Random Variables

Specializing the concepts of Chapter 7 to the case of real variables, this chapter introduces distribution functions, discrete and continuous distributions, and describes examples such as the binomial, uniform, Gaussian, Cauchy, and gamma distributions. It then treats multivariate distributions and the concept of independence.

Statistics and optimization of random pan stacking

We consider a minimal model of random pan stacking. A single pan consists of a V-shaped object characterized by its internal angle α. The stack is constructed by piling up N pans with different angles in a given, random order. The set of pans is generated by sampling from various kinds of distributions of the pan angles: discrete or continuous, uniform or optimized. For large N the mean height depends principally on the average of the distance between the bases of two consecutive pans, and the effective compaction of the stack, compared with the unstacked pans, is 2 log 2/π. We also obtain the discrete and continuous distributions that maximize the mean stack height. With only two types of pans, the maximum occurs for equal probabilities, while when many types of pans are available, the optimum distribution strongly favours those with the most acute and the most obtuse angles. With a continuous distribution of angles, while one never finds two identical pans, the behaviour is similar to a system with a large number of discrete angles.

Synthetic topology in Homotopy Type Theory for probabilistic programming

The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial σ-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.

Technical Perspective

The paper entitled "Probabilistic Data with Continuous Distributions" overviews recent work on the foundations of infinite probabilistic databases [3, 2]. Prior work on probabilistic databases (PDBs) focused almost exclusively on the finite case: A finite PDB represents a discrete probability distribution over a finite set of possible worlds [4]. In contrast, an infinite PDB models a continuous probability distribution over an infinite set of possible worlds. In both cases, each world is a finite relational database instance. Continuous distributions are essential and commonplace tools for reasoning under uncertainty in practice. Accommodating them in the framework of probabilistic databases brings us closer to applications that naturally rely on both continuous distributions and relational databases.