General properties of general Bayesian learning

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. 1 Review of main results In this paper we investigate the general properties of general Bayesian learning. By “general Bayesian learning” we mean inferring a probability measure from another that is regarded as evidence, and where the inference is conditionalizing the probability measure representing the evidence using the conditional expectation determined by a reference probability measure that is interpreted as representing the background subjective degrees of belief of a Bayesian Agent performing the inference. The investigation is motivated by the observation that the properties of Bayesian learning we wish to determine do not seem to have been analyzed in the literature on Bayesianism on the level of generality we aim at here. (For monographic works on Bayesianism we refer to [22], [3], [44]; for papers discussing basic aspects of Bayesianism see [21], [19], [20]; the recent paper by Weisberg [43] provides a compact review of Bayesianism). In particular, in this paper we take the position that the proper general technical device to perform Bayesian conditioning is the theory of conditional ∗MTA Rényi Institute of Mathematics, Budapest, Hungary, [email protected] †Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK, [email protected]