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Has anyone developed a "cheat sheet" of sorts that describes the appropriate use of distribution types for different types of data? For example, beta for coin-type data (e.g. Therapy versus control), poisson for counts...

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    $\begingroup$ That's not a strictly Bayesian problem; it's a general statistics modeling methodology. $\endgroup$
    – Jon
    Aug 25 '16 at 17:17
  • $\begingroup$ I agree - but, I'm looking at this from the perspective of designing bayesian analyses by the types of data that I have. $\endgroup$
    – MattCrow
    Aug 25 '16 at 17:18
  • $\begingroup$ That shouldn't matter. What will matter are your choices of priors, such as conjugate priors vs reference priors. $\endgroup$
    – Jon
    Aug 25 '16 at 17:35
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A lot of these exist. Here are a couple:

This one is a flowchart for choosing a distribution.

This is a summary of common distributions.

Wikipedia often has applications for various distributions on their pages.

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  • $\begingroup$ These are perfect. Thank you! Perhaps my Google search terms were not adequate for my query in looking for these. $\endgroup$
    – MattCrow
    Aug 25 '16 at 17:06
  • $\begingroup$ Glad I could help. $\endgroup$ Aug 25 '16 at 17:28
  • $\begingroup$ One thing to note is that these tables/charts are not strictly for Bayesians for Bayesian methodology. They are general statistical models. $\endgroup$
    – Jon
    Aug 25 '16 at 17:39
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Check the Statistics 110: Probability course by Joe Blitzstein (Harvard University). Materials and lectures are freely available online as well as his handbook Introduction to Probability. Blitzstein provides many examples of common probability distribution and "stories" behind them, that make it easier to memorize what is the general idea behind them.

You can check also the paper

Lawrence, M. and McQueston, J.T. (2008). Univariate Distribution Relationships. American Statistician, 62(1): 45–53.

or this diagram by John D. Cook and Compendium of Conjugate Priors by Daniel Fink (1997).

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