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A Characterization of the Language Invariant Families satisfying Spectrum Exchangeability in Polyadic Inductive Logic
, 2008
"... A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap ..."
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A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and NixParis Continua.
Dirichlet Mixtures for Query Estimation in Information Retrieval
, 2005
"... Treated as small samples of text, user queries require smoothing to better estimate the probabilities of their true model. Traditional techniques to perform this smoothing include automatic query expansion and local feedback. This paper applies the bioinformatics smoothing technique, Dirichlet mixtu ..."
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Treated as small samples of text, user queries require smoothing to better estimate the probabilities of their true model. Traditional techniques to perform this smoothing include automatic query expansion and local feedback. This paper applies the bioinformatics smoothing technique, Dirichlet mixtures, to the task of query estimation. We discuss Dirichlet mixtures ’ relation to relevance models, probabilistic latent semantic indexing, and other information retrieval techniques. We describe how Dirichlet mixtures give insight into the value of retaining the original query in query expansion. On the task of adhoc retrieval, query estimation by Dirichlet mixtures generally does not perform well, but aspects of its behavior show promise. Experiments where the original query is mixed with the models estimated by relevance models and Dirichlet mixtures confirms that query estimation methods should not fully discount the prior information held in a query.
Plausibilities of plausibilities’: an approach through circumstances. Being part I of “From ‘plausibilities of plausibilities’ to stateassignment methods” (2006), eprint arXiv:quantph/0607111
"... Probabilitylike parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of ‘circumstance’, a term which stands for any piece of knowledge that is useful in assigning a probability and that satisfies some additional logical properties. The ..."
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Probabilitylike parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of ‘circumstance’, a term which stands for any piece of knowledge that is useful in assigning a probability and that satisfies some additional logical properties. The idea, which can be traced to Laplace and Jaynes, is that the usual inferential reasonings about the probabilitylike parameters of a statistical model can be conceived as reasonings about equivalence classes of ‘circumstances ’ — viz., real or hypothetical pieces of knowledge, like e.g. physical hypotheses, that are useful in assigning a probability and satisfy some additional logical properties — that are uniquely indexed by the probability distributions they lead to. PACS numbers: 02.50.Cw,02.50.Tt,01.70.+w MSC numbers: 03B48,62F15,60A05 If you can’t join ’em, join ’em together. 0
On the Emergence of Reasons in Inductive Logic
, 2001
"... We apply methods of abduction derived from propositional probabilistic reasoning to predicate probabilistic reasoning, in particular inductive logic, by treating finite predicate knowledge bases as potentially infinite propositional knowledge bases. It is shown that for a range of predicate knowled ..."
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We apply methods of abduction derived from propositional probabilistic reasoning to predicate probabilistic reasoning, in particular inductive logic, by treating finite predicate knowledge bases as potentially infinite propositional knowledge bases. It is shown that for a range of predicate knowledge bases (such as those typically associated with inductive reasoning) and several key propositional inference processes (in particular the Maximum Entropy Inference Process) this procedure is well defined, and furthermore yields an explanation for the validity of the induction in terms of `reasons'.
Generating Degrees of Belief from Statistical Information: An Overview
, 1993
"... Consider an agent (or expert system) with a knowledge base KB that includes statistical information (such as "90% of patients with jaundice have hepatitis"), firstorder information ("all patients with hepatitis have jaundice"), and default information ("patients with jau ..."
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Consider an agent (or expert system) with a knowledge base KB that includes statistical information (such as "90% of patients with jaundice have hepatitis"), firstorder information ("all patients with hepatitis have jaundice"), and default information ("patients with jaundice typically have a fever"). A doctor with such a KB may want to assign a degree of belief to an assertion ' such as "Eric has hepatitis". Since the actions the doctor takes may depend crucially on this degree of belief, we would like to specify a mechanism by which she can use her knowledge base to assign a degree of belief to ' in a principled manner. We have been investigating a number of techniques for doing so; in this paper we give an overview of one of them. The method, which we call the random worlds method, is a natural one: For any given domain size N , we consider the fraction of models satisfying ' among models of size N satisfying KB . If we do not know the domain size N , but know that it is large, we can approximate the degree of belief in ' given KB by taking the limit of this fraction as N goes to infinity. As we show, this approach has many desirable features. In particular, in many cases that arise in practice, the answers we get using this method provably match heuristic assumptions made in many standard AI systems.
Statistics as inductive inference, in
"... This chapter1 concerns the relation between statistics and inductive logic. I start by describing induction in formal terms, and I introduce a general notion of probabilistic inductive inference. This provides a setting in which statistical procedures and inductive logics can be captured. Specifica ..."
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This chapter1 concerns the relation between statistics and inductive logic. I start by describing induction in formal terms, and I introduce a general notion of probabilistic inductive inference. This provides a setting in which statistical procedures and inductive logics can be captured. Specifically, I discuss three statistical procedures (hypotheses testing, parameter estimation, and Bayesian statistics) and I show to what extend they can be captured by certain inductive logics. I end with some suggestions on how inductive logic can be developed so that its ties with statistics are strengthened. 1 Statistical procedures as inductive logics An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on all possible data. Statistics, on the other hand, is a mathematical discipline that de
The development of the Hintikka program
 Handbook of the History of Logic
, 2009
"... years later Hintikka published a twodimensional continuum of inductive probability measures (see Hintikka, 1966), and ten years later he announced an axiomatic system with K ≥2 parameters (see [Hintikka and Niiniluoto, 1976]). These new original results showed once and for all the possibility of s ..."
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years later Hintikka published a twodimensional continuum of inductive probability measures (see Hintikka, 1966), and ten years later he announced an axiomatic system with K ≥2 parameters (see [Hintikka and Niiniluoto, 1976]). These new original results showed once and for all the possibility of systems of inductive logic where genuine universal generalizations have nonzero probabilities in an infinite universe. Hintikka not only disproved Karl Popper’s thesis that inductive logic is inconsistent (see [Popper, 1959; 1963]), but he also gave a decisive improvement of the attempts of Rudolf Carnap to develop inductive logic as the theory of partial logical implication (see [Carnap, 1945; 1950; 1952]). Hintikka’s measures have later found rich applications in semantic information theory, theories of confirmation and acceptance, cognitive decision theory, analogical inference, theory of truthlikeness, and machine learning. The extensions and applications have reconfirmed — pace the early evaluation of Imre Lakatos [1974] — the progressive nature of this research program in formal methodology and philosophy of science. 1 INDUCTIVE LOGIC AS A METHODOLOGICAL RESEARCH PROGRAM
INDUCTIVE LOGIC AND STATISTICS
"... There are strong parallels between statistics and inductive logic. An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on data. Statistics, on ..."
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There are strong parallels between statistics and inductive logic. An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on data. Statistics, on the other hand, is a mathematical discipline that describes procedures for deriving results about a population from sample data. These results include decisions on rejecting or accepting a hypothesis about the population, the determination of probability assignments over such hypotheses, predictions on future samples, and so on. Both inductive logic and statistics are calculi for getting from the given data to propositions or results that transcend the data. Despite this fact, inductive logic and statistics have evolved more or less separately. This is partly because there are objections to viewing statistics, especially classical statistical procedures, as inferential. A more important reason, to my mind, is that inductive logic has been dominated by the Carnapian programme, and that statisticians have perhaps not recognised Carnapian inductive logic as a
The LaplaceJaynes approach to induction. Being part II of “From ‘plausibilities of plausibilities’ to stateassignment methods
"... An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion o ..."
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An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve ‘unknown probabilities ’ or ‘propensities’. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed. PACS numbers: 02.50.Cw,02.50.Tt,01.70.+w MSC numbers: 03B48,60G09,60A05 Note, to head off a common misconception, that this is in no way to introduce a “probability of a probability”. It is simply convenient to index our hypotheses by parameters [...] chosen to be numerically equal to the probabilities assigned by those hypotheses; this avoids a doubling of our notation. We could easily restate everything so that the misconception could not arise; it would only be rather clumsy notationally and tedious verbally.
From “plausibilities of plausibilities” to stateassignment methods: I. “Plausibilities of plausibilities”: an approach through circumstances (2006), eprint arXiv:quantph/0607111
"... An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion o ..."
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An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve ‘unknown probabilities ’ or ‘propensities’. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed. PACS numbers: 02.50.Cw,02.50.Tt,01.70.+w MSC numbers: 03B48,60G09,60A05 Note, to head off a common misconception, that this is in no way to introduce a “probability of a probability”. It is simply convenient to index our hypotheses by parameters [...] chosen to be numerically equal to the probabilities assigned by those hypotheses; this avoids a doubling of our notation. We could easily restate everything so that the misconception could not arise; it would only be rather clumsy notationally and tedious verbally.