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We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability. |
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We present six significant open problems in Pure Inductive Logic, together with their background and current status, with the intention of raising awareness and leading ultimately to their resolution. |
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In this paper we investigate some mathematical consequences of the Equivocation Principle, and the Maximum Entropy models arising from that, for first order languages. We study the existence of Maximum Entropy models for these theories in terms of the quantifier complexity of the theory and will investigate some invariance and structural properties of such models. |
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Information theory and statistical mechanics II. |
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Additionally, the text shows how to develop computationally feasible methods to mesh with this framework. |
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We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms. |
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In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...) |
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The mathematical theory of communication. |
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Objective Bayesianism says that the strengths of one’s beliefs ought to be probabilities, calibrated to physical probabilities insofar as one has evidence of them, and otherwise sufficiently equivocal. These norms of belief are often explicated using the maximum entropy principle. In this paper we investigate the extent to which one can provide a unified justification of the objective Bayesian norms in the case in which the background language is a first-order predicate language, with a view to applying the resulting formalism (...) |
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The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage (...) |
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This essay is, primarily, a discussion of four results about the principle of maximizing entropy (MAXENT) and its connections with Bayesian theory. Result 1 provides a restricted equivalence between the two: where the Bayesian model for MAXENT inference uses an "a priori" probability that is uniform, and where all MAXENT constraints are limited to 0-1 expectations for simple indicator-variables. The other three results report on an inability to extend the equivalence beyond these specialized constraints. Result 2 established a sensitivity of (...) |
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This paper addresses a data integration problem: given several mutually consistent datasets each of which measures a subset of the variables of interest, how can one construct a probabilistic model that fits the data and gives reasonable answers to questions which are under-determined by the data? Here we show how to obtain a Bayesian network model which represents the unique probability function that agrees with the probability distributions measured by the datasets and otherwise has maximum entropy. We provide a general (...) |
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This paper is a sequel to an earlier result of the authors that in making inferences from certain probabilistic knowledge bases the maximum entropy inference process, ME, is the only inference process respecting “common sense.” This result was criticized on the grounds that the probabilistic knowledge bases considered are unnatural and that ignorance of dependence should not be identified with statistical independence. We argue against these criticisms and also against the more general criticism that ME is representation dependent. In a (...) |
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This paper addresses the problem of finding a Bayesian net representation of the probability function that agrees with the distributions of multiple consistent datasets and otherwise has maximum entropy. We give a general algorithm which is significantly more efficient than the standard brute-force approach. Furthermore, we show that in a wide range of cases such a Bayesian net can be obtained without solving any optimisation problem. |
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This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete and (...) |
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We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains. |
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