An influential body of work in recent cognitive science makes use of structured Bayesian models combining logic and statistics to understand activities as diverse as reasoning, vision, learning, social cognition, and language processing. In recent joint work with Noah Goodman I have applied this framework to language understanding, combining it with insights from both generative and cognitive linguistics. The structured Bayesian approach makes it possible to formalize graded meaning and other phenomena in which uncertainty plays an important role in communication while remaining within the framework of compositional model-theoretic semantics. It also gives insight into the effects of context and background knowledge on interpretation and, I’ll suggest, allow us to formalize the main insights of Gricean pragmatics as domain-general social reasoning. I will describe Bayesian inference and interpretation in non-technical terms and then consider three test cases: (1) predictions about the effects of background expectations on ambiguity resolution, (2) how the model derives graded implicatures as probabilistic inferences about speaker intentions, and (3) how context influences the information conveyed by vague and semantically context-sensitive language.