Coherent sets of desirable gamble sets is used as a model for representing an agents opinions and choice preferences under uncertainty. In this paper we provide some results about the axioms required for coherence and the natural extension of a given set of desirable gamble sets. We also show that coherent sets of desirable gamble sets can be represented by a proper filter of coherent sets of desirable gambles. This paper was primarily written in 2021, overlapping with my finishing up Campbell-Moore (2021). There is some overlap between this paper and de Cooman et al. (2023); the results of this paper were shown independently.

We present Uni-Fusion, a universal continuous mapping framework for surfaces, surface properties (color, infrared, etc.) and more (latent features in CLIP embedding space, etc.). We propose the first universal implicit encoding model that supports encoding of both geometry and different types of properties (RGB, infrared, features, etc.) without requiring any training. Based on this, our framework divides the point cloud into regular grid voxels and generates a latent feature in each voxel to form a Latent Implicit Map (LIM) for geometries and arbitrary properties. Then, by fusing a local LIM frame-wisely into a global LIM, an incremental reconstruction is achieved. Encoded with corresponding types of data, our Latent Implicit Map is capable of generating continuous surfaces, surface property fields, surface feature fields, and all other possible options. To demonstrate the capabilities of our model, we implement three applications: (1) incremental reconstruction for surfaces and color (2) 2D-to-3D transfer of fabricated properties (3) open-vocabulary scene understanding by creating a text CLIP feature field on surfaces. We evaluate Uni-Fusion by comparing it in corresponding applications, from which Uni-Fusion shows high-flexibility in various applications while performing best or being competitive. The project page of Uni-Fusion is available at https://jarrome.github.io/Uni-Fusion/ .

Post-selection inference (PoSI) is a statistical technique for obtaining valid confidence intervals and p-values when hypothesis generation and testing use the same source of data. PoSI can be used on a range of popular algorithms including the Lasso. Data carving is a variant of PoSI in which a portion of held out data is combined with the hypothesis generating data at inference time. While data carving has attractive theoretical and empirical properties, existing approaches rely on computationally expensive MCMC methods to carry out inference. This paper's key contribution is to show that pivotal quantities can be constructed for the data carving procedure based on a known parametric distribution. Specifically, when the selection event is characterized by a set of polyhedral constraints on a Gaussian response, data carving will follow the sum of a normal and a truncated normal (SNTN), which is a variant of the truncated bivariate normal distribution. The main impact of this insight is that obtaining exact inference for data carving can be made computationally trivial, since the CDF of the SNTN distribution can be found using the CDF of a standard bivariate normal. A python package sntn has been released to further facilitate the adoption of data carving with PoSI.

We study how to infer new choices from previous choices in a conservative manner. To make such inferences, we use the theory of choice functions: a unifying mathematical framework for conservative decision making that allows one to impose axioms directly on the represented decisions. We here adopt the coherence axioms of De Bock and De Cooman (2019). We show how to naturally extend any given choice assessment to such a coherent choice function, whenever possible, and use this natural extension to make new choices. We present a practical algorithm to compute this natural extension and provide several methods that can be used to improve its scalability.

Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions.

Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules to imprecise-probabilistic uncertainty models. We provide them with a clear interpretation in terms of attitudes towards gambling, borrowing ideas from the theory of sets of desirable gambles, and we use this interpretation to derive a set of basic axioms. We show that these axioms lead to a full-fledged theory of coherent choice functions, which includes a representation in terms of sets of desirable gambles, and a conservative inference method.

We present a new approach to credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. Instead of applying the commonly used notion of strong independence, we replace it by the weaker notion of epistemic irrelevance. We show how assessments of epistemic irrelevance allow us to construct a global model out of given local uncertainty models and mention some useful properties. The main results and proofs are presented using the language of sets of desirable gambles, which provides a very general and expressive way of representing imprecise probability models.