TY - JOUR
TI - Denotational validation of higher-order Bayesian inference
AU - Ścibior, Adam
AU - Kammar, Ohad
AU - Vákár, Matthijs
AU - Staton, Sam
AU - Yang, Hongseok
AU - Cai, Yufei
AU - Ostermann, Klaus
AU - Moss, Sean K.
AU - Heunen, Chris
AU - Ghahramani, Zoubin
T2 - Proceedings of the ACM on Programming Languages
AB - We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.
DA - 2017/12/27/
PY - 2017
DO - 10.1145/3158148
DP - arXiv.org
VL - 2
IS - POPL
SP - 1
EP - 29
J2 - Proc. ACM Program. Lang.
SN - 24751421
UR - http://arxiv.org/abs/1711.03219
Y2 - 2019/10/10/11:49:49
ER -
TY - JOUR
TI - A Convenient Category for Higher-Order Probability Theory
AU - Heunen, Chris
AU - Kammar, Ohad
AU - Staton, Sam
AU - Yang, Hongseok
T2 - arXiv:1701.02547 [cs, math]
AB - Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.
DA - 2017/01/10/
PY - 2017
DP - arXiv.org
UR - http://arxiv.org/abs/1701.02547
Y2 - 2019/10/10/11:48:09
ER -
TY - JOUR
TI - A Formal Semantics of Influence in Bayesian Reasoning
AU - Jacobs, Bart
AU - Zanasi, Fabio
T2 - Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany
AB - This paper proposes a formal deﬁnition of inﬂuence in Bayesian reasoning, based on the notions of state (as probability distribution), predicate, validity and conditioning. Our approach highlights how conditioning a joint entwined/entangled state with a predicate on one of its components has ‘crossover’ inﬂuence on the other components. We use the total variation metric on probability distributions to quantitatively measure such inﬂuence. These insights are applied to give a rigorous explanation of the fundamental concept of d-separation in Bayesian networks.
DA - 2017///
PY - 2017
DO - 10/ggdgbc
DP - DataCite
LA - en
UR - http://drops.dagstuhl.de/opus/volltexte/2017/8089/
Y2 - 2019/11/24/12:11:15
KW - Bayesianism
KW - Categorical probability theory
KW - Programming language theory
KW - Semantics
ER -
TY - JOUR
TI - A Predicate/State Transformer Semantics for Bayesian Learning
AU - Jacobs, Bart
AU - Zanasi, Fabio
T2 - Electronic Notes in Theoretical Computer Science
T3 - The Thirty-second Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXII)
AB - This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These definitions are illustrated in many examples in discrete and continuous probability theory and also in quantum theory.
DA - 2016/10/05/
PY - 2016
DO - 10/ggdgbb
DP - ScienceDirect
VL - 325
SP - 185
EP - 200
J2 - Electronic Notes in Theoretical Computer Science
LA - en
SN - 1571-0661
UR - http://www.sciencedirect.com/science/article/pii/S1571066116300883
Y2 - 2019/11/24/12:04:12
KW - Bayesianism
KW - Categorical ML
KW - Categorical probability theory
KW - Effectus theory
KW - Programming language theory
KW - Semantics
ER -
TY - JOUR
TI - Distributing probability over non-determinism
AU - Varacca, Daniele
AU - Winskel, Glynn
T2 - Mathematical Structures in Computer Science
DA - 2006/02/21/
PY - 2006
DO - 10/czs9sx
DP - Crossref
VL - 16
IS - 01
SP - 87
LA - en
SN - 0960-1295, 1469-8072
UR - http://www.journals.cambridge.org/abstract_S0960129505005074
Y2 - 2019/11/26/20:30:24
KW - Categorical probability theory
KW - Denotational semantics
KW - Programming language theory
ER -
TY - CONF
TI - Bisimulation for probabilistic transition systems: A coalgebraic approach
AU - de Vink, E. P.
AU - Rutten, J. J. M. M.
A2 - Degano, Pierpaolo
A2 - Gorrieri, Roberto
A2 - Marchetti-Spaccamela, Alberto
T3 - Lecture Notes in Computer Science
AB - The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendier in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation.
C1 - Berlin, Heidelberg
C3 - Automata, Languages and Programming
DA - 1997///
PY - 1997
DO - 10/fcqzmk
DP - Springer Link
SP - 460
EP - 470
LA - en
PB - Springer
SN - 978-3-540-69194-5
ST - Bisimulation for probabilistic transition systems
KW - Categorical probability theory
KW - Coalgebras
KW - Denotational semantics
KW - Probabilistic transition systems
KW - Transition systems
ER -