Probabilistic Interpretation Research Articles

Bayesian Decision Curve Analysis With Bayesdca

ABSTRACTClinical decisions are often guided by clinical prediction models or diagnostic tests. Decision curve analysis (DCA) combines classical assessment of predictive performance with the consequences of using these strategies for clinical decision‐making. In DCA, the best decision strategy is the one that maximizes the net benefit: the net number of true positives (or negatives) provided by a given strategy. Here, we employ Bayesian approaches to DCA, addressing four fundamental concerns when evaluating clinical decision strategies: (i) which strategies are clinically useful, (ii) what is the best available decision strategy, (iii) which of two competing strategies is better, and (iv) what is the expected net benefit loss associated with the current level of uncertainty. While often consistent with frequentist point estimates, fully Bayesian DCA allows for an intuitive probabilistic interpretation framework and the incorporation of prior evidence. We evaluate the methods using simulation and provide a comprehensive case study. Software implementation is available in the bayesDCA R package. Ultimately, the Bayesian DCA workflow may help clinicians and health policymakers adopt better‐informed decisions.

Improving ViT interpretability with patch-level mask prediction

Vision Transformers (ViTs) have demonstrated remarkable performances on various computer vision tasks. Attention scores are often used to explain the decision-making process of ViTs, showing which tokens are more important than others. However, the attention scores have several limitations as an explanation for ViT, such as conflicting with other explainable methods or highlighting unrelated tokens. In order to address this limitation, we propose a novel method for generating a visual explanation map from ViTs. Unlike previous approaches that rely on attention scores, our method leverages ViT features and conducts a single forward pass through our Patch-level Mask prediction (PM) module. Our visual explanation map provides class-dependent and probabilistic interpretation that can identify crucial regions of model decisions. Experimental results demonstrate that our approach outperforms previous techniques in both classification and interpretability aspects. Additionally, it can be applied to the weakly-supervised object localization (WSOL) tasks using pseudo mask labels. Our method requires no extra parameters and necessitates minimal locality supervision, utilizing less than 1% of the ImageNet-1k training dataset.

Large deviation principle for backward stochastic differential equations with a stochastic Lipschitz condition on z

In this paper, we provide a probabilistic interpretation of the viscosity solution of a parabolic partial differential equation through the solution of a class of quadratic backward stochastic differential equations (BSDEs). Additionally, we demonstrate the convergence and the large deviation principle for the solutions of these quadratic BSDEs, which are linked to a family of Markov processes with diffusion coefficients that tend toward zero.

Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers

Abstract Interpreting the representation and generalization powers has been a long-standing issue in the field of machine learning (ML) and artificial intelligence. This work contributes to uncovering the emergence of universal scaling laws in quantum-probabilistic ML. We take the generative tensor network (GTN) in the form of a matrix product state as an example and show that with an untrained GTN (such as a random TN state), the negative logarithmic likelihood (NLL) $L$ generally increases linearly with the number of features $M$, i.e., $L \simeq k M + const$. This is a consequence of the so-called “catastrophe of orthogonality,” which states that quantum many-body states tend to become exponentially orthogonal to each other as $M$ increases. We reveal that while gaining information through training, the linear scaling law is suppressed by a negative quadratic correction, leading to $L \simeq \beta M - \alpha M^2 + const$. The scaling coefficients exhibit logarithmic relationships with the number of training samples and the number of quantum channels $\chi$. The emergence of the quadratic correction term in NLL for the testing (training) set can be regarded as evidence of the generalization (representation) power of GTN. Over-parameterization can be identified by the deviation in the values of $\alpha$ between training and testing sets while increasing $\chi$. We further investigate how orthogonality in the quantum feature map relates to the satisfaction of quantum probabilistic interpretation, as well as to the representation and generalization powers of GTN. The unveiling of universal scaling laws in quantum-probabilistic ML would be a valuable step toward establishing a white-box ML scheme interpreted within the quantum probabilistic framework.

An Introduction to Bayesian Approaches to Trial Design and Statistics for Stroke Researchers.

While the majority of stroke researchers use frequentist statistics to analyze and present their data, Bayesian statistics are becoming more and more prevalent in stroke research. As opposed to frequentist approaches, which are based on the probability that data equal specific values given underlying unknown parameters, Bayesian approaches are based on the probability that parameters equal specific values given observed data and prior beliefs. The Bayesian paradigm allows researchers to update their beliefs with observed data to provide probabilistic interpretations of key parameters, for example, the probability that a treatment is effective. In this review, we outline the basic concepts of Bayesian statistics as they apply to stroke trials, compare them to the frequentist approach using exemplary data from a randomized trial, and explain how a Bayesian analysis is conducted and interpreted.

Anomalous Random Flights and Time-Fractional Run-and-Tumble Equations

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

Multiteam Semantics for Interventionist Counterfactuals: Probabilities and Causation

In (Barbero and Sandu 2020 Journal of Philosophical Logic, 50, 471-521), we showed that languages encompassing interventionist counterfactuals and causal notions based on them (as e.g. in Pearl’s and Woodward’s manipulationist approaches to causation) as well as information-theoretic notions (such as learning and dependence) can be interpreted in a semantic framework which combines the traditions of structural equation modeling and of team semantics. We now present a further extension of this framework (causal multiteams) which allows us to talk about probabilistic causal statements. We analyze the expressivity resources of two causal-probabilistic languages, one finitary and one infinitary. We show that many causal-probabilistic notions from the literature on causal inference can already be expressed in the finitary language, and we prove a normal form theorem that throws a new light on Pearl’s “ladder of causation”. In addition, we provide an exact semantic characterization of the infinitary language, which shows that this language captures precisely those causal-probabilistic statements that do not commit us to any specific interpretation of probability; and we prove that no usual, countable language is apt for this task.

Making sense of ghosts

Ghosts have been a stumbling block in the development of a UV complete quantum field theory for gravity. We discuss how difficulties associated with ghosts are overcome in the context of 0+1d QFT. Obtaining a probability interpretation is the key issue, and for this we discuss how an appropriate inner product can be constructed to define a sensible Born rule. Ghost theories are intrinsically unitary and perturbatively stable. They can also display nonperturbative stability even when the corresponding normal theory does not. The spectra and propagators are numerically obtained at both weak and strong coupling. Normalizable wave functions are obtained for the energy eigenstates and they show a violation of normal parity. We discuss connections to PT-symmetric quantum mechanics.

Competition between slicing and buckling underlies the erratic nature of paper cuts.

By enabling the dissemination and storage of information, paper has been central to human culture for more than a millennium. Its use is, however, associated with a common injury: the paper cut. Surprisingly, the physics underpinning a flexible sheet of paper slicing into soft tissues remains unresolved. In particular, the unpredictable occurrence of paper cuts, often restricted to a limited thickness range, has not been explained. Here we visualize and quantify the motion, deformation, and stresses during paper cuts, uncovering a remarkably complex relationship between cutting, geometry, and material properties. A model based on the hypothesis that a competition between slicing and buckling controls the probability of initiating a paper cut is developed and successfully validated. This explains why paper with a specific thickness is most hazardous (65µm, corresponding, e.g., to dot matrix paper) and suggests a probabilistic interpretation of irregular occurrence of paper cuts. Stimulated by these findings, we finally show how a recyclable cutting tool can harness the surprising power of paper.

Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples

The fact that the famous basic reproduction number R0, i.e., the largest eigenvalue of the next generation matrix FV−1, sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, −V, is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F. Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R0 with respect to explicit “age kernels” a(t), which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.

Extending Binary Neural Networks to Bayesian Neural Networks with Probabilistic Interpretation of Binary Weights

Extending Binary Neural Networks to Bayesian Neural Networks with Probabilistic Interpretation of Binary Weights

FEIR: Quantifying and Reducing Envy and Inferiority for Fair Recommendation of Limited Resources

Recommendation in settings such as e-recruitment and online dating involves distributing limited opportunities, which differs from recommending practically unlimited goods such as in e-commerce or music recommendation. This setting calls for novel approaches to quantify and enforce fairness. Indeed, typical recommender systems recommend each user their top relevant items, such that desirable items may be recommended simultaneously to more and to less qualified individuals. This is arguably unfair to the latter. Indeed, when they pursue such a desirable recommendation (e.g., by applying for a job), they are unlikely to be successful. To quantify fairness in such settings, we introduce inferiority : a novel (un)fairness measure that quantifies the competitive disadvantage of a user for their recommended items. Inferiority is complementary to envy : a previously-proposed fairness notion that quantifies the extent to which a user prefers other users’ recommendations over their own. We propose to use both inferiority and envy in combination with an accuracy-related measure called utility : the aggregated relevancy scores of the recommended items. Unfortunately, none of these three measures are differentiable, making it hard to optimize them, and restricting their immediate use to evaluation only. To remedy this, we reformulate them in the context of a probabilistic interpretation of recommender systems, resulting in differentiable versions. We show how these loss functions can be combined in a multi-objective optimization problem that we call FEIR (Fairness through Envy and Inferiority Reduction), used as a post-processing of the scores from any standard recommender system. Experiments on synthetic and real-world data show that the proposed approach effectively improves the trade-offs between inferiority, envy and utility, compared to the naive recommendation and the state-of-the-art method for the related problem of congestion alleviation in job recommendation. We discuss and enhance the practical impact of our findings on a wide range of real-world recommendation scenarios, and we offer implementations of visualization tools to render the envy and inferiority metrics more accessible.

Two semantic interpretations of probabilities in description logics of typicality

Abstract In this work we focus on extensions of Description Logics (DLs) of typicality by means of probabilities. We introduce a novel extension of the logic of typicality $\mathcal{ALC}+\mathbf{T}_{\mathbf{R}}$, able to represent and reason about typical properties and defeasible inheritance in DLs. The novel logic ($\mathcal{ALC}\mathbf{T}^{\textsf{P}}$: Typical $\mathcal{ALC}$ with Probabilities as Proportions) allows inclusions of the form $\mathbf{T}(C) \sqsubseteq ^{p} D$, with probability $p$ representing a proportion, meaning that “all the typical $C$s are $D$s, and the probability that a $C$ element is not a $D$ element is $1-p$”. We also compare and confront this novel logic with a similar one already presented in the literature ($\mathbf{T}^{\textsf{CL}}$, introduced in Lieto and Pozzato (2020, J. Exp. Theor. Artif. Intell., 32, 769–804)), inspired by the DISPONTE semantics and that allows inclusions of the form $p \:: \ \mathbf{T}(C) \sqsubseteq D$ with probability $p$, where $p$ represents a degree of belief, whose meaning is that “we believe with a degree $p$ that typical $C$s’ are also $D$s.”. We then show that the proposed $\mathcal{ALC}\mathbf{T}^{\textsf{P}}$ extension (like the previous $\mathbf{T}^{\textsf{CL}}$) can be applied in order to tackle a specific and challenging problem in the field of common-sense reasoning, namely the combination of prototypical concepts, that have been shown to be problematic to model for other symbolic approaches like fuzzy logic. We show that, for the proposed extension, the complexity of reasoning remains ExpTime-complete as for the underlying standard monotonic DL $\mathcal{ALC}$.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝ d for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝ d . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

Vývoj a interpretace pojmu pravděpodobnost

Considerations related to randomness appear relatively late in European history, at the turn of the Middle Ages and the modern era. They initially concern to the chances of winning in various games or situations and later move on to introduce classical and geometric probabilities. From a mathematical point of view, the probability calculus is completed by Kolmogorov’s axiomatic theory. However, many open questions, problems and paradoxes remain in the way probability is perceived and interpreted. The four main directions in the concept of probability (logical, frequentist, subjective and propensity) are closely related to the way of perceiving randomness (epistemological or ontological). Reflections on the evolution of the perception of randomness and the interpretation of probability bring the ability to navigate the basic principles and findings of science and contributes to a deeper understanding of the entire issue.

A machine-learning-based alternative to phylogenetic bootstrap.

Currently used methods for estimating branch support in phylogenetic analyses often rely on the classic Felsenstein's bootstrap, parametric tests, or their approximations. As these branch support scores are widely used in phylogenetic analyses, having accurate, fast, and interpretable scores is of high importance. Here, we employed a data-driven approach to estimate branch support values with a probabilistic interpretation. To this end, we simulated thousands of realistic phylogenetic trees and the corresponding multiple sequence alignments. Each of the obtained alignments was used to infer the phylogeny using state-of-the-art phylogenetic inference software, which was then compared to the true tree. Using these extensive data, we trained machine-learning algorithms to estimate branch support values for each bipartition within the maximum-likelihood trees obtained by each software. Our results demonstrate that our model provides fast and more accurate probability-based branch support values than commonly used procedures. We demonstrate the applicability of our approach on empirical datasets. The data supporting this work are available in the Figshare repository at https://doi.org/10.6084/m9.figshare.25050554.v1, and the underlying code is accessible via GitHub at https://github.com/noaeker/bootstrap_repo.

Searching for the General Science of Evidence: Venn on Probability and Induction

In this paper Venn's account of probability inference and induction is examined, tracing their differences as well as how they ‘co-operate’ in inferences from particulars to particulars. We discuss the role of mathematical idealizations in making probability inferences, the celebrated rule of succession and we delve into the nature of the reference class problem arguing that for Venn it is a common problem for both induction and inference in probability. Our approach is both historical and philosophical attempting to sketch Venn's position both in the philosophy of probability and induction of his time and in relation to the twentieth-century frequentism.

Entropy production for diffusion processes across a semipermeable interface

The emerging field of stochastic thermodynamics extends classical ideas of entropy, heat, and work to nonequilibrium systems. One notable finding is that the second law of thermodynamics typically only holds after taking appropriate averages with respect to an ensemble of stochastic trajectories. The resulting average rate of entropy production then quantifies the degree of departure from thermodynamic equilibrium. In this paper we investigate how the presence of a semipermeable interface increases the average entropy production of a single diffusing particle. Starting from the Gibbs-Shannon entropy for the particle probability density, we show that a semipermeable interface or membrane S increases the average rate of entropy production by an amount that is equal to the product of the flux through the interface and the logarithm of the ratio of the probability density on either side of the interface, integrated along S. The entropy production rate thus vanishes at thermodynamic equilibrium, but can be nonzero during the relaxation to equilibrium, or if there exists a nonzero stationary equilibrium state (NESS). We then give a probabilistic interpretation of the interfacial entropy production rate using so-called snapping out Brownian motion. This also allows us to construct a stochastic version of entropy production. Finally, we illustrate the theory using the example of diffusion with stochastic resetting on a circle, and find that the average rate of interfacial entropy production is a nonmonotonic function of the resetting rate and the permeability. Published by the American Physical Society 2024

A probabilistic interpretation of corrosion state through half-cell potential and electrical resistivity measures: The Flaminio Bridge in Rome

During the last decades, non-destructive investigation approaches have played an essential role in preserving structural integrity, capacity, and finding early defects in reinforced concrete (RC) heritage affected by time-dependent degradation. One of the most widespread structural RC damages belongs to corrosion phenomena. Electro-chemical approaches stand out among the non-destructive techniques presently available for identifying corrosive problems in-situ. The latter gives some knowledge in terms of physical quantities aiming to evaluate structural conditions by collecting indirect measures. Nevertheless, results are correlated to strong interpretative scale codes, which may not always provide appropriate interpretation of the defect's identity. In the paper a probabilistic degradation model, based on a statistical interpretation of the in situ electro-chemical measurements, has been proposed with the aim to give a contribution for structural diagnosis. Finally, results are validated through structural visual inspection.

Interpretation of probability in structural safety – A philosophical conundrum

The term probability is essential in the domain of structural safety and yet its interpretation is, even after more than 50 years of application, still a subject of discussion. For instance, the probability of failure of structures belonging to the same cohort for a specific period of time, is often understood in a pure frequentist way as an observable average number of failure events for that period and portfolio. By contrast, the Bayesian interpretation considers probability as a degree of belief and a reflection of the state of information to the best belief or knowledge of the decision maker. In the field of structural reliability, depending on the type of decision problem, probabilities are often referred to as nominal (or notional) measures of uncertainty to emphasize that these values are conditional on a model and available observations. Probabilistic methods then serve primarily to undertake the book-keeping required to assign probabilities to different possible outcomes or decisions in consistency with models, available observations and other relevant information. This paper discusses the background of these interpretations and the degree to which correspondence between reliability calculations and observations of failures can be expected and/or achieved. Recommendations corresponding to the JCSS line of thinking will be summarized in Section 8.