Amos Azaria

In standard probability theory, events of probability zero may still occur. That is, a real number $x$ can be sampled uniformly at random from $[0,1]$ despite having $\mathbb{P}(X=x)=0$. This goes against the intuitive principle that probability $0$ means impossible. In this paper, we make this tension explicit and argue that the usual resolution is backwards. We begin by analyzing two common claims: 1. (Zero) Probability $0$ means impossible for any possible outcome. 2. (Uniform) It is possible to sample a real number: there exists an experiment whose possible outcomes are all $x\in[0,1]$, distributed according to the uniform (Lebesgue) measure, and which returns a completed real on each run. These two claims are incompatible. Standard probability theory keeps (Uniform) and abandons (Zero), allowing events of probability zero to occur. We argue for the opposite choice. We retain (Zero) and deny that there is any literal experiment that samples a completed real from $[0,1]$ with the uniform law. The main tool is definability. We propose that to count as a number at all, an object must be definable by a finite description in a fixed language. Undefinable ``real numbers'' then do not exist as numbers; they are set theoretic artifacts rather than genuine numerical quantities. Under any continuous distribution on $[0,1]$, the set of definable reals has probability zero, so an idealized ``random real'' is almost surely an undefinable pseudo object. On our view this is a sign that the usual continuum picture is being applied outside its proper domain. We develop this finite information perspective across several domains: definable reals and complex numbers, definable infinitesimals and infinite quantities in hyperreal fields, definable distributions such as the Dirac delta, and definable infinities in extended number systems. We stress the abundance of definable numerical objects: giving up undefinable reals does not impoverish the numerical universe but focuses attention on a rich and extensible world of definable numbers. We also revisit Cantor's diagonal argument. We accept its conclusion as a theorem of set theory about the uncountability of the set theoretic reals, but argue that its anti diagonal construction presupposes the existence of undefinable real codes. Once we restrict attention to definable numbers, Cantor's theorem is about a larger background universe than the domain of genuine numerical quantities. Finally, we articulate a process based view of sampling on the continuum. No physical or computational procedure ever produces an infinite real; what exists in practice is a sampling process that yields arbitrarily fine but always finite information. In this operational setting, events with probability zero that refer to completed infinite outcomes never occur, and the principle that probability $0$ means impossible holds for all physically meaningful events.

Creating agents that proficiently interact with people is critical for many applications. Towards creating these agents, models are needed that effectively predict people's decisions in a variety of problems. To date, two approaches have been suggested to generally describe people's decision behavior. One approach creates a-priori predictions about people's behavior, either based on theoretical rational behavior or based on psychological models, including bounded rationality. A second type of approach focuses on creating models based exclusively on observations of people's behavior. At the forefront of these types of methods are various machine learning algorithms. This paper explores how these two approaches can be compared and combined in different types of domains. In relatively simple domains, both psychological models and machine learning yield clear prediction models with nearly identical results. In more complex domains, the exact action predicted by psychological models is not even clear, and machine learning models are even less accurate. Nonetheless, we present a novel approach of creating hybrid methods that incorporate features from psychological models in conjunction with machine learning in order to create significantly improved models for predicting people's decisions. To demonstrate these claims, we present an overview of previous and new results, taken from representative domains ranging from a relatively simple optimization problem and complex domains such as negotiation and coordination without communication