A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.
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Flamenco, recognized by UNESCO as part of the Intangible Cultural Heritage of Humanity, is a profound expression of cultural identity rooted in Andalusia, Spain. However, there is a lack of quantitative studies that help identify characteristic patterns in this long-lived music tradition. In this work, we present a computational analysis of Flamenco lyrics, employing natural language processing and machine learning to categorize over 2000 lyrics into their respective Flamenco genres, termed as $\textit{palos}$. Using a Multinomial Naive Bayes classifier, we find that lexical variation across styles enables to accurately identify distinct $\textit{palos}$. More importantly, from an automatic method of word usage, we obtain the semantic fields that characterize each style. Further, applying a metric that quantifies the inter-genre distance we perform a network analysis that sheds light on the relationship between Flamenco styles. Remarkably, our results suggest historical connections and $\textit{palo}$ evolutions. Overall, our work illuminates the intricate relationships and cultural significance embedded within Flamenco lyrics, complementing previous qualitative discussions with quantitative analyses and sparking new discussions on the origin and development of traditional music genres.
Purpose: Most studies evaluating artificial intelligence (AI) models that detect abnormalities in neuroimaging are either tested on unrepresentative patient cohorts or are insufficiently well-validated, leading to poor generalisability to real-world tasks. The aim was to determine the diagnostic test accuracy and summarise the evidence supporting the use of AI models performing first-line, high-volume neuroimaging tasks. Methods: Medline, Embase, Cochrane library and Web of Science were searched until September 2021 for studies that temporally or externally validated AI capable of detecting abnormalities in first-line CT or MR neuroimaging. A bivariate random-effects model was used for meta-analysis where appropriate. PROSPERO: CRD42021269563. Results: Only 16 studies were eligible for inclusion. Included studies were not compromised by unrepresentative datasets or inadequate validation methodology. Direct comparison with radiologists was available in 4/16 studies. 15/16 had a high risk of bias. Meta-analysis was only suitable for intracranial haemorrhage detection in CT imaging (10/16 studies), where AI systems had a pooled sensitivity and specificity 0.90 (95% CI 0.85 - 0.94) and 0.90 (95% CI 0.83 - 0.95) respectively. Other AI studies using CT and MRI detected target conditions other than haemorrhage (2/16), or multiple target conditions (4/16). Only 3/16 studies implemented AI in clinical pathways, either for pre-read triage or as post-read discrepancy identifiers. Conclusion: The paucity of eligible studies reflects that most abnormality detection AI studies were not adequately validated in representative clinical cohorts. The few studies describing how abnormality detection AI could impact patients and clinicians did not explore the full ramifications of clinical implementation.
We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.
In the realm of cost-sharing mechanisms, the vulnerability to Sybil strategies -- also known as false-name strategies, where agents create fake identities to manipulate outcomes -- has not yet been studied. In this paper, we delve into the details of different cost-sharing mechanisms proposed in the literature, highlighting their non-Sybil-resistant nature. Furthermore, we prove no deterministic, anonymous, truthful, Sybil-proof, upper semicontinuous, and individually rational cost-sharing mechanism for public excludable goods is better than $\Omega(n)$-approximate. This finding reveals an exponential increase in the worst-case social cost in environments where agents are restricted from using Sybil strategies. To circumvent these negative results, we introduce the concept of \textit{Sybil Welfare Invariant} mechanisms, where a mechanism does not decrease its welfare under Sybil strategies when agents choose weak dominant strategies and have subjective prior beliefs over other players' actions. Finally, we prove that the Shapley value mechanism for symmetric and submodular cost functions holds this property, and so deduce that the worst-case social cost of this mechanism is the $n$th harmonic number $\mathcal H_n$ under equilibrium with Sybil strategies, matching the worst-case social cost bound for cost-sharing mechanisms. This finding suggests that any group of agents, each with private valuations, can fund public excludable goods both permissionless and anonymously, achieving efficiency comparable to that of non-anonymous domains, even when the total number of participants is unknown.ess and anonymously, achieving efficiency comparable to that of permissioned and non-anonymous domains, even when the total number of participants is unknown.
We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The notion of data separability is not needed, which is in contrast to the classical set up of multi-class logistic regression in which each data sample belongs to one class. We also provide a general and constructive estimate of the convergence rate to the maximum likelihood estimate when gradient descent is used as the optimizer. Our estimate involves bounding the condition number of the Hessian of the maximum likelihood function. The approaches used in this article rely on a simple operator-theoretic framework.
Evolutionary game theory assumes that individuals maximize their benefits when choosing strategies. However, an alternative perspective proposes that individuals seek to maximize the benefits of others. To explore the relationship between these perspectives, we develop a model where self- and other-regarding preferences compete in public goods games. We find that other-regarding preferences are more effective in promoting cooperation, even when self-regarding preferences are more productive. Cooperators with different preferences can coexist in a new phase where two classic solutions invade each other, resulting in a dynamical equilibrium. As a consequence, a lower productivity of self-regarding cooperation can provide a higher cooperation level. Our results, which are also valid in a well-mixed population, may explain why other-regarding preferences could be a viable and frequently observed attitude in human society.
An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.
The Robbins estimator is the most iconic and widely used procedure in the empirical Bayes literature for the Poisson model. On one hand, this method has been recently shown to be minimax optimal in terms of the regret (excess risk over the Bayesian oracle that knows the true prior) for various nonparametric classes of priors. On the other hand, it has been long recognized in practice that Robbins estimator lacks the desired smoothness and monotonicity of Bayes estimators and can be easily derailed by those data points that were rarely observed before. Based on the minimum-distance distance method, we propose a suite of empirical Bayes estimators, including the classical nonparametric maximum likelihood, that outperform the Robbins method in a variety of synthetic and real data sets and retain its optimality in terms of minimax regret.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
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