard deviation \sigma, thus I could assign a positive uniform prior on the hyperparameter \omega?

Thanks so much for your help!

If you know the mean and the variance then for any given \omega you can solve for the other parameters (e.g. by solving the system of two equations for two unknowns given by the formulate for the mean and the variance here Skew normal distribution - Wikipedia). Thus, if you want you can put a prior on \omega and express the other parameters in terms of \omega, \mu, and \sigma.

For what it’s worth, this seems like a very roundabout way to express your prior on \theta. Are you sure you cannot capture a sufficiently similar statement of domain expertise using a less exotic way of building the prior?

It can’t hurt to try to simulate what the distribution of the prior would look like depending on the distribution of the hyperparameter.