Consequences of mapping data or parameters in Bayesian common-cause analysis

• We map the common-cause alpha factor model down to a smaller group.
• Treating mapped data like observed data is much too conservative.
• Mapped alpha factors have a joint posterior distribution that cannot be Dirichlet.
• We approximate the mapped alpha factors' complicated posterior distribution.
• Bayesian mapping up is also possible, but highly sensitive to the prior.

When mapping the common-cause alpha factor model from a group of one size to one of another size, the following facts are shown: (1) mapping data down and treating the mapped data like observed data is much too conservative; (2) mapping alpha factors down puts restrictions on the resulting alphas, so their joint distribution cannot be Dirichlet; (3) if the mapped alpha factors' posterior distributions are moderately bell-shaped, the joint distribution can be approximated well by using correlated logistic-normal conditional probabilities and (4) Bayesian mapping up is possible, but highly sensitive to the prior distribution in the top group.