A friend of mine is collaborating on a project using a Bayesian hierarchical model. One of the parameters in her model is error precision (inverse variance), which she’s giving a gamma-distributed prior. The other day she came to me with a question about the distribution, and how it could be interpreted. This struck me as less than intuitive, so I thought I’d make a simple interactive application to show how the prior parameters would translate into the precision distribtuion, as well as something more intuitive like standard deviation.

Here’s what I came up with:

Gamma density for precisions

Insights

Before making these plots I recognized that selecting prior parameters that pushed precisions nearer to zero would lead to larger standard deviations, but I had no idea how that would look in terms of distribution. I was surprised by how hard it was to push the standard deviations far from zero (I was afraid they might blow up).

Standard deviation density

Just to show my math a little more clearly, here is how I obtained the standard deviation density:

Let \(X_1\) be precision, \(X_2\) be standard deviation. Then

\[ X_2 = g(X_1) = \frac{1}{\sqrt{X_1}} \]

\[ X_1 = g^{-1}(X_2) = \frac{1}{X_2^2} \]

\[ f_{X_2}(x) = f_{X_1}(g^{-1}(x)) * \big| \frac{d}{d x} g^{-1}(x) \big| \]

In this case, this works out to be:

\[ f_{X_2}(x) = f_{X_1}(x^{-2}) * \frac{2}{x^3} \]

So if we get \(f_{X_1}(x)\) via dgamma(x1, shape = input$shape, rate = input$rate), we can get \(f_{X_2}(x)\) via dgamma(x2^-2, shape = input$shape, rate = input$rate) * 2 * x2^-3).