Proceedings of Machine Learning Research

Function-coherent gambles with non-additive sequential dynamics

Tue, 20 May 2025 00:00:00 +0000

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.

Function-coherent gambles

Tue, 20 May 2025 00:00:00 +0000

The desirable gambles framework provides a foundational approach to imprecise probability theory but relies heavily on linear utility assumptions. This paper introduces function-coherent gambles, a generalization that accommodates non-linear utility while preserving essential rationality properties. We establish core axioms for function-coherence and prove a representation theorem that characterizes acceptable gambles through continuous linear functionals. The framework is then applied to analyze various forms of discounting in intertemporal choice, including hyperbolic, quasi-hyperbolic, scale-dependent, and state-dependent discounting. We demonstrate how these alternatives to constant-rate exponential discounting can be integrated within the function-coherent framework. This unified treatment provides theoretical foundations for modeling sophisticated patterns of time preference within the desirability paradigm, bridging a gap between normative theory and observed behavior in intertemporal decision-making under genuine uncertainty.

A robust Bayesian model to quantify and adjust for study quality and conflict of interest in meta-analyses

Tue, 20 May 2025 00:00:00 +0000

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.

Rank-based decision theory and conditional deontic logic

Tue, 20 May 2025 00:00:00 +0000

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.

Random walks on graphs with interval weights as a model of reversible imprecise Markov chains

Tue, 20 May 2025 00:00:00 +0000

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.

Elicitation for sets of probabilities and distributions

Tue, 20 May 2025 00:00:00 +0000

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.

Towards conservative inference in credal networks using belief functions: the case of credal chains

Tue, 20 May 2025 00:00:00 +0000

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.

Dynamic $α$-DS mixture pricing in a market with bid-ask spreads

Tue, 20 May 2025 00:00:00 +0000

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 $\alpha \in [0,1]$ 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 $\alpha$.

Distorting lower probabilities using common distortion models

Tue, 20 May 2025 00:00:00 +0000

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.

On the closure of aggregation rules for imprecise probabilities

Tue, 20 May 2025 00:00:00 +0000

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.

Decision-theoretic properties of possibilistic inferential models

Tue, 20 May 2025 00:00:00 +0000

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.

On the value of varied evidence for imprecise probabilities

Tue, 20 May 2025 00:00:00 +0000

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 IP we 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.

Precise and imprecise Bayesianism applied to gas-solid reactions

Tue, 20 May 2025 00:00:00 +0000

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.

Time-slice Bayesianism as a potential solution to the problem of dilation and reflection for imprecise probabilities

Tue, 20 May 2025 00:00:00 +0000

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 $E_{i-1,i}$ 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.

Robustness quantification: a new method for assessing the reliability of the predictions of a classifier

Tue, 20 May 2025 00:00:00 +0000

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.

Conditioning through indifference in quantum mechanics

Tue, 20 May 2025 00:00:00 +0000

We can learn (more) about the state that a quantum system is in through measurements. We look at how to describe the uncertainty about a quantum system’s state conditional on executing such measurements. We show that by exploiting the interplay between desirability, coherence and indifference, a general rule for conditioning can be derived. We then apply this rule to conditioning on measurement outcomes, and show how it generalises to conditioning on a set of measurement outcomes.

A convenient characterisation of convergent upper transition operators

Tue, 20 May 2025 00:00:00 +0000

Motivated by its connection to the limit behaviour of imprecise Markov chains, we introduce and study the so-called convergence of upper transition operators: the condition that for any function, the orbit resulting from iterated application of this operator converges. In contrast, the existing notion of ‘ergodicity’ requires convergence of the orbit to a constant. We derive a very general (and practically verifiable) sufficient condition for convergence in terms of accessibility and lower reachability, and prove that this sufficient condition is also necessary whenever (i) all transient states are absorbed or (ii) the upper transition operator is finitely generated.

The marginal problem for sets of desirable gamble sets

Tue, 20 May 2025 00:00:00 +0000

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.

Dealing with cycles in graph-based probabilistic models: the case of Logical Credal Networks

Tue, 20 May 2025 00:00:00 +0000

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.

Conditioning and AGM-like belief change in the Desirability-Indifference framework

Tue, 20 May 2025 00:00:00 +0000

We show how the AGM framework for belief change (expansion, revision, contraction) can be extended to deal with conditioning in the so-called Desirability-Indifference framework, based on abstract notions of accepting and rejecting options, as well as on abstract notions of events. This level of abstraction allows us to deal simultaneously with classical and quantum probability theory.

Credal discrete classifier

Tue, 20 May 2025 00:00:00 +0000

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.

Distribution-free possibilistic inference on conditional quantities

Tue, 20 May 2025 00:00:00 +0000

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.

Conformal prediction regions are imprecise highest density regions

Tue, 20 May 2025 00:00:00 +0000

Recently, Cella and Martin proved how, under an assumption called consonance, a credal set (i.e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we establish a new relationship between Conformal Prediction and Imprecise Probability (IP) theories, via the IP concept of a cloud. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

Optimal transport for $ε$-contaminated credal sets

Tue, 20 May 2025 00:00:00 +0000

We present generalized versions of Monge’s and Kantorovich’s optimal transport problems with the probabilities being transported replaced by lower probabilities. We show that, when the lower probabilities are the lower envelopes of $\epsilon$-contaminated sets, then our version of Monge’s, and a restricted version of our Kantorovich’s problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich’s optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $\epsilon$-contaminations the lower probability versions of Monge’s and Kantorovich’s optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

D-separation for the strong extension and the main natural extension of a credal network

Tue, 20 May 2025 00:00:00 +0000

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.

Coherent rejection functions for arbitrary things

Tue, 20 May 2025 00:00:00 +0000

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.

The AI off-switch problem as a signalling game: bounded rationality and incomparability

Tue, 20 May 2025 00:00:00 +0000

The off-switch problem is a critical challenge in AI control: if an AI system resists being switched off, it poses a significant risk. In this paper, we model the off-switch problem as a signalling game, where a human decision-maker communicates its preferences about some underlying decision problem to an AI agent, which then selects actions to maximise the human’s utility. We assume that the human is a bounded rational agent and explore various bounded rationality mechanisms. Using real machine learning models, we reprove prior results and demonstrate that a necessary condition for an AI system to refrain from disabling its off-switch is its uncertainty about the human’s utility. We also analyse how message costs influence optimal strategies and extend the analysis to scenarios involving incomparability.