We consider the problem of aggregating a number of imprecise probability models into a joint one, and compare four aggregation rules: conjunction, disjunction, mixture and Pareto. We investigate for which particular cases of imprecise probability models these operators are closed, meaning that the output belongs to the same family as the inputs. Specifically, we analyse this problem for comparative probability models, 2-monotone capacities, probability intervals, belief functions, p-boxes and minitive measures.
This paper investigates how to characterize (axiomatize) coherent rejection functions for arbitrary objects. In this very general setting, we assume that there is some sensible notion of preference over these objects, the existence of an "objective" background order treated as a constraint on preferences, and that the preferences constitute a strict partial order; we assume nothing else about the structure of the objects. The first insight is that we can represent binary comparisons of objects by ordered pairs - which are just another kind of thing; it is simple to represent coherent preference orders directly in terms of these higher-order objects. Once we have coherence axioms for sets of desirable things of this kind, we can immediately see what the corresponding coherence axioms are for sets of desirable sets (SDS) of these things. But these are not in one-to-one correspondence with rejection functions for the original things; they express more. The second insight is that rejection functions do correspond (almost) exactly to objects we will call "set preferences". So I give coherence axioms for set preferences, which equivalently fully characterize coherence for rejection functions. I present two main results: (1) coherence axioms for set preferences, and (2) the connection between coherent rejection functions for the original things and coherence for SDS of the ordered-pair things.
This paper faces the problem of pricing a European derivative contract inside a discrete-time market with frictions in the form of bid-ask spreads. To this aim, we use a Markov and time-homogeneous multiplicative binomial process under Dempster-Shafer uncertainty for modeling the bid price of a non-dividend paying stock. Next, by taking alpha-mixtures of bid-ask prices, where alpha is an element of[0, 1] acts like a pessimism index, we propose a dynamic pricing rule consisting in the recursive one-step alpha-mixture of upper and lower conditional Choquet expectations. We provide a dynamic pricing rule that has a closed-form for monotonic contract functions. Finally, we perform a calibration procedure on market data, complying with the tuning of alpha.
Distortion or neighbourhood models are useful tools in the imprecise probability theory allowing to robustify a probabilistic model by considering a neighbourhood around a given probability measure. In this work, we tackle the more general problem of distorting a lower probability. This problem can be interesting when we believe that a given lower probability is too precise, or in coalitional game theory when the set of solutions is empty. Our main purpose is to investigate how the linear vacuous and pari mutuel models can be defined for the distortion of lower probabilities, and for this aim we address the problem in a more general manner: we extend the vertical barrier models, which include the linear vacuous and pari mutuel models, and investigate the properties they satisfy.
We study the marginal problem for sets of desirable gamble sets (SoDGSes), which is equivalent to studying this problem for choice functions. More specifically, given a number of marginal SoDGSes on overlapping domains, we establish conditions under which they are compatible in the sense that they can be derived from a common joint SoDGS. We do so for SoDGSes that admit a concrete finite representation. Our main result is that such SoDGSes are compatible if they are pairwise compatible and if a running intersection property is satisfied.
It has long been considered a truism that we can learn more from a variety of sources than from highly correlated sources. This truism is captured by the Variety of Evidence Thesis. To the surprise of many, this thesis turned out to fail in a number of Bayesian settings. In other words, replication can trump variation. Translating the thesis into IPwe obtain two distinct, a priori plausible formulations in terms of 'increased confirmation' and 'uncertainty reduction', respectively. We investigate both formulations, which both fail for different parameters and different reasons, that cannot be predicted prior to formal analysis. The emergence of two distinct formulations distinguishing confirmation increase from uncertainty reduction, which are conflated in the Bayesian picture, highlights fundamental differences between IP and Bayesian reasoning.
We examine the consequences of directed cycles in graph-based representations of joint distributions, investigating the effect of cycles on Markov conditions and on Gibbs factorizations. We focus on Logical Credal Networks, a flexible and general formalism, showing that Koster's theory of Directed-Undirected Mixed Graphs (DUMGs) leads to an interesting Gibbs factorization. We show that inferences with DUMGs lead to multilinear programs. We also study the failure of global Markov conditions in cyclic structural equation models, connecting that failure to probabilistic imprecision under interventions.
This paper presents a novel extension of the discrete Bayesian classifier (DBC) into a set-valued classification framework grounded in imprecise probability theory. The standard DBC framework, which relies on partitioning the input space into profiles and estimating class-conditional probabilities, may not be very robust to distribution changes or imperfections in observed data. In the hope to mitigate such issues, we introduce the Credal Discrete Classifier (CDC), an imprecise-probabilistic extension of the traditional Bayesian approach. By representing uncertainties in the estimated probabilities through belief functions, CDC offers interval-valued risks and set-valued decisions, thereby enhancing robustness. Experimental results on several benchmark datasets demonstrate that CDC effectively balances accuracy and determinacy by allowing for set-valued predictions in uncertain contexts, often outperforming or matching traditional precise classifiers.
One of the main objections against an imprecise probabilistic framework is the apparent absurdity of dilation when seemingly irrelevant evidence makes your belief in a proposition much less certain than it intuitively ought to be. In this work, after critically analysing an argument by White and refined by Topey, as well as responses by imprecise probabilists, I argue that one way to greatly alleviate the tension this type of case poses is to adopt a form of 'time-slice' Bayesianism. In the form I envision it, it means that our degrees of belief in A at time t(i) are no longer ontologically defined as the result of updating our degrees of belief at time t(i-1) with the evidence E-i-1,E-i we obtained in between, but as a function of our total evidence available at time t(i) and a fundamental prior set of credences. I explain why this move, which forces us to regard all probabilities as conditional probabilities outside time, greatly diminishes the intuitive appeal of dilation-based counterexamples to the soundness of imprecise Bayesianism.
Gas-solid reactions play a crucial role in sustainability, yet very few studies have focused on the uncertainty of their chemical kinetic parameters and its propagation. In this pioneering work, based on a numerically generated synthetic dataset of conversion profiles, we address the uncertainty arising from variations in powder particle size between any two small powder samples, which impacts experimental conversion profiles. This variation is assumed to follow a log-normal distribution and is propagated into the uncertainty of the activation energy, which subsequently affects the uncertainty of the delay time at which the chemical conversion reaches a desired value under other conditions. Both precise and imprecise Bayesian approaches were compared. The results indicate that precise Bayesian methods struggle to differentiate effectively between varying levels of knowledge. In contrast, the imprecise Bayesian method based on a set of truncated normal distributions proved efficient and significantly more useful than the one based on uniform priors for this purpose. Finally, we provide suggestions on how to apply this methodology to more realistic settings.
In the paper, we consider two possible extensions of a credal network: the strong extension and the main natural extension. We prove that for both extensions the condition of the d-separation is preserved. The proof is based on some properties of conditional independence in such credal networks.
Ranking theory is a well-established account of rational belief (= taking to be true) and its dynamics. Each formal representation of epistemic states should be extendable to a decision theory; ultimately, beliefs manifest themselves in rational action. However, the decision-theoretic extension of ranking theory is wanting. Giang and Shenoy's "A Qualitative Linear Utility Theory for Spohn's Theory of Epistemic Beliefs" (2000) is the only proposal so far. This paper will modify this proposal. The modification allows building a bridge to the field of (conditional) deontic logic, indeed advancing this field and thus instructing legal theory. This is to show that the modification is a fruitful one deserving further investigation and application.
Uncertainty quantification for conditional quantities-i.e., unknown quantities related to the conditional distribution of a response variable given covariates-is a fundamental problem. Existing methods often rely on restrictive parametric assumptions or smoothness conditions and typically only provide set estimates for the unknown quantities. This paper introduces Inferential Models (IMs) that offer possibilistic uncertainty quantification for conditional quantities, going beyond the simple provision of set estimates. Unlike traditional approaches, the proposed IMs are fully distribution-free and can handle both random and fixed conditional quantities. Moreover, they satisfy a marginal validity criterion, ensuring proper calibration of all IMs' outputs when averaged over the covariates distribution. Illustrations of this framework are provided for both random and fixed conditional quantities-specifically, a future response and the conditional median, respectively.
We investigate techniques for applying strictly proper scoring rules to elicit arbitrary sets of probabilities for an event and for eliciting sets of countably additive (finite dimensional) joint distributions. We contrast E-admissibility, Maximality, and Gamma-Maximin as three IP decision rules for these elicitations. The techniques we investigate apply with sets of probabilities that need not be convex or even connected, and with distributions that may lack moments. We address some challenges to applying these techniques for eliciting merely finitely additive probability distributions.
Inferential models (IMs) are data-dependent, imprecise-probabilistic structures designed to quantify uncertainty about unknowns. As the name suggests, the focus has been on uncertainty quantification for inference and on its reliability properties in that context. The present paper develops an IM framework for decision making, and investigates the decision-theoretic implications of the IM's reliability guarantees. I show that the IM's assessment of an action's quality, defined by a Choquet integral, will not be too optimistic compared to that of an oracle. This ensures that the IM tends not to favor actions that the oracle doesn't also favor, hence a IM is reliable for decision making too. In a certain special class of structured statistical models, further connections can be made between the IM's recommended actions and those recommended by Bayesian/fiducial frameworks, from which certain optimality conclusions can be drawn.
Meta-analyses are vital for synthesizing evidence in medical research, but conflicts of interest can introduce research bias, undermining the reliability of the synthesized findings. This paper proposes a new robust Bayesian meta-analysis model. The model inflates uncertainty of low-quality studies and incorporates a bias term for studies subject to conflicts of interest. Using a random-effects model and sensitivity analysis with bounded probabilities, the model enables robust adjustments for conflicts of interest in meta-analytic contexts. A case study on antidepressant trials illustrates the potential application of the model.
We consider random walks on weighted graphs where the edge weights are interval-valued, reflecting uncertainty in the relationships between vertices. We study this model in the framework of reversible imprecise Markov chains by viewing them as sets of precise inhomogeneous Markov chains. We define and analyze the notion of reversibility for such sets by extending classical reversibility concepts to the imprecise setting. These concepts are then applied to interval-weighted random walks, where the individual weight functions may not be symmetric but their sets exhibit symmetry. Our approach provides a basis for analyzing random walks in environments with uncertain or incomplete information.
This paper explores belief inference in credal networks using Dempster-Shafer theory. By building on previous work, we propose a novel framework for propagating uncertainty through a subclass of credal networks, namely chains. The proposed approach efficiently yields conservative intervals through belief and plausibility functions, combining computational speed with robust uncertainty representation. Key contributions include formalizing belief-based inference methods and comparing belief-based inference against classical sensitivity analysis. Numerical results highlight the advantages and limitations of applying belief inference within this framework, providing insights into its practical utility for chains and for credal networks in general.
Based on existing ideas in the field of imprecise probabilities, we present a new approach for assessing the reliability of the individual predictions of a generative probabilistic classifier. We call this approach robustness quantification, compare it to uncertainty quantification, and demonstrate that it continues to work well even for classifiers that are learned from small training sets that are sampled from a shifted distribution.
The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate non-linear utility. However, when repeated gambles are played over time -- especially in intertemporal choice where rewards compound multiplicatively -- the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.
