22 research outputs found
Partial exchangeability and related topics.
Thesis (Ph.D.)-University of Natal, Durban, 1991.Partial exchangeability is the fundamental building block in the subjective
approach to the probability of multi-type sequences which replaces the independence
concept of the objective theory.
The aim of this thesis is to present some theory for partially exchangeable
sequences of random variables based on well-known results for exchangeable
sequences.
The reader is introduced to the concepts of partially exchangeable events,
partially exchangeable sequences of random variables and partially exchangeable
o-fields, followed by some properties of partially exchangeable
sequences of random variables.
Extending de Finetti's representation theorem for exchangeable random
variables to hold for multi-type sequences, we obtain the following result
to be used throughout the thesis:
There exists a o-field, conditional upon which, an infinite partially exchangeable
sequence of random variables behaves like an independent sequence
of random variables, identically distributed within types.
Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker
requirement (the selection property) than partial exchangeability on the
infinite multi-type sequence of random variables, we obtain results related
to de Finetti's representation theorem for partially exchangeable sequences
of random variables.
Regarding partially exchangeable sequences as mixtures of independent and
identically distributed (within types) sequences, we (i) give three possible
expressions for the directed random measures of the partially exchangeable
sequence and (ii) look at three possible expressions for the o-field mentioned
in de Finetti's representation theorem.
By manipulating random measures and using de Finetti's representation
theorem, we point out some concrete ways of constructing partially exchangeable
sequences.
The main result of this thesis follows by extending de Finetti's represen.
tation theorem in conjunction with the Chatterji principle to obtain the
following result:
Given any a.s. limit theorem for multi-type sequences of independent random
variables, identically distributed within types, there exists an analogous
theorem satisfied by all partially exchangeable sequences and by all
sub-subsequences of some subsequence of an arbitrary dependent infinite
multi-type sequence of random variables, tightly distributed within types.
We finally give some limit theorems for partially exchangeable sequences of
random variables, some of which follow from the above mentioned result
Probability models for information retrieval based on divergence from randomness
This thesis devises a novel methodology based on probability theory, suitable for the construction of term-weighting models of Information Retrieval. Our term-weighting functions are created within a general framework made up of three components. Each of the three components is built independently from the others. We obtain the term-weighting functions from the general model in a purely theoretic way instantiating each component with different probability distribution forms.
The thesis begins with investigating the nature of the statistical inference involved in Information Retrieval. We explore the estimation problem underlying the process of sampling. De Finetti’s theorem is used to show how to convert the frequentist approach into Bayesian inference and we display and employ the derived estimation techniques in the context of Information Retrieval.
We initially pay a great attention to the construction of the basic sample spaces of Information Retrieval. The notion of single or multiple sampling from different populations in the context of Information Retrieval is extensively discussed and used through-out the thesis. The language modelling approach and the standard probabilistic model are studied under the same foundational view and are experimentally compared to the divergence-from-randomness approach.
In revisiting the main information retrieval models in the literature, we show that even language modelling approach can be exploited to assign term-frequency normalization to the models of divergence from randomness. We finally introduce a novel framework for the query expansion. This framework is based on the models of divergence-from-randomness and it can be applied to arbitrary models of IR, divergence-based, language modelling and probabilistic models included. We have done a very large number of experiment and results show that the framework generates highly effective Information Retrieval models
Ranked Fragmentations
In this paper we define and study self-similar ranked fragmentations. We
first show that any ranked fragmentation is the image of some partition-valued
fragmentation, and that there is in fact a one-to-one correspondence between
the laws of these two types of fragmentations. We then give an explicit
construction of homogeneous ranked fragmentations in terms of Poisson point
processes. Finally we use this construction and classical results on records of
Poisson point processes to study the small-time behavior of a ranked
fragmentation.Comment: 31 page
Models beyond the Dirichlet process
Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of
exibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and ecient simulation techniques that make nonparametric models amenable to concrete use in a number of applied statistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process.
A Bayesian Approach to Causality Assessment
1 online resource (PDF, 79 pages