In a statistical decision problem, the use of approximate or estimated priors leads naturally to the question of the continuity of riskR(\theta, d_{G})in the variableG. This continuity condition is explored, and in the finite parameter state case, a characterization in terms of the differentiability of the Bayes envelope is given.

When a given statistical decision problem occurs repeatedly, there is interest in the strategy that plays Bayes versus the empirical distribution of past parameter values. We point out some implications that some previously investigated continuity conditions have on the stability of average loss, risk, and average risk across the sequence of decision problems. This is done first for arbitrary parameter sequences and then in the context of independent and identically distributed parameters.