Fisher's fiducial argument is widely viewed as a failed version of Neyman's theory of confidence limits. But Fisher's goal -- Bayesian-like probabilistic uncertainty quantification without priors -- was more ambitious than Neyman's, and it's not out of reach. I've recently shown that reliable, prior-free probabilistic uncertainty quantification must be grounded in the theory of imprecise probability, and I've put forward a possibility-theoretic solution that achieves it. This has been met with resistance, however, in part due to statisticians' singular focus on confidence limits. Indeed, if imprecision isn't needed to perform confidence-limit-related tasks, then what's the point? In this paper, for a class of practically useful models, I explain specifically why the fiducial argument gives valid confidence limits, i.e., it's the "best probabilistic approximation" of the possibilistic solution I recently advanced. This sheds new light on what the fiducial argument is doing and on what's lost in terms of reliability when imprecision is ignored and the fiducial argument is pushed for more than just confidence limits.
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We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, and we establish properties of such orbitopes in nonparametric settings. We also show how a simple device called a cocycle can be used to reduce different forms of symmetry to a single problem. As applications, we obtain results on invariant kernel mean embeddings and a Monge-Kantorovich theorem on optimality of transport plans under symmetry constraints. We also explain connections to the Hunt-Stein theorem on invariant tests.
Kahn Process Networks (KPNs) are a deterministic Model of Computation (MoC) for distributed systems. KPNs supports non-blocking writes and blocking reads, with the consequent assumption of unbounded buffers between processes. Variants such as Finite FIFO Platforms (FFP) have been developed, which enforce boundedness. One issue with existing models is that they mix process synchronisation with process execution. In this paper we address how these two facets may be decoupled. This paper explores a recent alternative called bittide, which decouples the execution of a process from the control needed for process synchronisation, and thus preserves determinism and boundedness while ensuring pipelined execution for better throughput. Our intuition is that such an approach could leverage not only determinism and buffer boundedness but may potentially offer better overall throughput. To understand the behavior of these systems we define a formal model -- a deterministic MoC called Logical Synchrony Networks (LSNs). LSNs describes a network of processes modelled as a graph, with edges representing invariant logical delays between a producer process and the corresponding consumer process. We show that this abstraction is satisfied by KPNs. Subsequently, we show that both FFPs and bittide faithfully implement this abstraction. Thus, we show for the first time that FFPs and bittide offer two alternative ways of implementing deterministic distributed systems with the latter being more performant.
In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra.
P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for Quantum Logic what Boolean algebras are for Classical Logic.The closed subspaces of a separable Hilbert space form a P-algebra under orthogonal complementation and projection of a subspace onto another one. P-algebras are complemented orthomodular posets that are not lattices. Atomic algebras are defined and their main properties are studied. A substructural logic of sequents is proved to be sound and complete for the logic of P-algebras.
We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.
Physics-Informed Neural Networks (PINNs) have emerged as an iconic machine learning approach for solving Partial Differential Equations (PDEs). Although its variants have achieved significant progress, the empirical success of utilising feature mapping from the wider Implicit Neural Representations studies has been substantially neglected. We investigate the training dynamics of PINNs with a feature mapping layer via the limiting Conjugate Kernel and Neural Tangent Kernel, which sheds light on the convergence and generalisation of the model. We also show the inadequacy of commonly used Fourier-based feature mapping in some scenarios and propose the conditional positive definite Radial Basis Function as a better alternative. The empirical results reveal the efficacy of our method in diverse forward and inverse problem sets. This simple technique can be easily implemented in coordinate input networks and benefits the broad PINNs research.
We show that Edge Multiway Cut (also called Multiterminal Cut) and Node Multiway Cut are NP-complete on graphs of maximum degree $3$ (also known as subcubic graphs). This improves on a previous degree bound of $11$. Our NP-completeness result holds even for subcubic graphs that are planar.
Polysomnography (PSG) data is recorded and stored in various formats depending on the recording software. Although the PSG data can usually be exported to open formats, such as the European Data Format (EDF), they are limited in data types, validation, and readability. Moreover, the exported data is not harmonized, which means different datasets need customized preprocessing to conduct research on multiple datasets. In this work, we designed and implemented an open format for storage and processing of PSG data, called the Sleeplab format (SLF), which is both human and machine-readable, and has built-in validation of both data types and structures. SLF provides tools for reading, writing, and compression of the PSG datasets. In addition, SLF promotes harmonization of data from different sources, which reduces the amount of work needed to apply the same analytics pipelines to different datasets. SLF is interoperable as it utilizes the file system and commonly used file formats to store the data. The goal of developing SLF was to enable fast exploration and experimentation on PSG data, and to streamline the workflow of building analytics and machine learning applications that combine PSG data from multiple sources. The performance of SLF was tested with two open datasets of different formats (EDF and HDF5). SLF is fully open source and available at //github.com/UEF-SmartSleepLab/sleeplab-format.
In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting, reformulate the latter in the abstract additive Schwarz framework and prove its convergence to the finite element solution. We provide an implementation and test it on various examples of interest.
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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