| Mon 10th |
Classification
Imprecise Probability as a new perspective on basic statistical tasks
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Paul Fink: |
Influencing the predictive ability of (bags of) imprecise trees by
restrictions and aggregation rules
In a first simulation the impact of restrictions on the tree growing algorithm
[Abellan and Moral (2003)], varying values of the crucial imprecise dirichlet parameter 's' and a stopping rule induced by a minimal leaf size, are studied.
The second simulation deals with different aggregation rules to combine a bag of imprecise trees. Both rules on the predicted classes and the predicted class probability intervals are considered and compared.
Moreover, for both a bag and single imprecise tree are grown and they are compared alongside.
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Richard Crossman: |
Ensembles of Classification Trees with Interval Entropy
I want to discuss a generalised classification tree method based on
the Abellan/Moral/Masegosa method, and I want to talk about adapting
elements Popatov's TWIX ensemble method to the
Abellan/Moral/Masegosa method, so we can tackle continuous variables
properly, rather than just discretising them.
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Sébastien Destercke:
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Binary decomposition with imprecise probabilities
Decomposing a problem into binary classifiers is a seductive way to
transform a complex problem into a set of easier ones. It is used in
multiclass problems as well as in other classification problems such
as label ranking or multilabel prediction. In this talk, we review
the latest results about using binary classifiers with imprecise
probabilities, point out the possible remaining problems, and offer
some perspective on the use of such classifiers.
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Lev Utkin:
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An imprecise boosting-like approach to classification
It is shown that one of the reasons why the Adaboost algorithm in
classification overfits is its extreme imprecision, i.e., the
probabilities or weights for adaptation can be changed in the unit
simplex. A way for reducing the imprecision by means of exploiting
the well-known imprecise statistical models is proposed. This way is
called the imprecise AdaBoost. It is shown that this reduction
provides an efficient way for dealing with highly imbalanced
training data. Moreover, it is shown that the reduced sets of
probabilities can be changed at each iteration of AdaBoost by using
for example the imprecise Dirichlet model. Various numerical
experiments with the well-known data illustrate the peculiarities
and advantages of the imprecise AdaBoost.
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| Tue 11th |
Regression and support vector machines |
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Georg Schollmeyer:
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Linear models and partial identification: Imprecise linear
regression with interval data
In several areas of research like Economics, Engineering sciences,
or Geodesy, the aim of handling interval-valued observations to
reflect some kind of non-stochastic uncertainty is getting some
attention. In the special case of a linear model with
interval-valued dependent variables and precise independent
variables one can use the linear structure of the
least-squares-estimator to develop an appropriate, now set-valued
estimator, which is explicated seemingly independently in several
papers (Beresteanu and Molinari, 2008; Schön and Kutterer, 2005;
Cerny, Antoch, and Hladik, 2011). The geometric structure of the so
reached estimate is that of a zonotope, which is widely studied in
computational geometry. In this talk I want to introduce the
above-mentioned estimators, some of their properties, and two
different ways to construct confidence regions for them: One way is
to look at these estimators as set-valued point estimators and to
utilize random set theory, the other way is to see them as
collections of point estimators, for which one has to find
appropriate collections of confidence ellipsoids.
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Chel Hee Lee:
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Imprecise Probability Estimates for GLIM
We study imprecise priors for the generalized linear model to build
a framework for Walley's 1991 inferential paradigm that also
incorporates an effect of explanatory variables for quantifying
epistemic uncertainty. For easy exposition, we restrict ourselves to
Poisson sampling models giving an exponential family using the
canonical log-link function. Normal priors on the canonical
parameter of the Poisson sampling models lead to a three-parameter
exponential family of posteriors which includes the normal and
log-gamma as limiting cases. The canonical parameters simplify
dealing with families of priors as Bayesian updating corresponds to
a translation of the family in the canonical hyperparameter space.
The canonical link function creates a linear relationship between
regression coefficients of explanatory variables and the canonical
parameters of the sampling distribution. Thus, normal priors on the
regression coefficients induce normal priors on the canonical
parameters leading to a multi-parameter exponential family of
posteriors whose limiting cases are again normal or log-gamma. As an
implementation of the model we present a prototype for work-in-
progress of the project at the r-forge.r-project.org which is titled
`Imprecise Probability Estimates in GLM'.
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Lev Utkin:
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Imprecise statistical models and the robust SVM
A framework for constructing robust one-class classification models
is proposed in the paper. It is based on Walley's imprecise
extensions of contaminated models which produce a set of probability
distributions of data points instead of a single empirical
distribution. The minimax and minimin strategies are used to choose
an optimal probability distribution from the set and to construct
optimal separating functions. It is shown that an algorithm for
computing optimal parameters is determined by extreme points of the
probability set and is reduced to a finite number of standard SVM
tasks with weighted data points. Important special cases of the
models, including pari-mutuel, constant odd-ratio, contaminated
models and Kolmogorov-Smirnov bounds are studied. Experimental
results with synthetic and real data illustrate the proposed models.
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Andrea Wiencierz:
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Linear Likelihood-based Imprecise Regression (LIR) with interval data
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| Wed 12th |
Evaluation and comparison of imprecise methods and models
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Andrea Wiencierz and Alessandro Antonucci:
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Evaluation and comparison of imprecise methods and models — A short introduction
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Alessandro Antonucci:
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Evaluating imprecise classifiers: from discounted accuracy to utility-based measures
Imprecise classifiers can possibly assign more than a single class label to a test instance of the attributes. Accuracy can therefore characterize the performance only on instances labeled by single classes. The problem of evaluating an imprecise classifier on the whole dataset is discussed with a focus on a recently proposed utility-based approach. This produces a single measure which can be used to compare an imprecise classifier with others, either precise or imprecise.
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Sébastien Destercke:
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Comparing credal classifiers: ideas from Label Ranking
In this talk, we recall the basic scheme of the label ranking problem.
We then present some solutions recently in label ranking methods to measure the efficiency of classifiers returning a partial or incomplete answer. The use of such measurements to credal classifiers is then sketched briefly.
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Andrea Wiencierz:
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Evaluating imprecise regression
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Marco Cattaneo:
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Graphical comparison of imprecise methods
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Georg Schollmeyer:
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Evaluation and comparison of set-valued estimators: empirical and structural aspects
In this talk we investigate the problem of evaluation of set-valued estimators. We look at estimators as 'approximations of the truth', contrasting the goodness of these approximations in an empirical and in a structural manner respectively. We exemplify this along the lines of location-estimators and the problem of linear regression . Here it is useful to look also at set-domained, set-valued estimators.
Finally we try to motivate the need to satisfy structural properties at least in a practical sense and state a little lemma about the extension of undominated point-domained estimators to undominated set-domained estimators, which indicates the usefulness of set-monotonicity.
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| Thu 13th |
Learning and updating |
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Sébastien Destercke:
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Label ranking: interest for IP and problems (a short introduction)
In this talk, we present the label ranking problem and explain why
imprecise probabilities may be useful to deal with such a problem. We
also present some interesting challenges concerning decision and
statistical models used in such problems.
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Gero Walter:
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Boat or bullet: prior parameter set shapes and posterior imprecision
In generalized Bayesian inference based on sets of conjugate priors,
the prior credal set is taken as the convex hull of conjugate priors
whose parameters vary in a set. Even if equivalent in terms of prior
imprecision, different parameter set shapes may lead to different
updating behaviour, and thus influence posterior imprecision
significantly. Using a canonical parametrization of priors,
Walter & Augustin have proposed a simple set shape that leads to
additional posterior imprecision in case of prior-data conflict.
With the help of a different parametrization proposed by
Mik Bickis, Walter, Coolen & Bickis now have found a set shape that,
in addition to prior-data conflict sensitivity, also reduces imprecision
particularly when prior and data are in strong agreement.
In Bickis' parametrization, the set shape resembles a boat with a transom stern, or a bullet.
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Marco Cattaneo:
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On the estimation of conditional probabilities
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Roland Poellinger:
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Superimposing Imprecise Evidence onto Stable Causal Knowledge: Analyzing 'Prediction' in the Newcomb Case
Referring back to the physicist William NEWCOMB, Robert NOZICK (1969) elaborates on - as he calls it - Newcomb's problem, a decision-theoretic dilemma in which two principles of rational choice seemingly conflict each other, at least in numerous renditions in the vast literature on this topic: Dominance and the principle of maximum expected utility recommend different strategies in the plot of the game situation. While evidential decision theory (EDT) seems to be split over which principle to apply and how to interpret the principles in the first place, causal decision theory (CDT) seems to go for the solution recommended by dominance ("two-boxing").
In this talk I will prepare the ground for a understanding of causality that enables the causal decision theorist to answer NOZICK's challenge with the solution of one-boxing by drawing on the framework of causal knowledge patterns, i.e., Bayes net causal models built upon stable causal relations (cf. PEARL 1995 and 2000/2009) augmented by non-causal knowledge (epistemic contours). This rendition allows the careful re-examination of all relevant notions in the original story and facilitates approaching the following questions:
1. How may causality in general be understood to allow causal inference from hy-brid patterns encoding subjective knowledge?
2. How can the notion of prediction be analyzed - philosophically and formally?
3. If all relations given in the model represent stable causal knowledge, how can imprecise evidence be embedded formally? Or in other words: How can the unreliable predictor be modeled without discarding the core structure?
4. Finally, in what way could unreliable prediction be modeled with interval probability, as motivated by considerations in NOZICK's treatise? And what should be the interpretation of such a rendition?
References:
1. Nozick, R. in Rescher, N. (Ed.): Newcomb's Problem and Two principles of Choice. Essays in Honor of Carl G. Hempel, Dordrecht: Reidel, 1969, 114-146
2. Pearl, J.: Causality: Models, Reasoning, and Inference. Cambridge University Press, 2009
3. Pearl, J.: Causal diagrams for empirical research. Biometrika, 1995, 82, 669-688
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| Fri 14th |
Open topics |
|
Atiye Sarabi Jamab:
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A Comparison of Approximation Algorithms in Dempster Shafer Theory based on New Basis Dissimilarity Measures
Computational complexity of combining various independent pieces of evidence in Dempster Shafer theory (DST) motivates the development of approximation algorithms for simplification. In approximation algorithms, some approaches consider special types of evidence such as working on the quality of the belief functions to be combined. Another category of approaches is composed based on Monte-Carlo techniques, where the idea is to estimate exact values of belief and plausibility by comparing the different outcomes relative to randomly generated samples. The last category tries to reduce the growing number of focal sets during the combination process by simplification.
Many approaches are introduced to improve the efficiency of computational methods, and many analytical and numerical studies propose different distance measures and benchmarks to investigate and compare the approximation methods. While many distance measures can be found in the literature, the experiments show that the information content of these distance measures are highly over lapped. In this talk, first through a thorough analysis of dissimilarity measures, a set of more informative and less overlapping as the basis dissimilarity measures will be introduced. This basis will be used to investigate and compare the quality of approximation algorithms in Dempster Shafer Theory. To this end, three benchmarks along with the classic combination benchmark will be proposed. Existing Approximation methods will be compared on them and the overall qualitative performance will be summarized.
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Robert Schlicht:
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Dual Representation of Convex Sets of Probability Distributions
Sets of probability distributions appear in various contexts in both statistics (e.g. as parametric models) and probability theory (e.g. probability distributions determined by marginal constraints). They also present a form of interval probabilities with a particularly nice (linear) mathematical structure. Specifically, closed convex sets of probability distributions have equivalent representations as closed convex cones in a function space and, moreover, as preference relations between gain (or loss) functions.
In the first part of the talk, the mathematical background, essentially amounting to classical results from integration theory and functional analysis, is presented in a general form. Next, the relationship to imprecise probabilities and statistical decision theory is discussed. The last part of the talk explores several applications, including the Kolmogorov extension theorem, stochastic orders, transportation problems, and conditioning.
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Marco Cattaneo:
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Likelihood-based imprecise probabilities and decision making
Likelihood inference offers a general approach to the learning of imprecise probability models. In this talk, we consider properties of the statistical decisions resulting from these imprecise probability models, in connection with decisions based directly on the likelihood function.
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Damjan Škulj:
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Calculations of the solutions of interval matrix differential equations using bisection
Computation of the lower and upper bounds for the solution of interval matrix differential equation is generally computationally very expensive. Such equations usually appear in modelling continuous time imprecise Markov chains.
I will propose a method that in some important cases reduces this computational complexity.
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Andrey Bronevich:
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Measuring uncertainty and information of sign-based image representations
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| day |
subject |
organized by |
| Mon 10th |
Classification /
Imprecise Probability as a new perspective on basic statistical tasks
During the last two decades, research in imprecise probability (IP) has
seen a
big increase, including substantial attention to statistical inference.
The current
state-of-the-art consists mostly of methods which, effectively, use sets
of models
simultaneously. While this has led to sophisticated methods with
interesting properties,
it may not have too much to offer to applied statisticians, who may
consider the
IP-based methods even more complex to understand and to use than the more
established approaches.
It would be great if IP could in addition lead to statistical methods
that are better
and easier to apply, for non-experts, than the established methods, or
at least either
of these without being (substantially) worse on the other. During this
workshop day,
we look for ideas and discussions about such possible IP approaches,
with specific
focus on basic statistical problems.
One can think e.g. at the t-test or chi-square test, which are very
often applied
(even by non-experts) but have underlying assumptions that are often not
really
taken into account carefully. Could IP provide statistical methods that
can be applied,
and ideally understood, by non-experts and that are "better"?
A further possible area where IP-based statistical methods can have
substantial
impact is for more complex models, where allowing imprecision might lead to
simplifications that could be attractive in real-world applications. For
example,
one could consider IP models in which some aspects of more detailed models
are left unspecified, and the resulting inference may already be
sufficient or
could indicate that more detailed specification is required.
It will be crucial for wider success of IP-based statistical methods to
be attractive
to applied statisticians and non-expert users, we aim at some focus on
this in
presentations and discussions.
If you want to make a scheduled contribution to this day, please email
us by
Friday 31 August with a clear indication of your intended contribution,
that is which
challenge(s) you wish to address and how, what issues your presentation
will
raise for discussion, and how much time you would ideally have,
indicated as
"minutes for presentation + minutes for discussion". Also, if
participants can
prepare for your presentation and discussion, it would be useful to give an
indication how (e.g. some references or an explicit question).
|
Lev Utkin, Gero Walter /
Frank Coolen, Thomas Augustin
|
| Tue 11th |
Regression and support vector machines
|
Lev Utkin, Gero Walter
|
| Wed 12th |
Evaluation and comparison of imprecise methods and models
On this day of the workshop we will discuss how to evaluate and compare statistical methods, in general, and, more specifically, imprecise methods and models for classification and regression.
When based on imprecise methods, classification algorithms can possibly assign to an instance a set of class labels instead of a single one. Measuring the quality of such (partially) indeterminate results by single numbers is important to fairly compare different models. An open discussion about the results and the challenges in this field will be presented.
Furthermore, several imprecise methods for regression have been proposed in the recent years. These methods generalize precise regression in different ways. To evaluate and compare imprecise regression methods it is important to characterize them by their statistical properties like, e.g., coverage probability, consistency, and robustness. However, in some cases the notions are too narrow to be directly applied to the generalized method. We will discuss possible generalizations of these notions and explore the statistical properties of selected imprecise regression methods.
We will be very happy about anyone interested in taking part in the discussion. If you want to make a contribution to this day, please let us know by Friday 31 August 2012.
|
Alessandro Antonucci, Andrea Wiencierz
|
| Thu 13th |
Learning and updating
The problem of learning or estimating a statistical or a
probabilistic model is an old one. The estimated model can then be
used for various purposes (uncertainty propagation, regression and
classification, ...).
This day of the workshop is devoted to the problem of learning within
imprecise probability (IP) frameworks. Two particular questions that
arise in this situation are
(1) what is the interest of using IP in my
problem? and
(2) Is there an approach to handle the learning problem
efficiently?
These questions become even more critical when problems get complex,
i.e., when learning multidimensional models, models with an high
number of parameters (e.g., mixture models), models of structured data
(rankings, networks, ...) or when dealing with situations where
uncertainty or inconsistencies are particularly severe (e.g., rare
events, conflict between data and expert knowledge, uncertain data).
We aim at focusing at such questions on this day, with discussions and
presentations about practical and theoretical challenges. Discussions
about problems specific to the IP field (e.g., coherence in updating)
are also welcomed.
If you want to make a contribution to this day, please let us know by
Friday 31 August 2012 by providing us with a title and short
description of your intended contribution. An estimation of the time
you would like to have is also welcomed.
|
Sébastien Destercke, Georg Schollmeyer
|
| Fri 14th |
Open topics
|
Tahani Coolen-Maturi, Marco Cattaneo
|
| Sat 15th |
Excursion
|
Andrea Wiencierz
|
