Effective selection of prior distributions

Hoping someone might be able to provide a “cheat sheet” for effective selection of prior distributions, e.g. a list of available distributions with accompanying sentence explaining when each should be used. Or, alternatively, recommend a good resource. I have some general questions but also a specific model I am trying to specify. General questions:

  1. When should HalfNormal be chosen instead of HalfCauchy? Why would you choose one over the other?
  2. How might one choose between Normal, Gamma, Exponential and Poission prior distributions?
  3. I understand it is desirable for all random variables in the statistical model to have similar magnitude. If this is not possible for a particular parameter, modelling the logarithm of the parameter can be effective. Are there other ways of achieving this if using the logarithm is not suitable?

The model I am trying to specify is the following:

\begin{align} H(t) = \begin{cases} \frac{H_{ult}}{2} \big( 1- \exp(-B \: t^C) \big) & t < t_2 \\ \frac{H_{ult}}{2} \big( 1- \exp(-B \: t^C) \big) + \frac{H_{ult}}{2} \frac{t-t_2}{t-t_2+D} & t_2 \leq t \leq t_{end} \end{cases} \end{align}

I have data for H at four time values and wish to estimate the five parameters B, C, D, t_2, H_{ult}. I am using a continuous switch point for the change of curve as seen in Model with conditional parameters causing mass matrix to contain zeros on diagonal.

  1. All parameters are positive values. Is it therefore effective to use HalfNormal or HalfCauchy prior distributions?
  2. As t_2 lies somewhere between 0 and t_{end}, is it effective to use the Beta prior distribution with \alpha=1 and \beta=1 and scale it up by t_{end}?
  3. Should C have a special prior distribution as it is an exponent?

I usually start with https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations. And then you should follow https://arxiv.org/abs/1709.01449 to use prior predictive sampling to check your prior (and likelihood)


In addition to Junpeng’s recommendations, if you’ve got McElreath’s Rethinking, there is a whole section about the different kinds of priors (chapter on entropy and GLMs).