Weak Conditional Independence and Relative Invariance in Bayesian Statistics

In this chapter, the concept of invariance, standard in measure theory, is extended to the conditional case and is shown to provide a suitable framework to define invariant Bayesian experiments, even in the case of improper prior distributions. Also, the concept of conditional independence, standard in probability theory, is extended to the case of \(\sigma \) -finite (but unbounded) measures. Both extensions require, as a preliminary step, to work out necessary conditions for the existence of a well-defined “marginal-conditional” decomposition (actually, a desintegration) or a \(\sigma \) -finite measure. This framework is then used to handle invariance arguments in Bayesian statistics, with a particular emphasis on the search for mutually sufficient pairs of parameters and statistics.