In this paper, we introduce a fundamental framework to create a bridge
between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a
random experiment of selecting crisp elements with the criterion of having a
certain fuzzy attribute. To do so, we associate some specific crisp random
variables to the random experiment. Then, several formulas are presented, which
make it easier to compute different conditional probabilities and expected
values of these random variables. Also, we provide measure theoretical basis
for our probabilistic fuzzy logic framework. Note that in our theory, the
probability density functions of continuous distributions which come from the
aforementioned random variables include the Dirac delta function as a term.
Further, we introduce an application of our theory in Causal Inference.

该论文介绍了一个基础框架，用于创建概率论和模糊逻辑之间的桥梁，提出了一种将随机实验与具有某种模糊属性的清晰元素的选择联系起来的方法，并介绍了在因果推断中应用该框架的实例。

一种适用于因果推理的基本概率模糊逻辑框架

A Fundamental Probabilistic Fuzzy Logic Framework Suitable for Causal Reasoning

In standard reinforcement learning (RL), a learning agent seeks to optimize
the overall reward. However, many key aspects of a desired behavior are more
naturally expressed as constraints. For instance, the designer may want to
limit the use of unsafe actions, increase the diversity of trajectories to
enable exploration, or approximate expert trajectories when rewards are sparse.
In this paper, we propose an algorithmic scheme that can handle a wide class of
constraints in RL tasks: specifically, any constraints that require expected
values of some vector measurements (such as the use of an action) to lie in a
convex set. This captures previously studied constraints (such as safety and
proximity to an expert), but also enables new classes of constraints (such as
diversity). Our approach comes with rigorous theoretical guarantees and only
relies on the ability to approximately solve standard RL tasks. As a result, it
can be easily adapted to work with any model-free or model-based RL. In our
experiments, we show that it matches previous algorithms that enforce safety
via constraints, but can also enforce new properties that these algorithms do
not incorporate, such as diversity.

本文提出了一种能处理一大类 RL 任务约束的算法方案，这些约束需要某些向量测量（如行动使用）的期望值位于凸集中，可以捕获以前研究的约束（如安全和接近专家），也可以实现新类别的约束（如多样性）。

凸约束下的强化学习

Reinforcement Learning with Convex Constraints

This paper deals with the problem of quantifying the impact of model
misspecification when computing general expected values of interest. The
methodology that we propose is applicable in great generality, in particular,
we provide examples involving path dependent expectations of stochastic
processes. Our approach consists in computing bounds for the expectation of
interest regardless of the probability measure used, as long as the measure
lies within a prescribed tolerance measured in terms of a flexible class of
distances from a suitable baseline model. These distances, based on optimal
transportation between probability measures, include Wasserstein's distances as
particular cases. The proposed methodology is well-suited for risk analysis, as
we demonstrate with a number of applications. We also discuss how to estimate
the tolerance region non-parametrically using Skorokhod-type embeddings in some
of these applications.

本文提出了一种方法来计算特定概率模型下利益期望，并给出了为不同的概率模型设置偏差范围的基于最优输运的度量方法，同时应用于风险分析。