Probabilistic Programming Research Articles

Enhancing SMT-based Weighted Model Integration by structure awareness

The development of efficient exact and approximate algorithms for probabilistic inference is a long-standing goal of artificial intelligence research. Whereas substantial progress has been made in dealing with purely discrete or purely continuous domains, adapting the developed solutions to tackle hybrid domains, characterized by discrete and continuous variables and their relationships, is highly non-trivial. Weighted Model Integration (WMI) recently emerged as a unifying formalism for probabilistic inference in hybrid domains. Despite a considerable amount of recent work, allowing WMI algorithms to scale with the complexity of the hybrid problem is still a challenge. In this paper we highlight some substantial limitations of existing state-of-the-art solutions, and develop an algorithm that combines SMT-based enumeration, an efficient technique in formal verification, with an effective encoding of the problem structure. This allows our algorithm to avoid generating redundant models, resulting in drastic computational savings. Additionally, we show how SMT-based approaches can seamlessly deal with different integration techniques, both exact and approximate, significantly expanding the set of problems that can be tackled by WMI technology. An extensive experimental evaluation on both synthetic and real-world datasets confirms the substantial advantage of the proposed solution over existing alternatives. The application potential of this technology is further showcased on a prototypical task aimed at verifying the fairness of probabilistic programs.

Dynamic Game Theoretic Electric Vehicle Decision Making

<div>Real-world driving in diverse traffic must cope with dynamic environments including a multitude of agents with uncertain behaviors. This poses a challenging motion planning and decision-making problem, as suitable algorithms should manage to obtain optimal solutions considering nearby vehicles. The state-of-the-art way of environment and action generalization is built on mathematical modeling and probabilistic programming of idealistic incidents. In this article we present dynamic anytime decision making, a decision-making algorithm that takes advantage of natural evolutionary and developmental processes to make decisions for an autonomous vehicle navigating in traffic. The methodology to achieve multidimensional judgment under unforeseen circumstances is to enable stochastic Bayesian game theory when modeling interactive properties and scenario estimation. In particular, we employ a dual-layer optimization strategy that substantially enhances the capability to extract mutual influence into decision matrix that organically synthesize decision rules according to encountered situations. Furthermore, our decision-making algorithm incorporates incremental capability that computes modified control variable due to unexpected vehicle behavior or intention. This results in improved performance gain where dynamic conditions receive reasoning with more effectiveness. We produce swift policy correction by leveraging previous game mechanism, where cooperative characteristics of both the controlled vehicle and other agents are integrated without computation overload. Aiming to reach utmost accurate decision, our reward function is reinforced with weight-tuning function to deduce optimal hyper parameters in a wider spectrum of vehicle scenarios. Through coupled decision tree optimization and incremental trajectory control, economical commands are computed with enhanced precision under complex traffic pattern. To this end, our system provides a robust framework in terms of handling dynamic vehicle interactive control, as decision is computed by synergy between progressive interface and external rationalization. We demonstrate the algorithm performing preeminent passing maneuvers and merging in a number of different simulated scenarios.</div>

Research on Bayesian Statistics Teaching with R

Bayesian statistics uses the mathematical rules of probability to combine data with prior information to yield inferences which are always more precise than would be obtained by either source of information alone. The current Bayesian education has not yet attracted enough attention from the education community in China. The rise of MCMC and probabilistic programming languages has profoundly reshaped Bayesian statistics. Compared to precise mathematical analysis, the simulation-based teaching can help students to avoid tedious statistical calculations and cultivate Bayesian thinking.

Probabilistic Programming Interfaces for Random Graphs: Markov Categories, Graphons, and Nominal Sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.

Decalf: A Directed, Effectful Cost-Aware Logical Framework

We present decalf , a d irected, e ffectful c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs with effects. Like calf , the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike calf , the present language takes account of effects , such as probabilistic choice and mutable state. This extension requires a reformulation of calf ’s approach to cost accounting: rather than rely on a ”separable” notion of cost, here a cost bound is simply another program . To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a cost recurrence and readily extends to higher-order, effectful programs. The development proceeds by first introducing the decalf type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify decalf via a model in the topos of augmented simplicial sets.

Programmatic Strategy Synthesis: Resolving Nondeterminism in Probabilistic Programs

We consider imperative programs that involve both randomization and pure nondeterminism. The central question is how to find a strategy resolving the pure nondeterminism such that the so-obtained determinized program satisfies a given quantitative specification, i.e., bounds on expected outcomes such as the expected final value of a program variable or the probability to terminate in a given set of states. We show how memoryless and deterministic (MD) strategies can be obtained in a semi-automatic fashion using deductive verification techniques. For loop-free programs, the MD strategies resulting from our weakest precondition-style framework are correct by construction. This extends to loopy programs, provided the loops are equipped with suitable loop invariants - just like in program verification. We show how our technique relates to the well-studied problem of obtaining strategies in countably infinite Markov decision processes with reachability-reward objectives. Finally, we apply our technique to several case studies.

Asynchronous Probabilistic Couplings in Higher-Order Separation Logic

Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or "synchronizing" the sampling statements of the two programs which is not always possible. In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relation to reason about contextual refinement and equivalence of higher-order programs written in a rich language with a probabilistic choice operator, higher-order local state, and impredicative polymorphism. Finally, we demonstrate our approach on a number of case studies. All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework.

Inference of Probabilistic Programs with Moment-Matching Gaussian Mixtures

Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bounded program by means of Gaussian mixtures. It is parametrized by a map that associates each program location with the moment order to be matched in the approximation. We provide two main contributions. The first is a universal approximation theorem stating that, under mild conditions, Gaussian Semantics can approximate the exact semantics arbitrarily closely. The second is an approximation that matches up to second-order moments analytically in face of the generally difficult problem of matching moments of Gaussian mixtures with arbitrary moment order. We test our second-order Gaussian approximation (SOGA) on a number of case studies from the literature. We show that it can provide accurate estimates in models not supported by other approximation methods or when exact symbolic techniques fail because of complex expressions or non-simplified integrals. On two notable classes of problems, namely collaborative filtering and programs involving mixtures of continuous and discrete distributions, we show that SOGA significantly outperforms alternative techniques in terms of accuracy and computational time.

Positive Almost-Sure Termination: Complexity and Proof Rules

We study the recursion-theoretic complexity of Positive Almost-Sure Termination ( PAST ) in an imperative programming language with rational variables, bounded nondeterministic choice, and discrete probabilistic choice. A program terminates positive almost-surely if, for every scheduler, the program terminates almost-surely and the expected runtime to termination is finite. We show that PAST for our language is complete for the (lightface) co-analytic sets (Π 1 1 -complete). This is in contrast to the related notions of Almost-Sure Termination ( AST ) and Bounded Termination ( BAST ), both of which are arithmetical (Π 2 0 - and Σ 2 0 -complete respectively). Our upper bound implies an effective procedure to reduce reasoning about probabilistic termination to non-probabilistic fair termination in a model with bounded nondeterminism, and to simple program termination in models with unbounded nondeterminism. Our lower bound shows the opposite: for every program with unbounded nondeterministic choice, there is an effectively computable probabilistic program with bounded choice such that the original program is terminating if, and only if, the transformed program is PAST . We show that every program has an effectively computable normal form, in which each probabilistic choice either continues or terminates execution immediately, each with probability 1/2. For normal form programs, we provide a sound and complete proof rule for PAST . Our proof rule uses transfinite ordinals. We show that reasoning about PAST requires transfinite ordinals up to ω 1 CK ; thus, existing techniques for probabilistic termination based on ranking supermartingales that map program states to reals do not suffice to reason about PAST .

Strong Invariants Are Hard: On the Hardness of Strongest Polynomial Invariants for (Probabilistic) Programs

We show that computing the strongest polynomial invariant for single-path loops with polynomial assignments is at least as hard as the Skolem problem, a famous problem whose decidability has been open for almost a century. While the strongest polynomial invariants are computable for affine loops , for polynomial loops the problem remained wide open. As an intermediate result of independent interest, we prove that reachability for discrete polynomial dynamical systems is Skolem-hard as well. Furthermore, we generalize the notion of invariant ideals and introduce moment invariant ideals for probabilistic programs. With this tool, we further show that the strongest polynomial moment invariant is (i) uncomputable, for probabilistic loops with branching statements, and (ii) Skolem-hard to compute for polynomial probabilistic loops without branching statements. Finally, we identify a class of probabilistic loops for which the strongest polynomial moment invariant is computable and provide an algorithm for it.

Probabilistic programming versus meta-learning as models of cognition.

We summarize the recent progress made by probabilistic programming as a unifying formalism for the probabilistic, symbolic, and data-driven aspects of human cognition. We highlight differences with meta-learning in flexibility, statistical assumptions and inferences about cogniton. We suggest that the meta-learning approach could be further strengthened by considering Connectionist and Bayesian approaches, rather than exclusively one or the other.

Fractional programming for Stackelberg game under type-2 fuzzy environment

This study intends to present a Stackelberg game model for design of the fractional type-2 fuzzy programming. In achieving this aspiration, we develop a probabilistic fuzzy multi-objective fractional linear programming where the entire parameters are of type-2 fuzzy numbers apart from the right-hand side of the constraints are follow Weibull distribution. In the projected approach, the membership function allied with each objective function is generated by using the first-order Taylor series approximation and converted into a single objective function by assuming the weights of the objective functions are equal. Type conversion is made in two ways by existing methods, and using stochastic programming, the probabilistic constraints are transformed into a deterministic form. The accessible model incorporates the non-linear programming viewpoint of the decision-maker and is solved with the help of intuitionistic fuzzy programming (IFS). A comparison study on the optimum results by genetic algorithm (GA) and particle swarm optimization (PSO) with the LINGO 15.0 iterative scheme is offered to resolve the created bi-level programming problem (BLPP) in the course of the Stackelberg game. To make obvious the feasibility of the projected representation and solution methodology, realistic data are measured and results are presented through several discussions.

Enhancing neutrosophic fuzzy goal programming approach for solving multi-objective probabilistic linear programming problems

This article considers a stochastic multi-objective linear programming problem in a neutrosophic framework. The right-hand sides of the constraints are random variables associated with their means and variances, and the single-valued trapezoidal neutrosophic numbers are included in the coefficients of the objective functions to investigate the problem. The problem is successfully converted into the related deterministic programming model based on the formulation of the scoring function and several distributions, such as the Uniform, Exponential, and Gama distributions. Applying fuzzy goal programming, the deterministic problem is solved. An example is then solved to demonstrate the solution methodology.

Modeling PU learning using probabilistic logic programming

Modeling PU learning using probabilistic logic programming

A model of stochastic memoization and name generation in probabilistic programming: categorical semantics via monads on presheaf categories

Stochastic memoization is a higher-order construct of probabilistic programming languages that is key in Bayesian nonparametrics, a modular approach that allows us to extend models beyond their parametric limitations and compose them in an elegant and principled manner. Stochastic memoization is simple and useful in practice, but semantically elusive, particularly regarding dataflow transformations. As the naive implementation resorts to the state monad, which is not commutative, it is not clear if stochastic memoization preserves the dataflow property -- i.e., whether we can reorder the lines of a program without changing its semantics, provided the dataflow graph is preserved. In this paper, we give an operational and categorical semantics to stochastic memoization and name generation in the context of a minimal probabilistic programming language, for a restricted class of functions. Our contribution is a first model of stochastic memoization of constant Bernoulli functions with a non-enumerable type, which validates data flow transformations, bridging the gap between traditional probability theory and higher-order probability models. Our model uses a presheaf category and a novel probability monad on it.

Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming

The concept of updating a probability distribution in the light of new evidence lies at the heart of statistics and machine learning. Pearl's and Jeffrey's rule are two natural update mechanisms which lead to different outcomes, yet the similarities and differences remain mysterious. This paper clarifies their relationship in several ways: via separate descriptions of the two update mechanisms in terms of probabilistic programs and sampling semantics, and via different notions of likelihood (for Pearl and for Jeffrey). Moreover, it is shown that Jeffrey's update rule arises via variational inference. In terms of categorical probability theory, this amounts to an analysis of the situation in terms of the behaviour of the multiset functor, extended to the Kleisli category of the distribution monad.

Optimization of the hierarchical supply chain in the pharmaceutical industry

This paper discusses the modeling and solution of a hierarchical supply chain problem in the pharmaceutical industry. The considered supply chain includes the levels of suppliers, production centers, and distribution, which are taken into account in the decisions of the location of capacity facilities, optimal allocation of flow, and vehicle routing at the same time. Due to the indeterminacy of the problem environment, the two-stage probabilistic programming method has been used to control the model, and the new WOGA algorithm has been used to solve the problem. The presented algorithm is a combination of the Whale Optimization Algorithm (WOA) and Genetic Algorithm (GA) algorithms, which are used to minimize the costs of the entire designed network. The results obtained from the model analysis show that WOGA has a high efficiency in solving the developed mathematical model compared to GA and WOA. There was no significant difference between the averages of the objective function and the computational time between different solution methods. Since the perishability of the drug in transportation was considered in this article, it was observed that the cost of the entire network reaches its highest level if the period of perishability is 1. Because the production and distribution centers cannot have inventory in their warehouses and must meet the demand of pharmacies in every period.

Scalable Neural-Probabilistic Answer Set Programming

The goal of combining the robustness of neural networks and the expressiveness of symbolic methods has rekindled the interest in Neuro-Symbolic AI. Deep Probabilistic Programming Languages (DPPLs) have been developed for probabilistic logic programming to be carried out via the probability estimations of deep neural networks (DNNs). However, recent SOTA DPPL approaches allow only for limited conditional probabilistic queries and do not offer the power of true joint probability estimation. In our work, we propose an easy integration of tractable probabilistic inference within a DPPL. To this end, we introduce SLASH, a novel DPPL that consists of Neural-Probabilistic Predicates (NPPs) and a logic program, united via answer set programming (ASP). NPPs are a novel design principle allowing for combining all deep model types and combinations thereof to be represented as a single probabilistic predicate. In this context, we introduce a novel +/− notation for answering various types of probabilistic queries by adjusting the atom notations of a predicate. To scale well, we show how to prune the stochastically insignificant parts of the (ground) program, speeding up reasoning without sacrificing the predictive performance. We evaluate SLASH on various tasks, including the benchmark task of MNIST addition and Visual Question Answering (VQA).

LLM Multimodal Traffic Accident Forecasting.

With the rise in traffic congestion in urban centers, predicting accidents has become paramount for city planning and public safety. This work comprehensively studied the efficacy of modern deep learning (DL) methods in forecasting traffic accidents and enhancing Level-4 and Level-5 (L-4 and L-5) driving assistants with actionable visual and language cues. Using a rich dataset detailing accident occurrences, we juxtaposed the Transformer model against traditional time series models like ARIMA and the more recent Prophet model. Additionally, through detailed analysis, we delved deep into feature importance using principal component analysis (PCA) loadings, uncovering key factors contributing to accidents. We introduce the idea of using real-time interventions with large language models (LLMs) in autonomous driving with the use of lightweight compact LLMs like LLaMA-2 and Zephyr-7b-α. Our exploration extends to the realm of multimodality, through the use of Large Language-and-Vision Assistant (LLaVA)-a bridge between visual and linguistic cues by means of a Visual Language Model (VLM)-in conjunction with deep probabilistic reasoning, enhancing the real-time responsiveness of autonomous driving systems. In this study, we elucidate the advantages of employing large multimodal models within DL and deep probabilistic programming for enhancing the performance and usability of time series forecasting and feature weight importance, particularly in a self-driving scenario. This work paves the way for safer, smarter cities, underpinned by data-driven decision making.

A Deductive Verification Infrastructure for Probabilistic Programs

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs.