We propose a set of causal estimands that we call ``the mediated probabilities of causation.'' These estimands quantify the probabilities that an observed negative outcome was induced via a mediating pathway versus a direct pathway in a stylized setting involving a binary exposure or intervention, a single binary mediator, and a binary outcome. We outline a set of conditions sufficient to identify these effects given observed data, and propose a doubly-robust projection based estimation strategy that allows for the use of flexible non-parametric and machine learning methods for estimation. We argue that these effects may be more relevant than the probability of causation, particularly in settings where we observe both some negative outcome and negative mediating event, and we wish to distinguish between settings where the outcome was induced via the exposure inducing the mediator versus the exposure inducing the outcome directly. We motivate our quantities of interest by discussing applications to legal and medical questions of causal attribution.
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We explore different aspects of cognitive diversity and its effect on the success of group deliberation. To evaluate this, we use 500 dialogues from small, online groups discussing the Wason Card Selection task - the DeliData corpus. Leveraging the corpus, we perform quantitative analysis evaluating three different measures of cognitive diversity. First, we analyse the effect of group size as a proxy measure for diversity. Second, we evaluate the effect of the size of the initial idea pool. Finally, we look into the content of the discussion by analysing discussed solutions, discussion patterns, and how conversational probing can improve those characteristics. Despite the reputation of groups for compounding bias, we show that small groups can, through dialogue, overcome intuitive biases and improve individual decision-making. Across a large sample and different operationalisations, we consistently find that greater cognitive diversity is associated with more successful group deliberation. Code and data used for the analysis are available in the repository: //github.com/gkaradzhov/cognitive-diversity-groups-cogsci24.
Shannon defined the mutual information between two variables. We illustrate why the true mutual information between a variable and the predictions made by a prediction algorithm is not a suitable measure of prediction quality, but the apparent Shannon mutual information (ASI) is; indeed it is the unique prediction quality measure with either of two very different lists of desirable properties, as previously shown by de Finetti and other authors. However, estimating the uncertainty of the ASI is a difficult problem, because of long and non-symmetric heavy tails to the distribution of the individual values of $j(x,y)=\log\frac{Q_y(x)}{P(x)}$ We propose a Bayesian modelling method for the distribution of $j(x,y)$, from the posterior distribution of which the uncertainty in the ASI can be inferred. This method is based on Dirichlet-based mixtures of skew-Student distributions. We illustrate its use on data from a Bayesian model for prediction of the recurrence time of prostate cancer. We believe that this approach is generally appropriate for most problems, where it is infeasible to derive the explicit distribution of the samples of $j(x,y)$, though the precise modelling parameters may need adjustment to suit particular cases.
Making use of a newly developed package in the computer algebra system SageMath, we show how to perform a full asymptotic analysis by means of the Mellin transform with explicit error bounds. As an application of the method, we answer a question of B\'ona and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.
We present a template for the Promise Constraint Satisfaction Problem (PCSP) which is NP-hard but does not satisfy the current state-of-the-art hardness condition [ACMTCT'21]. We introduce a new "injective" condition based on the smooth version of the layered PCP Theorem and use this new condition to confirm that the problem is indeed NP-hard. In the second part of the article, we establish a dichotomy for Boolean PCSPs defined by templates with polymorphisms in the set of linear threshold functions. The reasoning relies on the new injective condition.
The Lamport diagram is a pervasive and intuitive tool for informal reasoning about "happens-before" relationships in a concurrent system. However, traditional axiomatic formalizations of Lamport diagrams can be painful to work with in a mechanized setting like Agda. We propose an alternative, inductive formalization -- the causal separation diagram (CSD) -- that takes inspiration from string diagrams and concurrent separation logic, but enjoys a graphical syntax similar to Lamport diagrams. Critically, CSDs are based on the idea that causal relationships between events are witnessed by the paths that information follows between them. To that end, we model happens-before as a dependent type of paths between events. The inductive formulation of CSDs enables their interpretation into a variety of semantic domains. We demonstrate the interpretability of CSDs with a case study on properties of logical clocks, widely-used mechanisms for reifying causal relationships as data. We carry out this study by implementing a series of interpreters for CSDs, culminating in a generic proof of Lamport's clock condition that is parametric in a choice of clock. We instantiate this proof on Lamport's scalar clock, on Mattern's vector clock, and on the matrix clocks of Raynal et al. and of Wuu and Bernstein, yielding verified implementations of each. The CSD formalism and our case study are mechanized in the Agda proof assistant.
We consider the statistical seriation problem, where the statistician seeks to recover a hidden ordering from a noisy observation of a permuted Robinson matrix. In this paper, we tightly characterize the minimax rate for this problem of matrix reordering when the Robinson matrix is bi-Lipschitz, and we also provide a polynomial time algorithm achieving this rate; thereby answering two open questions of [Giraud et al., 2021]. Our analysis further extends to broader classes of similarity matrices.
This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.
A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? We consider learning period three with random neural networks and report the universal property associated with it. We first show that the trained networks have a thermodynamic limit that depends on the choice of target data and network settings. Our analysis reveals that almost all learned periods are unstable and each network has its characteristic attractors (which can even be untrained ones). Here, we propose the concept of characteristic bifurcation expressing embeddable attractors intrinsic to the network, in which the target data points and the scale of the network weights function as bifurcation parameters. In conclusion, learning period three generates various attractors through characteristic bifurcation due to the stability change in latently existing numerous unstable periods of the system.
The pitch contours of Mandarin two-character words are generally understood as being shaped by the underlying tones of the constituent single-character words, in interaction with articulatory constraints imposed by factors such as speech rate, co-articulation with adjacent tones, segmental make-up, and predictability. This study shows that tonal realization is also partially determined by words' meanings. We first show, on the basis of a Taiwan corpus of spontaneous conversations, using the generalized additive regression model, and focusing on the rise-fall tone pattern, that after controlling for effects of speaker and context, word type is a stronger predictor of pitch realization than all the previously established word-form related predictors combined. Importantly, the addition of information about meaning in context improves prediction accuracy even further. We then proceed to show, using computational modeling with context-specific word embeddings, that token-specific pitch contours predict word type with 50% accuracy on held-out data, and that context-sensitive, token-specific embeddings can predict the shape of pitch contours with 30% accuracy. These accuracies, which are an order of magnitude above chance level, suggest that the relation between words' pitch contours and their meanings are sufficiently strong to be functional for language users. The theoretical implications of these empirical findings are discussed.
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.
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