On Filling-in Missing Conditional Probabilities in Causal Networks

Paris, Jeffrey

doi:10.1142/s021848850500345x

Cited by 11 publications

(6 citation statements)

References 18 publications

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“…From this point of view, then, formal principles of measure selection will have to be the same for subjective and statistical probabilities, and our subsequent considerations apply to both cases. We note, however, that Paris [12] holds an opposing view, and sees no reason why his rationality principles for measure selection, which were developed for subjective probability, should also apply to statistical probability. On the other hand, in support of our own position, it may be remarked that the measure selection principles Shore and Johnson [18] postulate are very similar to those of Paris and Vencovská [15], but they were formulated with statistical probabilities in mind.…”

Section: Equivariant Measure Selectionmentioning

confidence: 89%

A Representation Theorem and Applications

Jaeger

2003

Lecture Notes in Computer Science

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Abstract. We introduce a set of transformations on the set of all probability distributions over a finite state space, and show that these transformations are the only ones that preserve certain elementary probabilistic relationships. This result provides a new perspective on a variety of probabilistic inference problems in which invariance considerations play a role. Two particular applications we consider in this paper are the development of an equivariance-based approach to the problem of measure selection, and a new justification for Haldane's prior as the distribution that encodes prior ignorance about the parameter of a multinomial distribution.

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Section: Equivariant Measure Selectionmentioning

confidence: 89%

A Representation Theorem and Applications

Jaeger

2003

Lecture Notes in Computer Science

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“…Some other strands of work, which might be thought to be closely related, actually tackle rather different problems. For example, Paris (2005) considers the use of maximum entropy to determine conditional probabilities that are missing from a causal Bayesian network. In contrast, here we use non-causal Bayesian networks to represent maximum entropy distributions.…”

Section: Related Workmentioning

confidence: 99%

Objective Bayesian Nets for Integrating Consistent Datasets

Landes

Williamson

2022

jair

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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 algorithm, OBN-cDS, which offers substantial efficiency savings over the standard brute-force approach to determining the maximum entropy probability function. Furthermore, we develop modifications to the general algorithm which enable further efficiency savings but which are only applicable in particular situations. We show that there are circumstances in which one can obtain the model (i) directly from the data; (ii) by solving algebraic problems; and (iii) by solving relatively simple independent optimisation problems.

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“…It follows that if CM ∞ replaces CM as the form of equivocation over each P * i , then the result of the new norm is a convex combination of language invariant processes, which will itself be language invariant. For a longer discussion on these issues see [19,Section 4].…”

Section: Centre Of Mass Infinitymentioning

confidence: 99%

Invariant Equivocation

Landes

Masterton

2016

Erkenn

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Objective Bayesians hold that degrees of belief ought to be chosen in the set of probability functions calibrated with one's evidence. The particular choice of degrees of belief is via some objective, i.e., not agentdependent, inference process that, in general, selects the most equivocal probabilities from among those compatible with one's evidence. Maximising entropy is what drives these inference processes in recent works by Williamson and Masterton though they disagree as to what should have its entropy maximised. With regard to the probability function one should adopt as one's belief function, Williamson advocates selecting the probability function with greatest entropy compatible with one's evidence while Masterton advocates selecting the expected probability function relative to the density function with greatest entropy compatible with one's evidence. In this paper we discuss the significant relative strengths of these two positions. In particular, Masterton's original proposal is further developed and investigated to reveal its significant properties; including its equivalence to the centre of mass inference process and its ability to accommodate higher order evidence.Keywords: Maximum Entropy, Objective Bayesianism, Inference Process, Centre of Mass, Language Invariance 1 Introduction "How should one form graded beliefs?" is a question that has long fascinated philosophers. The answer to this question is highly relevant throughout science, law, operational research and policy-making. Intuitively, it is obvious that one's evidence ought to matter when forming beliefs, whether graded or binary. The best way of caching out this ubiquitous intuition is, however, a matter of lively disagreement. The main philosophical protagonists in this debate are: the subjective Bayesians, who can be further subdivided into the radical (de Finetti This paper is concerned with objective Bayesianism. Objective Bayesians, like their subjective brethren, hold that one's degrees of belief ought to be consistent with one's evidence. However, in cases where the evidential constraints are satisfied by more than one probability function the objective Bayesians differ from the subjective Bayesians in holding that the choice of a particular probability function from those consistent with the evidential constraints ought to made 1 in an objective-i.e., agent-independent-manner. Typically, such a choice is made by applying an inference process that picks from among the probability functions consistent with the evidential constraints that function which is, in some sense, maximally equivocal. 1 Objective Bayesianism is (still!) a minority view with a relatively small but dedicated group of advocates. Even in this relatively small group there are disagreements. One such recent disagreement is that between Williamson [27] and Masterton [16] on how to understand maximal equivocation.We shall herein first rehearse Williamson's account in Section 2.1 and 2.2. Then we improve on Masterton's approach by reformulating it, Section 2...

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On Filling-in Missing Conditional Probabilities in Causal Networks

Cited by 11 publications

References 18 publications

A Representation Theorem and Applications

A Representation Theorem and Applications

Objective Bayesian Nets for Integrating Consistent Datasets

Invariant Equivocation

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