All parameters in a Bayesian model require a prior specified; parameters are random variables.
\(y_{i} \sim \text{Binom}(N, p)\)
\(p \sim \text{Beta}(\alpha = 4, \beta=2)\)
curve(dbeta(x, 4,2),xlim=c(0,1),lwd=5)
The Prior
\(p \sim \text{Beta}(\alpha = 1, \beta=1)\)
curve(dbeta(x, 1,1),xlim=c(0,1),lwd=5)
The Prior
The prior describes what we know about the parameter before we collect any data
Priors can contain a lot of information (informative priors ) or very little (diffuse priors )
Well-constructed priors can also improve the behavior of our models (computational advantage)
No such thing as a ‘non-informative’ prior
Diffuse priors on one scale can be very informative on another
The Prior
Use diffuse priors as a starting point
It’s fine to use diffuse priors as you develop your model but you should always prefer to use “appropriate, well-contructed informative priors” (Hobbs & Hooten, 2015)
The Prior
Use your “domain knowledge”
We can often come up with weakly informative priors just by knowing something about the range of plausible values of our parameters.
The Prior
Dive into the literature
Find published estimates and use moment matching and other methods to convert published estimates into prior distributions
The Prior
Visualize your prior distribution
Be sure to look at the prior in terms of the parameters you want to make inferences about (use simulation!)
The Prior
Do a sensitivity analysis
Does changing the prior change your posterior inference? If not, don’t sweat it. If it does, you’ll need to return to point 2 and justify your prior choice
The Prior
Priors are mostly not invariant to transformation of scale
# Note how the data are the same, but the prior is changing.# Gibbs sampler post.1=rbeta(10000,alpha.prior1+sum(y1),beta.prior1+N1-sum(y1)) post.2=rbeta(10000,alpha.prior2+sum(y1),beta.prior2+N1-sum(y1)) post.3=rbeta(10000,alpha.prior3+sum(y1),beta.prior3+N1-sum(y1))
