Objective Bayesianism has been criticised for not allowing learning from experience: it is claimed that an agent must give degree of belief 1 to the next raven being black, however many other black ravens have 2 been observed. I argue that this objection can be overcome by appealing to objective Bayesian nets, a formalism for representing objective Bayesian degrees of belief. Under this account, previous observations exert an inductive influence on the next observation. I show how this approach can be used to capture the Johnson-Carnap continuum of inductive methods, as well as the Nix-Paris continuum, and show how inductive influence can

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

Abstract. “Privacy ” and “utility ” are words that frequently appear in the literature on statistical privacy. But what do these words really mean? In recent years, many problems with intuitive notions of privacy and utility have been uncovered. Thus more formal notions of privacy and utility, which are amenable to mathematical analysis, are needed. In this paper we present our initial work on an axiomatization of privacy and utility. We present two privacy axioms which describe how privacy is affected by post-processing data and by randomly selecting a privacy mechanism. We present three axioms for utility measures which also describe how measured utility is affected by postprocessing. Our analysis of these axioms yields new insights into the construction of privacy definitions and utility measures. In particular, we characterize the class of relaxations of differential privacy that can be obtained by changing constraints on probabilities; we show that the resulting constraints must be formed from concave functions. We also present several classes of utility metrics satisfying our axioms and explicitly show that measures of utility borrowed from statistics can lead to utility paradoxes when applied to statistical privacy. Finally, we show that the outputs of differentially private algorithms are best interpreted in terms of graphs or likelihood functions rather than query answers or synthetic data. 1.

We consider the problem of induction over languages with binary predicates and show that a natural generalization of Johnson’s Sufficientness Postulate eliminates all but two solutions. We discuss the historical context and connections to the unary case. 1

“Privacy ” and “utility ” are words that frequently appear in the literature on statistical privacy. But what do these words really mean? In recent years, many problems with intuitive notions of privacy and utility have been uncovered. Thus more formal notions of privacy and utility, which are amenable to mathematical analysis, are needed. In this paper we present our initial work on an axiomatization of privacy and utility. In particular, we study how these concepts are affected by randomized algorithms. Our analysis yields new insights into the construction of both privacy definitions and mechanisms that generate data according to such definitions. In particular, it characterizes a class of relaxations of differential privacy and shows that desirable outputs of a differentially private mechanism are best interpreted as certain graphs rather than query answers or synthetic data. 1.

Before a football match can begin the tradition is that the referee tosses a coin and one of the captains calls, heads or tails, whilst the coin is in the air. If the captain gets it right s/he picks which end to start playing at, or alternatively to have the kick off. There never seems to be an issue of which captain actually

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