9.1 What NOT to do when considering priors
You have a great deal of flexibility in choosing a prior, and there are many reasonable approaches. However, there are a few things that you should NOT do.
Do NOT choose a prior that assigns 0 probability/density to possible values of the parameter regardless of how initially implausible the values are. Even very stubborn priors can be overturned with enough data, but no amount of data can turn a prior probability of 0 into a positive posterior probability. Always consider the range of possible values of the parameter, and be sure the prior density is non-zero over that range of values.
Do NOT base the prior on the observed data. The prior reflects the degree of uncertainty about parameters before observing data. Adjusting the prior to reflect observed data to achieve some desired result is akin to “data snooping” or “p-hacking” and is bad statistics. (Of course, the posterior is based on the observed data. But not the prior.)
Do NOT feel like you have to find that one, perfect prior. The prior is just one assumption of the model and should be considered just like other assumptions. In practice, no assumption of a statistical model is ever satisfied exactly. We only hope that our set of assumptions provides a reasonable model for reality. No one prior will ever be just right for a situation, but some might be more reasonable than others. You are not only allowed but encouraged to try different priors to see how sensitive the results are to the choice of prior. (Remember, you should check the other assumptions too!) There is also no requirement that you have to choose a single prior. It’s possible to consider several models, each consisting of its own prior, and average over these models. (We’ll see a little more detail about model averaging later.)
Do NOT worry too much about the prior! In general, in Bayesian estimation the larger the sample size the smaller the role that the prior plays. But it is often desirable for the prior to play some role. You should not feel the need to apologize for priors when significant prior knowledge is available.