16 research outputs found
A desirability-based axiomatisation for coherent choice functions
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules to imprecise-probabilistic uncertainty models. We provide them with a clear interpretation in terms of attitudes towards gambling, borrowing ideas from the theory of sets of desirable gambles, and we use this interpretation to derive a set of basic axioms. We show that these axioms lead to a full-fledged theory of coherent choice functions, which includes a representation in terms of sets of desirable gambles, and a conservative inference method
A Desirability-Based Axiomatisation for Coherent Choice Functions
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that typically arise from applying decision
rules to imprecise-probabilistic uncertainty models. We provide them with a
clear interpretation in terms of attitudes towards gambling, borrowing ideas
from the theory of sets of desirable gambles, and we use this interpretation to
derive a set of basic axioms. We show that these axioms lead to a full-fledged
theory of coherent choice functions, which includes a representation in terms
of sets of desirable gambles, and a conservative inference method
Outer Approximations of Coherent Lower Probabilities Using Belief Functions
We investigate the problem of outer approximating a coherent lower probability with a more tractable model. In particular, in this work we focus on the outer approximations made by belief functions. We show that they can be obtained by solving a linear programming problem. In addition, we consider the subfamily of necessity measures, and show that in that case we can determine all the undominated outer approximations in a simple manner
Optimal control of a linear system subject to partially specified input noise
One of the most basic problems in control theory is that of controlling a discrete-time linear system subject to uncertain noise with the objective of minimising the expectation of a quadratic cost. If one assumes the noise to be white, then solving this problem is relatively straightforward. However, white noise is arguably unrealistic: noise is not necessarily independent and one does not always precisely know its expectation. We first recall the optimal control policy without assuming independence, and show that in this case computing the optimal control inputs becomes infeasible. In a next step, we assume only knowledge of lower and upper bounds on the conditional expectation of the noise, and prove that this approach leads to tight lower and upper bounds on the optimal control inputs. The analytical expressions that determine these bounds are strikingly similar to the usual expressions for the case of white noise
A frequentist framework of inductive reasoning
Reacting against the limitation of statistics to decision procedures, R. A.
Fisher proposed for inductive reasoning the use of the fiducial distribution, a
parameter-space distribution of epistemological probability transferred
directly from limiting relative frequencies rather than computed according to
the Bayes update rule. The proposal is developed as follows using the
confidence measure of a scalar parameter of interest. (With the restriction to
one-dimensional parameter space, a confidence measure is essentially a fiducial
probability distribution free of complications involving ancillary statistics.)
A betting game establishes a sense in which confidence measures are the only
reliable inferential probability distributions. The equality between the
probabilities encoded in a confidence measure and the coverage rates of the
corresponding confidence intervals ensures that the measure's rule for
assigning confidence levels to hypotheses is uniquely minimax in the game.
Although a confidence measure can be computed without any prior distribution,
previous knowledge can be incorporated into confidence-based reasoning. To
adjust a p-value or confidence interval for prior information, the confidence
measure from the observed data can be combined with one or more independent
confidence measures representing previous agent opinion. (The former confidence
measure may correspond to a posterior distribution with frequentist matching of
coverage probabilities.) The representation of subjective knowledge in terms of
confidence measures rather than prior probability distributions preserves
approximate frequentist validity.Comment: major revisio
Archimedean choice functions : an axiomatic foundation for imprecise decision making
If uncertainty is modelled by a probability measure, decisions are typically
made by choosing the option with the highest expected utility. If an imprecise
probability model is used instead, this decision rule can be generalised in
several ways. We here focus on two such generalisations that apply to sets of
probability measures: E-admissibility and maximality. Both of them can be
regarded as special instances of so-called choice functions, a very general
mathematical framework for decision making. For each of these two decision
rules, we provide a set of necessary and sufficient conditions on choice
functions that uniquely characterises this rule, thereby providing an axiomatic
foundation for imprecise decision making with sets of probabilities. A
representation theorem for Archimedean choice functions in terms of coherent
lower previsions lies at the basis of both results