Question: What is a second order prior distribution in Bayesian statistics? What are some good examples of this concept?

Answer: 
Here is a straightforward example from a blog entry from John Kruschke's Blog (<http://doingbayesiandataanalysis.blogspot.de/>).

![](https://2.bp.blogspot.com/-Fd0CXaBgx5M/Tja0ezCLIWI/AAAAAAAAAC4/E9A6VzEvyDk/s1600/StormTD2010bugsDiagram.jpg)

This is a Beta-Binomial Model (The sum of independent Bernoulli variables with equal probability is binomially distributed) where $y\_{ij}$ is the $i^{th}$ outcome (0,1) of the $j^{th}$ group with probability $\theta\_j$. The beta(a,b) distribution is placed as a conjugate prior on the $\theta\_j$'s. The *hyperparameters* $a$ and $b$ determine the strenght of shrinkage exerted on the $\theta\_j$'s. If this strength should be determined by the data themselves, it is necessary to place a *hyperprior* on those *hyperparameters*. I suppose this is what you mean by "second order prior distribution". In this example, the *hyperpriors* are not place directly on $a$ and $b$, but on a convenient reparametrization, namely $\mu = \frac{a}{a+b}$ (the mean of the beta distribution) and $\kappa = a+b$ ("sample size"). The parameters $A\_\mu$, $B\_\mu$, $S\_\kappa$, $R\_\kappa$ are the parameters of the *hyperprior* and are set to fixed values unless there would be another level in the data hierarchy such that the hyperparameters $a$ and $b$ would themselves be of interest.
n-order priors arise in "Hierarchical Bayesian Models". Second order refers to imposing a prior on a prior. This [paper](http://faculty.washington.edu/bajari/iosp07/rossi1.pdf) gives more detail on the subject.