priors.Rmd

# Priors

Priors for coefficients and scales 

- Improper uniform priors $(-\infty, +\infty)$ or $(0, +\infty)$.
- Uninformative Proper Priors such as $\sigma^2 \sim \dinvgamma(0.001, 0.001)$.
- Weakly Informative Priors
- Bounded Priors
- Informative Priors
- Conjugate priors
- Hyperparametrs - "priors on priors" These are usually specified in models.
    They should be specified to ensure that the posterior is proper and not
    sensitive statistically or computationally to wide tails in the priors.
- Boundary avoiding priors - In MAP, priors can be specified to keep parameters away from boundaries, e.g. 0 in a scale model.
    
See:

- [Stan Wiki](https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations) and the [rstanarm](https://cran.r-project.org/web/packages/rstanarm/vignettes/priors.html) vignette includes comprehensive advice for prior choice recommendations.
- @Betancourt2017a provides numerical simulation of how the shapes of weakly informative priors affects inferences.
- @Stan2016a for discussion of some types of priors in regression models
- @ChungRabe-HeskethDorieEtAl2013a discuss scale priors in penalized MLE models
- @GelmanJakulinPittauEtAl2008a discusses using Cauchy(0, 2.5) for prior distributions
- @Gelman2006a provides a prior distribution on variance parameters in hierarchial models.
- @PolsonScott2012a on using Half-Cauchy priors for scale parameters