Probabilistic programming (PP) is a programming paradigm in which probabilistic models are specified and inference for these models is performed automatically.[1] It represents an attempt to unify probabilistic modeling and traditional general purpose programming in order to make the former easier and more widely applicable.[2][3] It can be used to create systems that help make decisions in the face of uncertainty.
Programming languages used for probabilistic programming are referred to as "probabilistic programming languages" (PPLs).
Applications
Probabilistic reasoning has been used for a wide variety of tasks such as predicting stock prices, recommending movies, diagnosing computers, detecting cyber intrusions and image detection.[4] However, until recently (partially due to limited computing power), probabilistic programming was limited in scope, and most inference algorithms had to be written manually for each task.
Nevertheless, in 2015, a 50-line probabilistic computer vision program was used to generate 3D models of human faces based on 2D images of those faces. The program used inverse graphics as the basis of its inference method, and was built using the Picture package in Julia.[4] This made possible "in 50 lines of code what used to take thousands".[5][6]
The Gen probabilistic programming library (also written in Julia) has been applied to vision and robotics tasks.[7]
More recently, the probabilistic programming system Turing.jl has been applied in various pharmaceutical[8] and economics applications.[9]
Probabilistic programming in Julia has also been combined with differentiable programming by combining the Julia package Zygote.jl with Turing.jl. [10]
Probabilistic programming languages are also commonly used in Bayesian cognitive science to develop and evaluate models of cognition. [11]
Probabilistic programming languages
PPLs often extend from a basic language. For instance, Turing.jl[12] is based on Julia, Infer.NET is based on .NET Framework,[13] while PRISM extends from Prolog.[14] However, some PPLs, such as WinBUGS, offer a self-contained language that maps closely to the mathematical representation of the statistical models, with no obvious origin in another programming language.[15][16]
The language for WinBUGS was implemented to perform Bayesian computation using Gibbs Sampling and related algorithms. Although implemented in a relatively unknown programming language (Component Pascal), this language permits Bayesian inference for a wide variety of statistical models using a flexible computational approach. The same BUGS language may be used to specify Bayesian models for inference via different computational choices ("samplers") and conventions or defaults, using a standalone program WinBUGS (or related R packages, rbugs and r2winbugs) and JAGS (Just Another Gibbs Sampler, another standalone program with related R packages including rjags, R2jags, and runjags). More recently, other languages to support Bayesian model specification and inference allow different or more efficient choices for the underlying Bayesian computation, and are accessible from the R data analysis and programming environment, e.g.: Stan, NIMBLE and NUTS. The influence of the BUGS language is evident in these later languages, which even use the same syntax for some aspects of model specification.
Several PPLs are in active development, including some in beta test. Two popular tools are Stan and PyMC.[17]
Relational
A probabilistic relational programming language (PRPL) is a PPL specially designed to describe and infer with probabilistic relational models (PRMs).
A PRM is usually developed with a set of algorithms for reducing, inference about and discovery of concerned distributions, which are embedded into the corresponding PRPL.
Most approaches to probabilistic logic programming are based on the distribution semantics, which splits a program into a set of probabilistic facts and a logic program. It defines a probability distribution on interpretations of the Herbrand universe of the program.[18]
List of probabilistic programming languages
This list summarises the variety of PPLs that are currently available, and clarifies their origins.
Reasoning about variables as probability distributions causes difficulties for novice programmers, but these difficulties can be addressed through use of Bayesian network visualizations and graphs of variable distributions embedded within the source code editor.[52]
As many PPLs rely on the specification of priors on the variables of interest, specifying informed priors is often difficult for novices. In some cases, libraries such as PyMC provide automated methods to find the parameterization of informed priors[53].
↑ Gorinova, Maria I.; Sarkar, Advait; Blackwell, Alan F.; Syme, Don (January 1, 2016). "A Live, Multiple-Representation Probabilistic Programming Environment for Novices". Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems. CHI '16. New York, NY, USA: ACM. pp.2533–2537. doi:10.1145/2858036.2858221. ISBN9781450333627. S2CID3201542.
Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical rather than mathematical induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.
A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.
Programming languages can be grouped by the number and types of paradigms supported.
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution.
Solomonoff's theory of inductive inference proves that, under its common sense assumptions (axioms), the best possible scientific model is the shortest algorithm that generates the empirical data under consideration. In addition to the choice of data, other assumptions are that, to avoid the post-hoc fallacy, the programming language must be chosen prior to the data and that the environment being observed is generated by an unknown algorithm. This is also called a theory of induction. Due to its basis in the dynamical character of Algorithmic Information Theory, it encompasses statistical as well as dynamical information criteria for model selection. It was introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory.
Formal epistemology uses formal methods from decision theory, logic, probability theory and computability theory to model and reason about issues of epistemological interest. Work in this area spans several academic fields, including philosophy, computer science, economics, and statistics. The focus of formal epistemology has tended to differ somewhat from that of traditional epistemology, with topics like uncertainty, induction, and belief revision garnering more attention than the analysis of knowledge, skepticism, and issues with justification.
Probabilistic logic involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals.
Statistical relational learning (SRL) is a subdiscipline of artificial intelligence and machine learning that is concerned with domain models that exhibit both uncertainty and complex, relational structure. Typically, the knowledge representation formalisms developed in SRL use first-order logic to describe relational properties of a domain in a general manner and draw upon probabilistic graphical models to model the uncertainty; some also build upon the methods of inductive logic programming. Significant contributions to the field have been made since the late 1990s.
A probabilistic logic network (PLN) is a conceptual, mathematical and computational approach to uncertain inference. It was inspired by logic programming and it uses probabilities in place of crisp (true/false) truth values, and fractional uncertainty in place of crisp known/unknown values. In order to carry out effective reasoning in real-world circumstances, artificial intelligence software handles uncertainty. Previous approaches to uncertain inference do not have the breadth of scope required to provide an integrated treatment of the disparate forms of cognitively critical uncertainty as they manifest themselves within the various forms of pragmatic inference. Going beyond prior probabilistic approaches to uncertain inference, PLN encompasses uncertain logic with such ideas as induction, abduction, analogy, fuzziness and speculation, and reasoning about time and causality.
LaplacesDemon is an open-source statistical package that is intended to provide a complete environment for Bayesian inference. LaplacesDemon has been used in numerous fields. The user writes their own model specification function and selects a numerical approximation algorithm to update their Bayesian model. Some numerical approximation families of algorithms include Laplace's method, numerical integration, Markov chain Monte Carlo (MCMC), and variational Bayesian methods.
Stan is a probabilistic programming language for statistical inference written in C++. The Stan language is used to specify a (Bayesian) statistical model with an imperative program calculating the log probability density function.
Probabilistic Soft Logic (PSL) is a statistical relational learning (SRL) framework for modeling probabilistic and relational domains. It is applicable to a variety of machine learning problems, such as collective classification, entity resolution, link prediction, and ontology alignment. PSL combines two tools: first-order logic, with its ability to succinctly represent complex phenomena, and probabilistic graphical models, which capture the uncertainty and incompleteness inherent in real-world knowledge. More specifically, PSL uses "soft" logic as its logical component and Markov random fields as its statistical model. PSL provides sophisticated inference techniques for finding the most likely answer (i.e. the maximum a posteriori (MAP) state). The "softening" of the logical formulas makes inference a polynomial time operation rather than an NP-hard operation.
The following table compares notable software frameworks, libraries and computer programs for deep learning.
PyMC is a probabilistic programming language written in Python. It can be used for Bayesian statistical modeling and probabilistic machine learning.
ArviZ is a Python package for exploratory analysis of Bayesian models. It is specifically designed to work with the output of probabilistic programming libraries like PyMC, Stan, and others by providing a set of tools for summarizing and visualizing the results of Bayesian inference in a convenient and informative way. ArviZ also provides a common data structure for manipulating and storing data commonly arising in Bayesian analysis, like posterior samples or observed data.
Owl Scientific Computing is a software system for scientific and engineering computing developed in the Department of Computer Science and Technology, University of Cambridge. The System Research Group (SRG) in the department recognises Owl as one of the representative systems developed in SRG in the 2010s. The source code is licensed under the MIT License and can be accessed from the GitHub repository.
ProbLog is a probabilistic logic programming language that extends Prolog with probabilities. It minimally extends Prolog by adding the notion of a probabilistic fact, which combines the idea of logical atoms and random variables. Similarly to Prolog, ProbLog can query an atom. While Prolog returns the truth value of the queried atom, ProbLog returns the probability of it being true.
Probabilistic logic programming is a programming paradigm that combines logic programming with probabilities.
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