Nathanael Leedom Ackerman

Noncomputable Conditional Distributions

We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[Y|X] encodes the halting problem. Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinite-dimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.

Invariant measures concentrated on countable structures

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

Completeness in generalized ultrametric spaces

We analyze the relationship between four notions of completeness for Γ-ultrametric spaces. The notions we consider include Cauchy completeness, strong Cauchy completeness, spherical completeness and injectivity. In the process we show that the category of Γ-ultrametric spaces is equivalent to the category of flabby separated presheaves on Γop.

On computability and disintegration

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

Invariant measures via inverse limits of finite structures

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain first-order theories, including measures that do not assign positive measure to the isomorphism class of any single model. We also characterize those transitive Borel G-spaces admitting a G-invariant probability measure, when G is an arbitrary countable product of symmetric groups on a countable set.

Relativized Grothendieck topoi

Abstract In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.

Graph Turing Machines

We consider graph Turing machines, a model of parallel computation on a graph, which provides a natural generalization of several standard computational models, including ordinary Turing machines and cellular automata. In this extended abstract, we give bounds on the computational strength of functions that graph Turing machines can compute. We also begin the study of the relationship between the computational power of a graph Turing machine and structural properties of its underlying graph.

A classification of orbits admitting a unique invariant measure

Abstract We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are S ∞ -invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique S ∞ -invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order ( Q , ) .

Priors on exchangeable directed graphs

Directed graphs occur throughout statistical modeling of networks, and exchangeability is a natural assumption when the ordering of vertices does not matter. There is a deep structural theory for exchangeable undirected graphs, which extends to the directed case via measurable objects known as digraphons. Using digraphons, we first show how to construct models for exchangeable directed graphs, including special cases such as tournaments, linear orderings, directed acyclic graphs, and partial orderings. We then show how to construct priors on digraphons via the infinite relational digraphon model (di-IRM), a new Bayesian nonparametric block model for exchangeable directed graphs, and demonstrate inference on synthetic data.

The Beta-Bernoulli process and algebraic effects

In this paper we use the framework of algebraic effects from programming language theory to analyze the Beta-Bernoulli process, a standard building block in Bayesian models. Our analysis reveals the importance of abstract data types, and two types of program equations, called commutativity and discardability. We develop an equational theory of terms that use the Beta-Bernoulli process, and show that the theory is complete with respect to the measure-theoretic semantics, and also in the syntactic sense of Post. Our analysis has a potential for being generalized to other stochastic processes relevant to Bayesian modelling, yielding new understanding of these processes from the perspective of programming.