A type theory for probability density functions

Bhat, Sooraj; Agarwal, Ashish; Vuduc, Richard; Gray, Alexander G.

doi:10.1145/2103621.2103721

Cited by 19 publications

(35 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They implicitly pick measures on some measurable spaces used in the semantics. These measures are called reference measures, and constructed out of Lebesgue and counting measures [Bhat et al 2012[Bhat et al , 2013Hur et al 2015]. Then, the algorithms interpret commands as density functions with respect to these measures.…”

Section: Posterior Inference and Density Semanticsmentioning

confidence: 99%

Towards verified stochastic variational inference for probabilistic programs

Lee

Rival

et al. 2019

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

Probabilistic programming is the idea of writing models from statistics and machine learning using program notations and reasoning about these models using generic inference engines. Recently its combination with deep learning has been explored intensely, which led to the development of so called deep probabilistic programming languages, such as Pyro, Edward and ProbTorch. At the core of this development lie inference engines based on stochastic variational inference algorithms. When asked to find information about the posterior distribution of a model written in such a language, these algorithms convert this posterior-inference query into an optimisation problem and solve it approximately by a form of gradient ascent or descent. In this paper, we analyse one of the most fundamental and versatile variational inference algorithms, called score estimator or REINFORCE, using tools from denotational semantics and program analysis. We formally express what this algorithm does on models denoted by programs, and expose implicit assumptions made by the algorithm on the models. The violation of these assumptions may lead to an undefined optimisation objective or the loss of convergence guarantee of the optimisation process. We then describe rules for proving these assumptions, which can be automated by static program analyses. Some of our rules use nontrivial facts from continuous mathematics, and let us replace requirements about integrals in the assumptions, such as integrability of functions defined in terms of programs' denotations, by conditions involving differentiation or boundedness, which are much easier to prove automatically (and manually). Following our general methodology, we have developed a static program analysis for the Pyro programming language that aims at discharging the assumption about what we call model-guide support match. Our analysis is applied to the eight representative model-guide pairs from the Pyro webpage, which include sophisticated neural network models such as AIR. It finds a bug in two of these cases, and shows that the assumptions are met in the others.In this paper, we consider inference engines that lie at the core of so called deep probabilistic programming languages, such as Pyro [Bingham et al. 2019], Edward [Tran et al. 2018, 2016 and ProbTorch [Siddharth et al. 2017]. These languages let their users freely mix deep neural networks with constructs from probabilistic programming, in particular, those for writing Bayesian probabilistic models. In so doing, they facilitate the development of probabilistic deep-network models that may address the problem of measuring the uncertainty in current non-Bayesian deepnetwork models; a non-Bayesian model may predict that the price of energy goes up and that of a house goes down, but it cannot express, for instance, that the model is very confident with the first prediction but not the second.The primary inference engines for these deep probabilistic programming languages implement stochastic variational inference algorithms. Converting infer...

show abstract

Section: Posterior Inference and Density Semanticsmentioning

confidence: 99%

Towards verified stochastic variational inference for probabilistic programs

Lee

Rival

et al. 2019

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

show abstract

“…Bhat et al also stated that their eventual goal is the formal verification of such a compiler in a theorem prover [3]. This has the advantage of providing guaranteed correctness, i. e. the result of the compilation is provably a PDF for the source expression, according to the formal semantics.…”

Section: Motivationmentioning

confidence: 99%

“…This work is based on the work of Bhat et al [3,4], which presents a density compiler for probability spaces described by expressions in the language Fun. This is a small functional language with basic arithmetic, Boolean logic, product and sum types, conditionals, and a number of built-in discrete and continuous distributions.…”

Section: Related Workmentioning

confidence: 99%

“…It does, however, not support lists or recursion. The correctness proof is purely pen and paper, but the authors stated that formalisation in a proof assistant such as Coq is the ultimate goal [3].…”

Section: Related Workmentioning

confidence: 99%

“…David Cock formally proved a large number of results about the semantics of the language and developed mechanisms for refinement and program verification; however, the focus of this work was verification of probabilistic programs, not compiling them to density functions. Bhat et al mention that reconciling recursion with probability density functions is difficult [3], so a feature-rich language such as pGCL is probably not suited for developing a density compiler.…”

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

A Verified Compiler for Probability Density Functions

Eberl

Hölzl

Nipkow

2015

Programming Languages and Systems

View full text Add to dashboard Cite

Bhat et al. developed an inductive compiler that computes density functions for probability spaces described by programs in a simple probabilistic functional language. In this work, we implement such a compiler for a modified version of this language within the theorem prover Isabelle and give a formal proof of its soundness w.r.t. the semantics of the source and target language. Together with Isabelle's code generation for inductive predicates, this yields a fully verified, executable density compiler. The proof is done in two steps, using a standard refinement approach: first, an abstract compiler working with abstract functions modelled directly in the theorem prover's logic is defined and proven sound. Then, this compiler is refined to a concrete version that returns a target-language expression.Comment: Presented at ESOP 201

show abstract

Does a Program Yield the Right Distribution?

Chen

Katoen

Lutz

et al. 2022

Computer Aided Verification

View full text Add to dashboard Cite

We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a program generates exactly a specified distribution over its outputs (provided the program terminates almost-surely). The class of distributions that can be specified in our formalism consists of standard distributions (geometric, uniform, etc.) and finite convolutions thereof. Our method relies on representing these (possibly infinite-support) distributions as probability generating functions which admit effective arithmetic operations. We have automated our techniques in a tool called $$\textsc {Prodigy}$$ P R O D I G Y , which supports automatic invariance checking, compositional reasoning of nested loops, and efficient queries to the output distribution, as demonstrated by experiments.

show abstract

A type theory for probability density functions

Cited by 19 publications

References 15 publications

Towards verified stochastic variational inference for probabilistic programs

Towards verified stochastic variational inference for probabilistic programs

A Verified Compiler for Probability Density Functions

Does a Program Yield the Right Distribution?

Contact Info

Product

Resources

About