# Is there a difference between Bayesian and Classical sufficiency?

The title pretty much says it all. I wonder whether there is any difference in the way Bayesians understand sufficiency vs. the way orthodox statistics understands sufficiency, or are they equivalent? If there is a difference, what is it?

Here is one example of differentiation between classical and Bayesian statistics: when comparing two models $$\mathcal{M}_1$$ and $$\mathcal{M}_2$$, a statistic $$S(\cdot)$$ may be sufficient for both models, hence sufficient in a classical sense, but insufficient for model comparison as e.g. in Bayes factors, when the conditional distribution of the data given $$S$$ varies between models. The difference is due to the fact that the model index is a parameter from a Bayesian perspective but not a parameter from a classical one. (This is discussed further in our ABC model choice papers.)