A new two-parameter discrete distribution, namely the PoiG distribution is derived by the convolution of a Poisson variate and an independently distributed geometric random variable. This distribution generalizes both the Poisson and geometric distributions and can be used for modelling over-dispersed as well as equi-dispersed count data. A number of important statistical properties of the proposed count model, such as the probability generating function, the moment generating function, the moments, the survival function and the hazard rate function. Monotonic properties are studied, such as the log concavity and the stochastic ordering are also investigated in detail. Method of moment and the maximum likelihood estimators of the parameters of the proposed model are presented. It is envisaged that the proposed distribution may prove to be useful for the practitioners for modelling over-dispersed count data compared to its closest competitors.
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Two natural ways of modelling Formula 1 race outcomes are a probabilistic approach, based on the exponential distribution, and statistical regression modelling of the ranks. Both approaches lead to exactly soluble race-winning probabilities. Equating race-winning probabilities leads to a set of equivalent parametrisations. This time-rank duality is attractive theoretically and leads to new ways of dis-entangling driver and car level effects as well and a simplified Monte Carlo simulation algorithm. Results are illustrated by applications to the 2022 and 2023 Formula 1 seasons.
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for categories with families, categories with attributes, and natural models; in particular, all of them can be traced back to certain discrete Grothendieck fibrations. We extend this intuition to the case of general, non necessarily discrete, fibrations, so that over a given context one has not only a set but a category of types. We argue that the added structure can be attributed to a notion of subtyping that shares many features with that of coercive subtyping, in the sense that it is the product of thinking about subtyping as an abbreviation mechanism: we say that a given type $A'$ is a subtype of $A$ if there is a unique coercion from $A'$ to $A$. Whenever we need a term of type $A$, then, it suffices to have a term of type $A'$, which we can `plug-in' into $A$. For this version of subtyping we provide rules, coherences, and explicit models, and we compare and contrast it to coercive subtyping as introduced by Z. Luo and others. We conclude by suggesting how the tools we present can be employed in finding appropriate rules relating subtyping and certain type constructors.
Deep neural networks (DNNs) often fail silently with over-confident predictions on out-of-distribution (OOD) samples, posing risks in real-world deployments. Existing techniques predominantly emphasize either the feature representation space or the gradient norms computed with respect to DNN parameters, yet they overlook the intricate gradient distribution and the topology of classification regions. To address this gap, we introduce GRadient-aware Out-Of-Distribution detection in interpolated manifolds (GROOD), a novel framework that relies on the discriminative power of gradient space to distinguish between in-distribution (ID) and OOD samples. To build this space, GROOD relies on class prototypes together with a prototype that specifically captures OOD characteristics. Uniquely, our approach incorporates a targeted mix-up operation at an early intermediate layer of the DNN to refine the separation of gradient spaces between ID and OOD samples. We quantify OOD detection efficacy using the distance to the nearest neighbor gradients derived from the training set, yielding a robust OOD score. Experimental evaluations substantiate that the introduction of targeted input mix-upamplifies the separation between ID and OOD in the gradient space, yielding impressive results across diverse datasets. Notably, when benchmarked against ImageNet-1k, GROOD surpasses the established robustness of state-of-the-art baselines. Through this work, we establish the utility of leveraging gradient spaces and class prototypes for enhanced OOD detection for DNN in image classification.
Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs: namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.
We consider the accuracy of an approximate posterior distribution in nonparametric regression problems by combining posterior distributions computed on subsets of the data defined by the locations of the independent variables. We show that this approximate posterior retains the rate of recovery of the full data posterior distribution, where the rate of recovery adapts to the smoothness of the true regression function. As particular examples we consider Gaussian process priors based on integrated Brownian motion and the Mat\'ern kernel augmented with a prior on the length scale. Besides theoretical guarantees we present a numerical study of the methods both on synthetic and real world data. We also propose a new aggregation technique, which numerically outperforms previous approaches.
In discussions about the development and governance of AI, a false binary is often drawn between two groups: those most concerned about the existing, social impacts of AI, and those most concerned about possible future risks of powerful AI systems taking actions that don't align with human interests. In this piece, we (i) describe the emergence of this false binary, (ii) explain why the seemingly clean distinctions drawn between these two groups don't hold up under scrutiny and (iii) highlight efforts to bridge this divide.
In recent years, various interacting particle samplers have been developed to sample from complex target distributions, such as those found in Bayesian inverse problems. These samplers are motivated by the mean-field limit perspective and implemented as ensembles of particles that move in the product state space according to coupled stochastic differential equations. The ensemble approximation and numerical time stepping used to simulate these systems can introduce bias and affect the invariance of the particle system with respect to the target distribution. To correct for this, we investigate the use of a Metropolization step, similar to the Metropolis-adjusted Langevin algorithm. We examine Metropolization of either the whole ensemble or smaller subsets of the ensemble, and prove basic convergence of the resulting ensemble Markov chain to the target distribution. Our numerical results demonstrate the benefits of this correction in numerical examples for popular interacting particle samplers such as ALDI, CBS, and stochastic SVGD.
The ensemble Kalman inversion (EKI), a recently introduced optimisation method for solving inverse problems, is widely employed for the efficient and derivative-free estimation of unknown parameters. Specifically in cases involving ill-posed inverse problems and high-dimensional parameter spaces, the scheme has shown promising success. However, in its general form, the EKI does not take constraints into account, which are essential and often stem from physical limitations or specific requirements. Based on a log-barrier approach, we suggest adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. Finally, we showcase our results through two examples involving partial differential equations (PDEs).
In this study, a novel preconditioner based on the absolute-value block $\alpha$-circulant matrix approximation is developed, specifically designed for nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge from the numerical discretization of evolutionary equations. Our preconditioner is constructed by taking an absolute-value of a block $\alpha$-circulant matrix approximation to the BLTT matrix. To apply our preconditioner, the original BLTT linear system is converted into a symmetric form by applying a time-reversing permutation transformation. Then, with our preconditioner, the preconditioned minimal residual method (MINRES) solver is employed to solve the symmetrized linear system. With properly chosen $\alpha$, the eigenvalues of the preconditioned matrix are proven to be clustered around $\pm1$ without any significant outliers. With the clustered spectrum, we show that the preconditioned MINRES solver for the preconditioned system has a convergence rate independent of system size. To the best of our knowledge, this is the first preconditioned MINRES method with size-independent convergence rate for the dense BLTT system. The efficacy of the proposed preconditioner is corroborated by our numerical experiments, which reveal that it attains optimal convergence.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
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