Stochastic Lambda Calculus

A Stochastic Lambda Calculus is a Lambda Calculus that is a Stochastic Calculus.

• • …
• Counter-Example(s):
  • a Deterministic Lambda Calculus.
  • a Markov Logic (which is derived from first-order logic).
  • a Probabilistic Turing Machine.
• See: Probabilistic Programming Language.

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References

2012

• (Goodman, 2012). => Noah D. Goodman. (2012). http://www.nasslli2012.com/courses/stochastic-lambda-calculus
  • QUOTE: Logic and probability are key themes of cognitive science that have long had an uneasy coexistence. This course will introduce the stochastic lambda calculus, an extension of the standard lambda calculus into a system with probabilistic semantics. This provides a mathematical foundation for uniting logic and probability: compositional languages, that support reasoning by probabilistic inference. This general framework is realized in the probabilistic programming language Church, which we will use for hands on examples throughout the class.

     We will use Church to explore the basic principles of inference over structured probabilistic models, as they are relevant to modeling human cognition. These include explaining-away, the Bayesian Occam's razor, and learning-to-learn in hierarchical Bayesian models. Examples will be draw from various domains in cognitive science, including causal learning and language. We will then cover highlights from current frontiers in Bayesian modeling, including models of social cognition and rational process-level models. We will end with applications to natural language semantics and pragmatics.

2005

• (Park et al., 2005) ⇒ Sungwoo Park, Frank Pfenning, and Sebastian Thrun. (2005). “A Probabilistic Language based Upon Sampling Functions.” In: Proceedings of the 32nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ISBN:1-58113-830-X doi:10.1145/1047659.1040320
  • QUOTE: As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages that treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. In this paper, we present a probabilistic language, called [math]\displaystyle{ \lambda_\circ }[/math], which uniformly supports all kinds of probability distributions -- discrete distributions, continuous distributions, and even those belonging to neither group.

2003

• (Park, 2003) ⇒ Sungwoo Park. (2003). “A Calculus for Probabilistic Languages.” In: Proceedings of the 2003 ACM SIGPLAN international workshop on Types in languages design and implementation. ISBN:1-58113-649-8 doi:10.1145/640136.604180
  • QUOTE: As probabilistic computation plays an increasing role in diverse fields in computer science, researchers have designed new languages to facilitate the development of probabilistic programs. In this paper, we develop a probabilistic calculus by extending the traditional lambda calculus.

2002

• (Ramsey & Pfeffer, 2002) ⇒ Norman Ramsey, and Avi Pfeffer. (2002). “Stochastic Lambda Calculus and Monads of Probability Distributions.” In: Proceedings of the 29th ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ISBN:1-58113-450-9 doi:10.1145/503272.503288

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