Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers ‘black-and-white’ graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

New definitions in the theory of Type 1 computable topological spaces

Motivated by the two remarks, that the study of computability based on the use of numberings -- Type 1 computability -- does not have to be restricted to countable sets equipped with onto numberings, and that computable topologies have been in part developed with the implicit hypothesis that the considered spaces should be computably separable, we propose new definitions for Type 1 computable topological spaces. We define computable topological spaces without making reference to a basis. Following Spreen, we show that the use of a formal inclusion relation should be systematized, and that it cannot be avoided if we want computable topological spaces to generalize computable metric spaces. We also compare different notions of effective bases. The first one, introduced by Nogina, is based on an effective version of the statement "a set $O$ is open if for any point in $O$, there is a basic set containing that point and contained in $O$''. The second one, associated to Lacombe, is based on an effective version of "a set $O$ is open if it can be written as a union of basic open sets''. We show that neither of these notions of basis is completely satisfactory: Nogina bases do not permit to define computable topologies unless we restrict our attention to countable sets, and the conditions associated to Lacombe bases are too restrictive, and they do not apply to metric spaces unless we add effective separability hypotheses. We define a new notion of basis, based on an effective version of the Nogina statement, but adding to it several classically empty conditions, expressed in terms of formal inclusion relations. Finally, we obtain a new version of the theorem of Moschovakis which states that the Lacombe and Nogina approaches coincide on countable recursive Polish spaces, but which applies to sets equipped with non-onto numberings, and with effective separability as a sole hypothesis.Comment: 50 pages, 2 figure

Computability, inference and modeling in probabilistic programming

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 135-144).We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure.by Daniel M. Roy.Ph.D