9 research outputs found
Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory
Bishop's measure theory (BMT) is an abstraction of the measure theory of a
locally compact metric space , and the use of an informal notion of a
set-indexed family of complemented subsets is crucial to its predicative
character. The more general Bishop-Cheng measure theory (BCMT) is a
constructive version of the classical Daniell approach to measure and
integration, and highly impredicative, as many of its fundamental notions, such
as the integration space of -integrable functions , rely on
quantification over proper classes (from the constructive point of view). In
this paper we introduce the notions of a pre-measure and pre-integration space,
a predicative variation of the Bishop-Cheng notion of a measure space and of an
integration space, respectively. Working within Bishop Set Theory (BST), and
using the theory of set-indexed families of complemented subsets and
set-indexed families of real-valued partial functions within BST, we apply the
implicit, predicative spirit of BMT to BCMT. As a first example, we present the
pre-measure space of complemented detachable subsets of a set with the
Dirac-measure, concentrated at a single point. Furthermore, we translate in our
predicative framework the non-trivial, Bishop-Cheng construction of an
integration space from a given measure space, showing that a pre-measure space
induces the pre-integration space of simple functions associated to it.
Finally, a predicative construction of the canonically integrable functions
, as the completion of an integration space, is included.Comment: 29 pages; shortened and corrected versio
Densities of almost-surely terminating probabilistic programs are differentiable almost everywhere
We study the differential properties of higher-order statistical
probabilistic programs with recursion and conditioning. Our starting point is
an open problem posed by Hongseok Yang: what class of statistical probabilistic
programs have densities that are differentiable almost everywhere? To formalise
the problem, we consider Statistical PCF (SPCF), an extension of call-by-value
PCF with real numbers, and constructs for sampling and conditioning. We give
SPCF a sampling-style operational semantics a la Borgstrom et al., and study
the associated weight (commonly referred to as the density) function and value
function on the set of possible execution traces. Our main result is that
almost-surely terminating SPCF programs, generated from a set of primitive
functions (e.g. the set of analytic functions) satisfying mild closure
properties, have weight and value functions that are almost-everywhere
differentiable. We use a stochastic form of symbolic execution to reason about
almost-everywhere differentiability. A by-product of this work is that
almost-surely terminating deterministic (S)PCF programs with real parameters
denote functions that are almost-everywhere differentiable. Our result is of
practical interest, as almost-everywhere differentiability of the density
function is required to hold for the correctness of major gradient-based
inference algorithms
A program logic for union bounds
International audienceWe propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory. While the union bound is simple, it is an extremely common tool for analyzing randomized algorithms. In formal verification terms, the union bound allows flexible and compos-itional reasoning over possible ways an algorithm may go wrong. It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably, assertions in our logic are non-probabilistic, even though we can conclude probabilistic facts from the judgments. Our logic can also prove accuracy properties for interactive programs, where the program must produce intermediate outputs as soon as pieces of the input arrive, rather than accessing the entire input at once. This setting also enables adaptivity, where later inputs may depend on earlier intermediate outputs. We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive. 1998 ACM Subject Classification D.2.4 Software/Program Verification 1 Introduction Probabilistic computations arise naturally in many areas of computer science. For instance, they are widely used in cryptography, privacy, and security for achieving goals that lie beyond the reach of deterministic programs. However, the correctness of probabilistic programs can be quite subtle, often relying on complex reasoning about probabilistic events. Accordingly, probabilistic computations present an attractive target for formal verification. A long line of research, spanning more than four decades, has focused on expressive formalisms for reasoning about general probabilistic properties both for purely probabilistic programs and for programs that combine probabilistic and non-deterministic choice (see, e.g., [29, 34, 35]). More recent research investigates specialized formalisms that work with more restricted assertions and proof techniques, aiming to simplify formal verification. As perhaps the purest examples of this approach, some program logics prove probabilistic properties by working purely with non-probabilistic assertions; we call such systems lightweight logics. Examples include probabilistic relational Hoare logic [3] for proving the reductionist security of cryptographic constructions, and the related approximate probabilistic relational Hoare logic [4] for reasoning about differential privacy. These logics rely on the powerful abstraction of probabilistic couplings to derive probabilistic facts from non-probabilistic assertions [7]
Measure Transformer Semantics for Bayesian Machine Learning
The Bayesian approach to machine learning amounts to computing posterior
distributions of random variables from a probabilistic model of how the
variables are related (that is, a prior distribution) and a set of observations
of variables. There is a trend in machine learning towards expressing Bayesian
models as probabilistic programs. As a foundation for this kind of programming,
we propose a core functional calculus with primitives for sampling prior
distributions and observing variables. We define measure-transformer
combinators inspired by theorems in measure theory, and use these to give a
rigorous semantics to our core calculus. The original features of our semantics
include its support for discrete, continuous, and hybrid measures, and, in
particular, for observations of zero-probability events. We compile our core
language to a small imperative language that is processed by an existing
inference engine for factor graphs, which are data structures that enable many
efficient inference algorithms. This allows efficient approximate inference of
posterior marginal distributions, treating thousands of observations per second
for large instances of realistic models.Comment: An abridged version of this paper appears in the proceedings of the
20th European Symposium on Programming (ESOP'11), part of ETAPS 201