Probabilistic Reasoning across the Causal Hierarchy

We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of expressing quantitative probabilistic reasoning -- including conditional independence and Bayesian inference -- the second encoding do-calculus reasoning for causal effects, and the third capturing a fully expressive do-calculus for arbitrary counterfactual queries. We give a corresponding series of finitary axiomatizations complete over both structural causal models and probabilistic programs, and show that satisfiability and validity for each language are decidable in polynomial space.Comment: AAAI-2

Environmental bisimulations for probabilistic higher-order languages

Environmental bisimulations for probabilistic higher-order languages are studied. In contrastwith applicative bisimulations, environmental bisimulations are known to be more robust and do not require sophisticated techniques such as Howe's in the proofs of congruence. As representative calculi, call-by-name and call-by-value λ-calculus, and a (call-by-value) λ-calculus extended with references (i.e., a store) are considered. In each case, full abstraction results are derived for probabilistic environmental similarity and bisimilarity with respect to contextual preorder and contextual equivalence, respectively. Some possible enhancements of the (bi)simulations, as "up-to techniques," are also presented. Probabilities force a number of modifications to the definition of environmental bisimulations in nonprobabilistic languages. Some of thesemodifications are specific to probabilities, others may be seen as general refinements of environmental bisimulations, applicable also to non-probabilistic languages. Several examples are presented, to illustrate the modifications and the differences

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators.We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming.Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators. We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming. Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction.Comment: Extended version of LiCS 2023 pape

Behavioral Equivalences for Higher-Order Languages with Probabilities

Higher-order languages, whose paradigmatic example is the lambda-calculus, are languages with powerful operators that are capable of manipulating and exchanging programs themselves. This thesis studies behavioral equivalences for programs with higher-order and probabilistic features. Behavioral equivalence is formalized as a contextual, or testing, equivalence, and two main lines of research are pursued in the thesis. The first part of the thesis focuses on contextual equivalence as a way of investigating the expressiveness of different languages. The discriminating powers offered by higher-order concurrent languages (Higher-Order pi-calculi) are compared with those offered by higher-order sequential languages (à la lambda-calculus) and by first-order concurrent languages (à la CCS). The comparison is carried out by examining the contextual equivalences induced by the languages on two classes of first-order processes, namely nondeterministic and probabilistic processes. As a result, the spectrum of the discriminating powers of several varieties of higher-order and first-order languages is obtained, both in a nondeterministic and in a probabilistic setting. The second part of the thesis is devoted to proof techniques for contextual equivalence in probabilistic lambda-calculi. Bisimulation-based proof techniques are studied, with particular focus on deriving bisimulations that are fully abstract for contextual equivalence (i.e., coincide with it). As a first result, full abstraction of applicative bisimilarity and similarity are proved for a call-by-value probabilistic lambda-calculus with a parallel disjunction operator. Applicative bisimulations are however known not to scale to richer languages. Hence, more robust notions of bisimulations for probabilistic calculi are considered, in the form of environmental bisimulations. Environmental bisimulations are defined for pure call-by-name and call-by-value probabilistic lambda-calculi, and for a (call-by-value) probabilistic lambda-calculus extended with references (i.e., a store). In each case, full abstraction results are derived

On Applicative Similarity, Sequentiality, and Full Abstraction

International audienceWe study how applicative bisimilarity behaves when instantiated on a call-by-value probabilistic λ-calculus, endowed with Plotkin's parallel disjunction operator. We prove that congruence and coincidence with the corresponding context relation hold for both bisimilarity and similarity, the latter known to be impossible in sequential languages

Categories of Timed Stochastic Relations

AbstractStochastic behavior—the probabilistic evolution of a system in time—is essential to modeling the complexity of real-world systems. It enables realistic performance modeling, quality-of-service guarantees, and especially simulations for biological systems. Languages like the stochastic pi calculus have emerged as effective tools to describe and reason about systems exhibiting stochastic behavior. These languages essentially denote continuous-time stochastic processes, obtained through an operational semantics in a probabilistic transition system. In this paper we seek a more descriptive foundation for the semantics of stochastic behavior using categories and monads. We model a first-order imperative language with stochastic delay by identifying probabilistic choice and delay as separate effects, modeling each with a monad, and combining the monads to build a model for the stochastic language

A coherent differential PCF

The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential linear logic are concerned, these models feature finite non-determinism and indeed these languages are essentially non-deterministic. In a previous paper we introduced a categorical framework for differentiation which does not require additivity and is compatible with deterministic models such as coherence spaces and probabilistic models such as probabilistic coherence spaces. Based on this semantics we develop a syntax of a deterministic version of the differential lambda-calculus. One nice feature of this new approach to differentiation is that it is compatible with general fixpoints of terms, so our language is actually a differential extension of PCF for which we provide a fully deterministic operational semantics